Research ArticleA Modified Spectral PRP Conjugate GradientProjection Method for Solving Large-Scale MonotoneEquations and Its Application in Compressed Sensing

Jie Guo and ZhongWan

School of Mathematics and Statistics Central South University Hunan Changsha China

Correspondence should be addressed to Zhong Wan wanmath163com

Received 24 November 2018 Accepted 24 March 2019 Published 8 April 2019

Academic Editor Higinio Ramos

Copyright copy 2019 Jie Guo and Zhong Wan This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

In this paper we develop an algorithm to solve nonlinear system of monotone equations which is a combination of a modifiedspectral PRP (Polak-Ribiere-Polyak) conjugate gradient method and a projection methodThe search direction in this algorithm isproved to be sufficiently descent for any line search rule A line search strategy in the literature is modified such that a better steplength is more easily obtained without the difficulty of choosing an appropriate weight in the original one Global convergence ofthe algorithm is proved under mild assumptions Numerical tests and preliminary application in recovering sparse signals indicatethat the developed algorithm outperforms the state-of-the-art similar algorithms available in the literature especially for solvinglarge-scale problems and singular ones

1 Introduction

In many fields of sciences and engineering solution of anonlinear system of equations is a fundamental problemFor example in [1 2] both a Nash economic equilibriumproblem and a signal processing problem were formulatedinto a nonlinear system of equations Owing to complexityin the past five decades numerous algorithms and somesoftware packages in virtue of those powerful algorithmshave been developed for solving the nonlinear system ofequations See for example [1 3ndash16] and the referencestherein Nevertheless in practice no any algorithm canefficiently solve all the systems of equations arising fromsciences and engineering It is significant to develop a specificalgorithm to solve the problems with different analytic andstructural features [17 18]

In this paper we consider the following nonlinear systemof monotone equations

119865 (119909) = 0 (1)

where 119865 119877119899 997888rarr 119877119899 is a continuous and monotone functionthat is to say 119865 satisfies(119865 (119909) minus 119865 (119910))119879 (119909 minus 119910) ⩾ 0 forall119909 119910 isin 119877119899 (2)

It has been shown that the solution set of problem (1) isconvex if it is nonempty [3] In addition throughout thepaper the space 119877119899 is equipped with the Euclidean norm sdot and the inner product ⟨119909 119910⟩ = 119909119879119910 for 119909 119910 isin 119877119899

Aiming at solution of problem (1)many efficientmethodswere developed recently Only by incomplete enumerationwe here mention the trust region method [19] the Newtonand the quasi-Newton methods [4ndash6 19] the Gauss-Newtonmethods [7 8] the Levenberg-Marquardt methods [20ndash22]the derivative-free methods and its modified versions [9ndash16 23ndash27] the derivative-free conjugate gradient projectionmethod [14] the modified PRP (Polak-Ribiere-Polyak) con-jugate gradient method [11] the TPRP method [10] thePRP-type method [28] the projection method [23] theFR-type method [9] and the modified spectral conjugategradient projection method [13] Summarily the spectral

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 5261830 17 pageshttpsdoiorg10115520195261830

2 Mathematical Problems in Engineering

gradient methods and the conjugate gradient methods aremore popular in solving a large-scale nonlinear system ofequations than the Newton and the quasi-Newton methodsOne of the formerrsquos advantages lies in that there is norequirement of computing and storing the Jacobian matrix orits approximation

Specifically Li introduced a class of methods for large-scale nonlinear monotone equations in [10] which includethe SG-liked method the MPRP method and the TPRPmethod Chen [28] proposed a PRP method for large-scalenonlinear monotone equations A descent modified PRPmethod and FR-type methods were presented in [9 11]respectively Liu and Li proposed a projection method forconvex constrained monotone nonlinear equations in [23]Two derivative-free conjugate gradient projection methodswere presented for such a system in [14] Three extensionsof conjugate gradient algorithms were developed in [24ndash26]respectively Based on the projection technique in [29] Wanand Liu proposed a modified spectral conjugate gradientprojection method (MSCGP) to solve a nonlinear monotonesystem of symmetric equations in [13] Then in [2] MSCGPwas successfully applied into recovering sparse signals andrestoring blurred images

It is noted that the main idea of [13 23] is to constructa search direction by projection technique such that it issufficiently descent In virtue of derivative-free and lowstorage properties numerical experiments indicated that thedeveloped algorithm in [13] is more efficient to solve large-scale nonlinear monotone systems of equations than thesimilar ones available in the literature

In [30] Yang et al proposed a modified spectral PRPconjugate gradient method for solving unconstrained opti-mization problem It was proved that the search direction ateach iteration is a descent direction of objective function andglobal convergence was established under mild conditionsOur research interest in this paper is to study how to extendthismethod to solution of problem (1) Specifically we shouldaddress the following issues

(1)Without need of derivative information of the function119865 how to determine the spectral and conjugate parameters toget a sufficiently descent search direction at each iteration

(2) To ensure global convergence of algorithm howto choose an appropriate step length for the given searchdirection Particularly monotonicity of 119865 should be utilizedto design a new iterate scheme

(3) What about the numerical performance of newiteration scheme Especially whether it is more efficient ornot than the similar algorithms available in the literature

Note that a new line search rule was proposed in [31]for solving nonlinear monotone equations with convex con-straints and it was shown that in virtue of this line searchthe developed algorithm has good numerical performanceHowever the presented line search is involved with choice ofa weight Since it may be difficult to choose an appropriateweight in the practical implementation of algorithm weattempt to overcome this difficulty in this paper

Summarily we intend to propose a modified spectralPRP conjugate gradient derivative-free projectionmethod forsolving large-scale nonlinear equations Global convergence

of this method will be proved and numerical tests willbe conducted by implementing the developed algorithm tosolve benchmark large-scale test problems and to reconstructsparse signals in compressive sensing

The rest of this paper is organized as follows In Section 2we first state the idea to propose a new spectral PRP conjugategradient methodThen a new algorithm is developed Globalconvergence is established in Section 3 Section 4 is devotedto numerical experiments Preliminary application of thealgorithm is presented in Section 5 Some conclusions aredrawn in Section 6

2 Development of New Algorithm

In this section we will state how to develop a new algorithmin detail

21 Projection Method Generally to solve (1) we need toconstruct an iterative format as follows119909119896+1 = 119909119896 + 120572119896119889119896 (3)

where 120572119896 gt 0 is called a step length and 119889119896 is a searchdirection Let 119911119896 = 119909119896 + 120572119896119889119896 If 119911119896 satisfies119865 (119911119896)119879 (119909119896 minus 119911119896) gt 0 (4)

then a projection method can be obtained for solving Prob-lem (1) Actually by monotonicity of 119865 it holds that

119865 (119911119896)119879 (119909lowast minus 119911119896) = (119865 (119911119896) minus 119865 (119909lowast))119879 (119909lowast minus 119911119896) le 0 (5)

for any solution of (1) 119909lowast With such a 119911119896 we define ahyperplane

119867119896 = 119909 isin 119877119899 | 119865 (119911119896)119879 (119909 minus 119911119896) = 0 (6)

From (4) and (5) it is clear that119867119896 strictly separates the iteratepoint 119909119896 from the solution 119909lowastThus the projection of 119909119896 onto119867119896 is closer to 119909lowast than 119909119896 Consequently the iterative format

119909119896+1 = 119909119896 minus 119865 (119911119896)119879 (119909119896 minus 119911119896)1003817100381710038171003817119865 (119911119896)10038171003817100381710038172 119865 (119911119896) (7)

is referred to as the projection method proposed in [29]Both analytic properties and numerical results have shownefficiency and robustness of the projection-based algorithmsformonotone systemof equations [10 13 14] In this paper weintend to propose a new spectral conjugate gradient methodalso in virtue of the above projection technique

22 A Modified Spectral PRP Conjugate Gradient Method Inthe projection method (7) it is noted that 119911119896 must satisfy (4)That is to say 119889119896 should be a search direction satisfying

119865 (119911119896)119879 119889119896 lt 0 (8)

Mathematical Problems in Engineering 3

Very recently Wan et al [13] proposed a modified spectralconjugate gradient projection method for solving nonlinearmonotone symmetric equations where 119889119896 was chosen by

119889119896 = minus119865119896 if 119896 = 0minus120579119896119865119896 + 120573119896119889119896minus1 if 119896 gt 0 (9)

and 120573119896 and 120579119896 are computed by

120579119896 = 119889119879119896minus1 (119910119896minus1 minus 119865119896119865119879119896 119904119896minus1 100381710038171003817100381711986511989610038171003817100381710038172)119889119879119896minus1

(119868 minus 119865 (119909119896) 119865 (119909119896)119879 1003817100381710038171003817119865 (119909119896)10038171003817100381710038172) 119910119896minus1 120573119896 = 119865119879119896 (119910119896minus1 minus 119904119896minus1)119889119879

119896minus1(119868 minus 119865 (119909119896) 119865 (119909119896)119879 1003817100381710038171003817119865 (119909119896)10038171003817100381710038172) 119910119896minus1

(10)

respectively 119904119896minus1 = 119909119896 minus 119909119896minus1 and 119910119896minus1 = 119865(119909119896) minus 119865(119909119896minus1) Itwas proved in [13] that 119889119896 given by (9) and (10) is sufficientlydescent and satisfies 119865119879119896 119889119896 = minus1198651198962

Note that a modified spectral PRP conjugate gradientmethod was proposed for solving unconstrained optimiza-tion problems in [30] Similar to the idea in [13] we canextend the method in [30] to solution of problem (1)Specifically we compute 120573119896 and 120579119896 in (9) by

120579119896 = 119889119879119896minus1119910119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 minus 119889119879119896minus1119865119896119865119879119896 119865119896minus1100381710038171003817100381711986511989610038171003817100381710038172 1003817100381710038171003817119865119896minus110038171003817100381710038172 120573119896 = 119865119879119896 119910119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172

(11)

respectively Although (11) gives different choices of 120573119896 and 120579119896from (10) we can also prove the following result

Proposition 1 Let 119889119896 be given by (9) and (11) en for any119896 ge 0 the following equality holds119865119879119896 119889119896 = minus 100381710038171003817100381711986511989610038171003817100381710038172 (12)

Proof For 119896 = 0 we have1198651198790 1198890 = 1198651198790 (minus1198650) = minus 1003817100381710038171003817119865010038171003817100381710038172 (13)

For 119896 = 1 we have1198651198791 1198891 = 1198651198791 (minus12057911198651 + 12057311198890)= minus1198891198790 (1198651 minus 1198650)1003817100381710038171003817119865010038171003817100381710038172 1198651198791 1198651 + 119889119879011986511198651198791 11986501003817100381710038171003817119865110038171003817100381710038172 10038171003817100381710038171198650100381710038171003817100381721198651198791 1198651

+ 1198651198791 (1198651 minus 1198650)1003817100381710038171003817119865010038171003817100381710038172 11988911987901198651 = 1198891198790119865010038171003817100381710038171198650100381710038171003817100381721198651198791 1198651= 10038171003817100381710038171198651100381710038171003817100381721003817100381710038171003817119865010038171003817100381710038172 (minus 1003817100381710038171003817119865010038171003817100381710038172) = minus 1003817100381710038171003817119865110038171003817100381710038172

(14)

We now prove that if119865119879119896minus1119889119896minus1 = minus 1003817100381710038171003817119865119896minus110038171003817100381710038172 (15)

holds for 119896 minus 1 (119896 gt 1) then (12) also holds for 119896

Actually119865119879119896 119889119896 = 119865119879119896 (minus120579119896119865119896 + 120573119896119889119896minus1)= minus119889119879119896minus1 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119865119879119896 119865119896

+ 119889119879119896minus1119865119896119865119879119896 119865119896minus1100381710038171003817100381711986511989610038171003817100381710038172 1003817100381710038171003817119865119896minus110038171003817100381710038172 119865119879119896 119865119896+ 119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119889119879119896minus1119865119896 = 119889119879119896minus1119865119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 119865119879119896 119865119896

= minus 1003817100381710038171003817119865119896minus1100381710038171003817100381721003817100381710038171003817119865119896minus110038171003817100381710038172 100381710038171003817100381711986511989610038171003817100381710038172 = minus 100381710038171003817100381711986511989610038171003817100381710038172

(16)

where the forth equality follows condition (15) Consequentlyby mathematical induction (12) holds for any 119896

Proposition 1 ensures that the idea of projection methodcan be incorporated into design of iteration schemes to solve(1) as the search direction is determined by (9) and (11)

23 Modified Line Search Rule Since it is critical to choosean appropriate step length to improve the performance ofthe iterate scheme (3) as well as determination of searchdirections we now present an inexact line search rule todetermine 120572119896 in (3)

Very recently Ou and Li [31] presented a line search ruleas follows find a nonnegative step length 120572119896 = max119904120588119894 119894 =0 1 2 such that the following inequality is as followsminus119865 (119909119896 + 120572119896119889119896)119879 119889119896 ge 120590120572119896120574119896 100381710038171003817100381711988911989610038171003817100381710038172 (17)

where 119889119896 is a fixed search direction 119904 gt 0 is a given initialstep length 120588 isin (0 1) and 120590 gt 0 are two given constants and120574119896 is specified by120574119896 = 120582119896 + (1 minus 120582119896) 1003817100381710038171003817119865 (119909119896 + 120572119896119889119896)1003817100381710038171003817 (18)

In [31] it was required that 120582119896 in (18) satisfies 120582119896 isin[120582119898119894119899 120582119898119886119909] sube (0 1] Clearly 120582119896 and (1 minus 120582119896) are the weightsfor the values 1 and 119865(119909119896 + 120572119896119889119896) respectively In thepractical implementation it is may be difficult to choose anappropriate 120582119896 To overcome this difficulty we choose 120574119896 in(18) by 120574119896 = min 1 1003817100381710038171003817119865 (119909119896 + 120572119896119889119896)1003817100381710038171003817 (19)

It is sure that for the newmethod as a combination of (9) (11)(17) and (19) we need to establish its convergence theory andto further test its numerical performance

Remark 2 In [31] to ensure that well-defined the line search(17) is well-defined it is assumed that 119889119896 satisfies119865 (119909119896)119879 119889119896 le minus120591 1003817100381710038171003817119865 (119909119896)10038171003817100381710038172 (20)

where 120591 gt 0 is a given constant By Proposition 1 it is clearthat 119889119896 chosen by (9) and (11) satisfies (20) as 120591 = 1

4 Mathematical Problems in Engineering

InputAn initial point 1199090 isin 119877119899 positive constants 119896119898119886119909 120590 120576 119904 and 120588 isin (0 1)Begin119896 larr997888 01198650 larr997888 119865(1199090)While (119865119896 ge 120576 and 119896 lt 119896119898119886119909)Step 1 (Search direction)Compute 119889119896 by (9) and (11)Step 2 (Step length)120572 larr997888 119904Compute 119865119896 larr997888 119865(119909119896 + 120572119889119896)While (minus119865119879119896 119889119896 lt 1205901205721205741198961198891198962)120572 larr997888 120588120572Compute 119865119896 larr997888 119865(119909119896 + 120572119889119896)EndWhile120572119896 larr997888 120572Step 3 (Projection and update)119911119896 larr997888 119909119896 + 120572119896119889119896If 119865(119911119896) le 120576119909119896+1 larr997888 119911119896BreakEnd IfCompute 119909119896+1 by (7)119865119896+1 larr997888 119865(119909119896+1)119896 larr997888 119896 + 1EndWhileEnd

Algorithm 1 Modified spectral PRP derivative-free projection-based algorithm (MPPRP)

24 Development of New Projection-Based Algorithm Withthe above preparation we are in a position to develop analgorithm to solve problem (1) by combining the projectiontechnique and the new methods to determine a searchdirection and a step length

We now present the computer procedure of Algorithm 1

Remark 3 Since Algorithm 1 does not involve computing theJacobian matrix of 119865 or its approximation both informationstorage and computational cost of the algorithm are lower Invirtue of this advantage Algorithm 1 is helpful to solution oflarge-scale problems In next section we will prove that Algo-rithm 1 is applicable even if 119865 is nonsmooth Our numericaltests in Section 4will further show that Algorithm 1 can find asingular solution of problem (1) (see problem 5 and Table 5)

Remark 4 Compared with the algorithm developed in [13]problem (1) is not assumed to be a symmetric system ofequations

3 Convergence

In this section we are going to study global convergence ofAlgorithm 1

Apart from different choices of search direction and steplength Algorithm 1 can be treated as a variant of the projec-tion algorithm in [29] So similar to some critical points ofestablishing global convergence in [29] we attempt to prove

that Algorithm 1 is globally convergent Very recently locallylinear convergence was proved in [32] for some PRP-typeprojection methods

We first state the following mild assumptions

Assumption 5 The function 119865 is monotone on 119877119899Assumption 6 The solution set of problem (1) is nonempty

Assumption 7 The function 119865 is Lipschitz continuous on 119877119899namely there exists a positive constant 119871 such that for all119909 119910 isin 119877119899 1003817100381710038171003817119865 (119909) minus 119865 (119910)1003817100381710038171003817 le 119871 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 (21)

Under these assumptions we can prove that Algorithm 1has the following nice properties

Lemma 8 Let 119909119896 be a sequence generated by Algorithm 1 IfAssumptions 5 6 and 7 hold then

(1) for any 119909lowast such that 119865(119909lowast) = 01003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172 le 1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 (22)

(2) e sequence 119909119896 is bounded(3) If 119909119896 is a finite sequence then the last iterate point is

a solution of problem (1) otherwise119899sum119894=1

1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 lt infin (23)

Mathematical Problems in Engineering 5

and

lim119896997888rarrinfin

1003817100381710038171003817119909119896+1 minus 1199091198961003817100381710038171003817 = 0 (24)

(4) e sequence 119865(119909119896) is bounded Hence there existsa constant 119872119891 such that 119865(119909119896) le 119872119891Proof (1) Let 119909lowast be any point such that 119865(119909lowast) = 0 Then bymonotonicity of 119865 we have⟨119865 (119911119896) 119909lowast minus 119911119896⟩ le 0 (25)

From (7) it is also easy to verify that 119909119896+1 is the projection of119909119896 onto the halfspace119909 isin 119877119899 | ⟨119865 (119911119896) 119909 minus 119911119896⟩ le 0 (26)

Thus it follows from (25) that 119909lowast belongs to this halfspaceFrom the basic properties of projection operator [33] weknow that ⟨119909119896 minus 119909119896+1 119909119896+1 minus 119909lowast⟩ ge 0 (27)

Consequently1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 = 1003817100381710038171003817119909119896 minus 119909119896+110038171003817100381710038172 + 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172+ 2 ⟨119909119896 minus 119909119896+1 119909119896+1 minus 119909lowast⟩ge 1003817100381710038171003817119909119896 minus 119909119896+110038171003817100381710038172 + 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172 (28)

The desired result (22) is directly obtained from (28)(2) From (28) it is clear that the sequence 119909119896 minus 119909lowast is

nonnegative and decreasing Thus 119909119896minus119909lowast is a convergentsequence It is concluded that 119909119896 is bounded

(3) From (28) we know1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 le 1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172 (29)

Thus

119899sum119896=1

1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 le 119899sum119896=1

(1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172)= 10038171003817100381710038171199091 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119899+1 minus 119909lowast10038171003817100381710038172 (30)

Since the sequence 119909119896 is bounded the series sum119899119896=1 119909119896+1 minus1199091198962 is convergent Consequentlylim119896997888rarrinfin

1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 = 0 (31)

The third result has been proved(4) For any 119909119896 by Lipschitz continuity we have1003817100381710038171003817119865 (119909119896)1003817100381710038171003817 = 1003817100381710038171003817119865 (119909119896) minus 119865 (119909lowast)1003817100381710038171003817 le 119871 1003817100381710038171003817119909119896 minus 119909lowast1003817100381710038171003817 (32)

Since 119909119896 minus119909lowast is convergent we conclude that 119865(119909119896) isbounded Consequently for all 119896 isin 119873 there exists a constant119872119891 such that 119865(119909119896) le 119872119891

Lemma 8 indicates that for the sequence 119909119896 generatedby Algorithm 1 the sequence 119909119896 minus 119909lowast is decreasing andconvergent and the sequence 119909119896 is bounded where 119909lowast isany solution of problem (1)

Lemma 9 Suppose that Assumptions 5 6 and 7 hold Let119889119896 be a sequence generated by Algorithm 1 If there exists aconstant 1205760 gt 0 such that for any positive integer 11989610038171003817100381710038171198651198961003817100381710038171003817 ge 1205760 (33)

en the sequence of directions 119889119896 is bounded ie thereexists a constant 119872 gt 0 such that for any positive integer 11989610038171003817100381710038171198891198961003817100381710038171003817 le 119872 (34)

Proof From (9) (11) (12) and the results of Lemma 8 itfollows that10038171003817100381710038171198891198961003817100381710038171003817 = 1003817100381710038171003817minus120579119896119865119896 + 120573119896119889119896minus11003817100381710038171003817 = 1003817100381710038171003817100381710038171003817100381710038171003817minus119889119879119896minus1 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119865119896

+ 119889119879119896minus1119865119896119865119879119896 119865119896minus1100381710038171003817100381711986511989610038171003817100381710038172 1003817100381710038171003817119865119896minus110038171003817100381710038172 119865119896 + 119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119889119896minus11003817100381710038171003817100381710038171003817100381710038171003817 = 1003817100381710038171003817100381710038171003817100381710038171003817minus119865119896minus 119889119879119896minus1119865119896119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 100381710038171003817100381711986511989610038171003817100381710038172 119865119896 + 119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119889119896minus11003817100381710038171003817100381710038171003817100381710038171003817le 2 100381710038171003817100381711986511989610038171003817100381710038171003817100381710038171003817119865119896minus110038171003817100381710038172 1003817100381710038171003817119889119896minus11003817100381710038171003817 1003817100381710038171003817119910119896minus11003817100381710038171003817 + 10038171003817100381710038171198651198961003817100381710038171003817 le 211987111987211989112057620 1003817100381710038171003817119909119896minus 119909119896minus11003817100381710038171003817 1003817100381710038171003817119889119896minus11003817100381710038171003817 + 119872119891

(35)

From (24) we know that there exist a positive integer 1198961 anda positive number 1205761 (0 lt 1205761 lt 1) such that for all 119896 ge 1198961211987111987211989112057620 1003817100381710038171003817119909119896 minus 119909119896minus11003817100381710038171003817 lt 1205761 (36)

Hence10038171003817100381710038171198891198961003817100381710038171003817 le 1205761 1003817100381710038171003817119889119896minus11003817100381710038171003817 + 119872119891 le 12057621 1003817100381710038171003817119889119896minus21003817100381710038171003817 + 1205761119872119891 + 119872119891le 12057611989611 10038171003817100381710038171003817119889119896110038171003817100381710038171003817 + 1205761198961minus11 119872119891 + sdot sdot sdot + 1205761119872119891 + 119872119891 le 10038171003817100381710038171003817119889119896110038171003817100381710038171003817+ 119872119891 1 minus 120576119896111 minus 1205761 le 10038171003817100381710038171003817119889119896110038171003817100381710038171003817 + 1198721198911 minus 1205761

(37)

Let 1198721 = 1198891198961 + 119872119891(1 minus 1205761) Take119872 = max 100381710038171003817100381711988901003817100381710038171003817 100381710038171003817100381711988911003817100381710038171003817 10038171003817100381710038171003817119889119896110038171003817100381710038171003817 1198721 (38)

Then 119889119896 le 119872 holds for any positive integer 119896Lemma 10 Suppose that Assumptions 5 6 and 7 hold Let119909119896 and 119889119896 be two sequences generated by Algorithm 1enthe line search rule (17) of Step 2 in Algorithm 1 is well-defined

6 Mathematical Problems in Engineering

Proof Our aim is to show that the line search rule (17) termi-nates finitely with a positive step length 120572119896 By contradictionsuppose that for some iterate indexes such as 119896lowast condition(17) does not hold As a result for all 119898 isin 119873minus119865 (119909119896lowast + 119904120588119898119889119896lowast)119879 119889119896lowast lt 120590119904120588119898120574119896lowast 1003817100381710038171003817119889119896lowast10038171003817100381710038172 (39)

From (18) and the termination condition of Algorithm 1 itfollows that for all 119898 isin 1198730 le 120576 le 120574119896lowast le 1 (40)

By taking the limit as 119898 997888rarr infin in both sides of (39) we have

minus119865 (119909119896lowast)119879 119889119896lowast le 0 (41)

Equations (41) contradicts the fact that minus119865119879119896 119889119896 = minus1198651198962 gt 0for all 119896 That is to say the line search rule terminates finitelywith a positive step length 120572119896 ie the line search step ofAlgorithm 1 is well-defined

With the above preparation we are now state the conver-gence result of Algorithm 1

Theorem 11 Suppose that Assumptions 5 6 and 7 hold Let119909119896 be a sequence generated by Algorithm 1 en

lim inf119896997888rarrinfin

10038171003817100381710038171198651198961003817100381710038171003817 = 0 (42)

Proof For the sake of contradiction we suppose that theconclusion is not trueThen by the definition of inferior limitthere exists a constant 1205760 gt 0 such that for any 119896 isin 11987310038171003817100381710038171198651198961003817100381710038171003817 ge 1205760 (43)

Consequently from100381710038171003817100381711986511989610038171003817100381710038172 = minus119865119879119896 119889119896 le 10038171003817100381710038171198651198961003817100381710038171003817 10038171003817100381710038171198891198961003817100381710038171003817 (44)

it follows that 119889119896 ge 1205760 gt 0 for any 119896 isin 119873From (7) (17) and (40) we get

1003817100381710038171003817119909119896+1 minus 1199091198961003817100381710038171003817 = 100381610038161003816100381610038161003816119865 (119911119896)119879 (119909119896 minus 119911119896)1003816100381610038161003816100381610038161003817100381710038171003817119865 (119911119896)1003817100381710038171003817 = 100381610038161003816100381610038161003816120572119896119865 (119911119896)119879 1198891198961003816100381610038161003816100381610038161003817100381710038171003817119865 (119911119896)1003817100381710038171003817ge 1205901205722119896120574119896 1003817100381710038171003817119889119896100381710038171003817100381721003817100381710038171003817119865 (119911119896)1003817100381710038171003817 ge 1205901205761205722119896 100381710038171003817100381711988911989610038171003817100381710038172119872119891 ge 0 (45)

Combining (24) and (45) we obtain

lim119896997888rarrinfin

120572119896 10038171003817100381710038171198891198961003817100381710038171003817 = 0 (46)

Since 119889119896 ge 1205760 gt 0 for any 119896 isin 119873 we have

lim119896997888rarrinfin

120572119896 = 0 (47)

Clearly 120572lowast119896 = 120588minus1120572119896 does not satisfy in (17) It says that

minus119865 (119909119896 + 120572lowast119896119889119896)119879 119889119896 lt 120590120572lowast119896120574119896 100381710038171003817100381711988911989610038171003817100381710038172 (48)

By Lemmas 8 and 9 we know that the two sequences 119909119896and 119889119896 are bounded Without loss of generality we choosea subset 119870 isin 119873 such that

lim119896997888rarrinfin119896isin119870

119909119896 = 119909lowastlim119896997888rarrinfin119896isin119870

119889119896 = 119889lowast (49)

Taking the limit in the two sides of (48) as 119896 997888rarr infin (119896 isin 119870)it holds that

119865 (119909lowast)119879 119889lowast ge 0 (50)

On the other hand from (43) we know that

119865 (119909119896)119879 119889119896 = minus 100381710038171003817100381711986511989610038171003817100381710038172 le minus12057620 lt 0 (51)

By taking the limit 119896 997888rarr infin in the two sides of (51) for 119896 isin 119870we get

119865 (119909lowast)119879 119889lowast le minus1205760 lt 0 (52)

It contradicts (50) Thus the proof of Theorem 11 has beencompleted

Remark 12 (only with 120574119896 being generated by (19)) As 120574119896 isdetermined by (18) the proofs are similar

Since the proof of Theorem 11 does not involve differen-tiability of 119865 let alone nonsingularity of its Jacobian matrixwe know that the following result holds

Corollary 13 For any nonsmooth or singular function 119865 let119909119896 be a sequence generated as Algorithm 1 is used to solveproblem (1) Under Assumptions 5 6 and 7 it holds that

lim inf119896997888rarrinfin

10038171003817100381710038171198651198961003817100381710038171003817 = 0 (53)

It should be pointed out that the global convergenceof Algorithm 1 depends on assumption on monotonicity of119865 For nonmonotonic function 119865 Algorithm 1 may be notapplicable

4 Numerical Experiments

In this section by numerical experiments we are going tostudy the effectiveness and robustness of Algorithm 1 forsolving large-scale system of equations

We first collect some benchmark test problems availablein the literature

Problem 14 (see [5]) The elements of 119865(119909) are given by119865119894 (119909) = 2119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (54)

Problem 15 (see [5]) The elements of 119865(119909) are given by119865119894 (119909) = 2119909119894 minus 1003816100381610038161003816sin (119909119894)1003816100381610038161003816 119894 = 1 2 119899 minus 1 (55)

Mathematical Problems in Engineering 7

Problem 16 (see [15]) The elements of 119865(119909) are given by

1198651 (119909) = 1199091 minus 119890cos((1199091+1199092)(119899+1))119865119894 (119909) = 119909119894 minus 119890cos((119909119894minus1+119909119894+119909119894+1)(119899+1)) 119894 = 2 119899 minus 1119865119899 (119909) = 119909119899 minus 119890cos((119909119899minus1+119909119899)(119899+1)) (56)

Problem 17 (see [28]) The elements of 119865(119909) are given by

119865 (119909) = 119860119909 + 119887119887 = (minus1 minus1 minus1 minus1)119879

119860 = ((((((

52 11 52 1d d d

d d 11 52

))))))

(57)

Clearly problem 4 is a linear system of equations which isused to test Algorithm 1 in this special case

Problem 18 (see [15]) The elements of 119865(119909) are given by

119865119894 (119909) = 119909119894 minus 11198991199092119894 + 1119899 119899sum119894=1

119909119894 + 119894 119894 = 2 119899 minus 1 (58)

Problem 19 (see [15]) The elements of 119865(119909) are given by

1198651 (119909) = 1311990931 + 1211990922119865119894 (119909) = minus121199092119894 + 11989431199093119894 + 121199092119894+1 119894 = 2 119899 minus 1119865119899 (119909) = minus121199092119899 + 11989931199093119899

(59)

Clearly apart from problem 4 all the others are nonlinearsystem of equations Problem 15 is nonsmooth at point(0 0 0) The size of all the test problems is variable Ifthis size is larger than 1000 the problem can be regarded tobe large-scale We solve all the test problems with differentsizes by Algorithm 1 especially in comparison with theexisting seven similar algorithms such as those developedvery recently in [10 13ndash15 28]

All the algorithms are coded inMATLABR2010b and runon a personal computer with a 22GHZ CPU processor 8GB

memory and Windows 10 operation system The relevantparameters of algorithms are specified by119896119898119886119909 = 100000120590 = 03119904 = 05120588 = 07

(60)

120582119896 = 12058202 if 119896 = 1120582119896minus1 + 120582119896minus22 if 119896 gt 1 (61)

where 1205820 = 1 The termination condition is 119865119896 le 10minus4Numerical performance of all the algorithms is reported

in Tables 1 2 and 3 Table 1 shows the numerical performanceof all the eight algorithms with the fixed initial points 1199090 =(minus1 minus1)119879 for problems 1 5 and 6 1199090 = (1 1)119879 forproblems 2 and 4 1199090 = (10 10)119879 for problem 3 Table 2demonstrates the numerical performance of all the eightalgorithms with initial points randomly generated byMatlabrsquosCode ldquorand(n1)rdquo Table 3 shows the numerical performanceof our algorithm with dimension of 1000000

For simplification of statement we use the followingnotations

Dim the dimension of test problems

Ni the number of iterations

Nf the number of function evaluations

MPPRP-M the developed algorithm with 120574119896 determined by(19) in this paper

MPPRP-W the developed algorithmwith 120574119896 generated by (18)in this paper

MSDFPB the modified spectral derivative-free projection-based algorithm in [13]

PRP the PRP conjugate gradient derivative-free projection-based methods in [28]

MPRP the modified PRP conjugate gradient derivative-freeprojection-based methods in [10]

TPRP the two-term PRP conjugate gradient derivative-freeprojection-based methods in [10]

DFPB1 the steepest descent derivative-free projection-basedmethods in [14] with search direction as follows

119889119896 = minus1198650 if 119896 = 0minus119865119896 + 120573119896119889119896minus1 + 120579119896 if 119896 gt 0 (62)

where 120573119896 = 120573119875119877119875119896 = 119865119879119896 119910119896minus1119865119896minus12 120579119896 = (119865119879119896 119910119896minus1)120596119896minus12119865119896minus14 119910119896minus1 = 119865119896 minus 119865119896minus1 and 120596119896minus1 = 120572119896minus1119889119896minus1

8 Mathematical Problems in Engineering

Table 1 Numerical performance with fixed initial points

Problem Dim Method CPU-time Ni NfP1 10000 MPPRP-M 0049106 19 60

MPPRP-W 0083986 28 132MSDFPB 0207396 49 357

PRP 0210513 57 373MPRP 0225433 49 357TPRP 0205193 49 357DFPB1 0208191 52 360DFPB2 0180871 41 333MHS 0201207 49 357

P1 20000 MPPRP-M 0097751 20 63MPPRP-W 0202214 34 187MSDFPB 0538910 63 522

PRP 0601340 72 544MPRP 0578528 63 522TPRP 0567042 63 522DFPB1 0564317 67 534DFPB2 0538403 56 503MHS 0568248 63 522

P1 50000 MPPRP-M 0189582 20 63MPPRP-W 0776513 47 317MSDFPB 2349394 93 913

PRP 2426823 101 926MPRP 2373229 93 913TPRP 232399 93 913DFPB1 2421556 97 925DFPB2 2364173 86 894MHS 2601235 93 913

P1 100000 MPPRP-M 0519087 21 66MPPRP-W 2762991 61 478MSDFPB 9002493 127 1384

PRP 9108308 135 1402MPRP 9448788 127 1384TPRP 8986777 127 1384DFPB1 9100524 130 1391DFPB2 8819755 119 1360MHS 9419715 127 1384

P2 10000 MPPRP-M 0054535 19 60MPPRP-W 0088811 28 132MSDFPB 0217590 49 357

PRP 0218645 57 373MPRP 0226537 49 357TPRP 0226429 49 357DFPB1 0210470 52 360DFPB2 0204578 41 333MHS 0234442 49 357

P2 20000 MPPRP-M 0089917 20 63MPPRP-W 0198822 34 187MSDFPB 0563358 63 522

PRP 0604772 72 544

Mathematical Problems in Engineering 9

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0670294 63 522TPRP 0677819 63 522DFPB1 0679165 67 534DFPB2 0607357 56 503MHS 0589872 63 522

P2 50000 MPPRP-M 0191132 20 63MPPRP-W 0720108 47 317MSDFPB 2248779 93 913

PRP 2464597 101 926MPRP 2353173 93 913TPRP 2342777 93 913DFPB1 2363392 97 925DFPB2 2279782 86 894MHS 2384060 93 913

P2 100000 MPPRP-M 0516081 21 66MPPRP-W 2734834 61 478MSDFPB 9004299 127 1384

PRP 8763470 135 1402MPRP 8647855 127 1384TPRP 8626593 127 1384DFPB1 8626593 130 1391DFPB2 8476836 119 1360MHS 8905454 127 1384

P3 5000 MPPRP-M 0077881 22 69MPPRP-W 0955814 127 1360MSDFPB 2155000 242 3141

PRP 2346950 251 3167MPRP 2200878 242 3141TPRP 2204338 242 3141DFPB1 2246450 245 3148DFPB2 2209059 234 3117MHS 2236389 242 3141

P3 10000 MPPRP-M 0145481 23 72MPPRP-W 2590862 174 2059MSDFPB 6749566 337 4751

PRP 6892930 346 4767MPRP 6843273 337 4751TPRP 6885266 337 4751DFPB1 7222119 341 4763DFPB2 6881919 330 4732MHS 6992060 337 4751

P3 15000 MPPRP-M 0190225 23 72MPPRP-W 4894470 212 2656MSDFPB 12242514 411 6059

PRP 13494894 420 6078MPRP 13547676 411 6059TPRP 12255626 411 6059DFPB1 12473808 415 6071

10 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfDFPB2 12734751 404 6040MHS 12425026 411 6059

P3 20000 MPPRP-M 0239615 23 72MPPRP-W 7505229 242 3126MSDFPB 18305460 472 7144

PRP 19195911 481 7169MPRP 19482789 472 7144TPRP 19011112 472 7144DFPB1 19165800 475 7151DFPB2 19428512 464 7120MHS 18928398 472 7144

P4 10000 MPPRP-M 0061549 16 102MPPRP-W 0113153 19 131MSDFPB 0289696 68 555

PRP 0187397 39 391MPRP 0233670 48 462TPRP 0358863 89 658DFPB1 0657537 185 1142DFPB2 0213256 40 422MHS 0414446 62 803

P4 20000 MPPRP-M 0125760 17 108MPPRP-W 0256772 31 203MSDFPB 0703061 79 730

PRP 0558050 52 565MPRP 0663444 59 635TPRP 0822848 100 833DFPB1 1442003 203 1352DFPB2 0565771 51 596MHS 0674281 60 724

P4 50000 MPPRP-M 0295365 17 108MPPRP-W 0961757 38 273MSDFPB 2972641 102 1109

PRP 2655601 73 943MPRP 2738925 81 1007TPRP 3209823 122 1207DFPB1 4783178 234 1771DFPB2 2644189 75 981MHS 6106110 158 2717

P4 100000 MPPRP-M 0768978 17 108MPPRP-W 3113887 45 357MSDFPB 10575505 129 1568

PRP 9164701 101 1406MPRP 9756561 108 1465TPRP 11151034 146 1651DFPB1 16812871 275 2301DFPB2 9292510 99 1423MHS 21792651 179 3305

P5 100 MPPRP-M 0012253 18 77MPPRP-W 0018540 62 555MSDFPB 0049702 129 1522

PRP 0033290 144 1558

Mathematical Problems in Engineering 11

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0033017 139 1560TPRP 0034758 128 1518DFPB1 0026313 150 1601DFPB2 0029466 131 1538MHS 0040165 133 1539

P5 500 MPPRP-M 0016447 22 93MPPRP-W 0231399 544 9066MSDFPB 0772128 1313 25232

PRP 0727796 1332 25274MPRP 0585951 1372 25394TPRP 0577368 1315 25238DFPB1 0578794 1337 25315DFPB2 0589438 1317 25247MHS 0595316 1315 25241

P5 1000 MPPRP-M 0017438 23 97MPPRP-W 1072332 1505 29740MSDFPB 3218997 3680 81797

PRP 2889154 3696 81825MPRP 2963704 3735 81946TPRP 2948045 3682 81802DFPB1 3040253 3704 81880DFPB2 3045603 3683 81803MHS 3220028 3678 81791

P5 2000 MPPRP-M 0021613 25 106MPPRP-W 5944706 4206 95732MSDFPB 17227153 10341 260347

PRP 16073363 10355 260379MPRP 16960973 10394 260490TPRP 16826907 10343 260352DFPB1 16257104 10364 260424DFPB2 16739385 10345 260356MHS 17503252 10339 260342

P6 100 MPPRP-M 0217224 1182 3717MPPRP-W 0193153 1148 3605MSDFPB 0237793 1186 3706

PRP 0220389 1207 3766MPRP 0224211 1186 3658TPRP 0248104 1195 3722DFPB1 0236804 1201 3750DFPB2 0235470 1170 3620MHS 0236250 1198 3720

P6 500 MPPRP-M 0944828 1348 5050MPPRP-W 0794555 1278 4482MSDFPB 0797832 1185 3968

PRP 0911934 1322 4267MPRP 0845336 1258 3983TPRP 0912343 1326 4335DFPB1 0903852 1321 4300DFPB2 1020228 1299 4143MHS 0918611 1307 4147

P6 1000 MPPRP-M 2337405 1495 6667MPPRP-W 1680275 1381 4951MSDFPB 1536904 1331 4320

12 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfPRP 1851282 1389 4553MPRP 1647356 1277 4040TPRP 1686418 1273 4109DFPB1 1792886 1360 4409DFPB2 1812678 1343 4330MHS 1775351 1341 4283

P6 2000 MPPRP-M 6738143 1737 11016MPPRP-W 3257538 1440 4936MSDFPB 3065903 1381 4524

PRP 3574894 1390 4522MPRP 3381889 1316 4247TPRP 3565741 1378 4510DFPB1 3609619 1377 4511DFPB2 3647662 984 3232MHS 3608315 1362 4397

DFPB2 the steepest descent derivative-free projection-basedmethods in [14] with 120579119896 in (62) being replaced by

120579119896 = 119865119879119896 120596119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 + (119865119879119896 119910119896minus1) 1003817100381710038171003817119910119896minus1100381710038171003817100381721003817100381710038171003817119865119896minus110038171003817100381710038174 (63)

MHS the MHS-PRP conjugate gradient derivative-freeprojection-based methods in [15]

From the results in Tables 1 2 and 3 it follows that ouralgorithm (MPPRP) outperforms the other seven algorithmsno matter how to choose the initial points (see the italicizedresults) Especially it seems to more efficiently solve large-scale test problems Actually Table 3 shows that MPPRP-Mcan solve the first five Problems with dimension of 1000000in less time compared with the other algorithms

In order to further measure the efficiency difference ofall the eight algorithms we calculate the average numberof iteration the average consumed CPU time and theirstandard deviations respectively In Table 4 A-Ni and Std-Ni stand for the average number of iteration and its stan-dard deviation respectively A-CT and Std-CT represent theaverage consumed CPU time and its standard deviationrespectively The average number of function evaluation andits standard deviation are denoted by A-Nf and Std-Nfrespectively Clearly Std-Ni Std-Nf and Std-CT can showrobustness of all the algorithms

The results in Table 4 demonstrate that both ofMPPRP-Mand MPPRP-W outperform the other seven algorithms

In the end of this section we use our algorithm to solvea singular system of equations The next test problem is amodified version of problem 1

Problem 20 The elements of 119865(119909) are given by119865119894 (119909) = 119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (64)

The initial point is fixed as 1199090 = (1 1)119879

Note that problem 7 is singular since zero is its solutionand the Jacobian matrix 1198651015840(0) is singular We implementAlgorithm 1 to solve problem 7 with different dimen-sions to test whether it can find the singular solution ornot

The results in Table 5 indicate that Algorithm 1 is alsoefficient to solve the singular system of equations

5 Preliminary Applicationin Compressed Sensing

In this section we will apply our algorithm to solve anengineering problem originated from compressed sensing ofsparse signals

Let 119860 isin 119877119898times119899 (119898 ≪ 119899) be a linear operator let 119909 isin 119877119899be a sparse or a nearly sparse original signal and let 119887 isin119877119898 be an observed value which satisfies the following linearequations

119887 = 119860119909 (65)

The original signal 119909 is desirable to be reconstructedfrom the linear equations 119860119909 = 119887 Unfortunately itis often that this linear equation is underdetermined orill-conditioned in practice and has infinitely many solu-tions From the fundamental principles of compressed sens-ing it is a reasonable approach to seek the sparsest oneamong all the solutions which contains the least nonzerocomponents

In [2] it was shown that the compressed sensing of sparsesignals can be formulated the following nonlinear system ofequations

min 119911 (119867 + 119863) 119911 + 119888 = 0 (66)

Mathematical Problems in Engineering 13

Table 2 Efficiency with random initial points

Problem Dim Method CPU-time Ni NfP1 100000 MPPRP-M 0489058 20 63

MPPRP-W 2005818 42 268MSDFPB 4565133 79 717

PRP 4757480 87 825MPRP 4561151 79 796TPRP 4631983 79 796DFPB1 4763311 82 806DFPB2 4610199 74 776MHS 4879031 79 796

P2 100000 MPPRP-M 0483632 20 63MPPRP-W 2251160 42 268MSDFPB 4375745 79 717

PRP 4561437 87 825MPRP 4830143 79 796TPRP 4560123 79 796DFPB1 4427558 82 806DFPB2 4550071 74 776MHS 4428999 79 796

P3 100000 MPPRP-M 1187410 20 63MPPRP-W 5215137 46 306MSDFPB 10136405 79 710

PRP 10808303 87 816MPRP 10124779 79 789TPRP 10068535 79 789DFPB1 10372113 82 799DFPB2 9458892 71 757MHS 10137711 79 789

P4 100000 MPPRP-M 9152258 253 1306MPPRP-W 11371204 260 1423MSDFPB 13374780 293 1837

PRP 8352463 170 1352MPRP 9223706 184 1474TPRP 14279675 302 2184DFPB1 15329979 336 2386DFPB2 11244725 231 1770MHS 17738408 216 2902

P5 2000 MPPRP-M 0019793 25 106MPPRP-W 7148874 4207 95776MSDFPB 16293374 10362 260980

PRP 16689685 10375 271376MPRP 16112906 10415 271538TPRP 16006655 10364 271349DFPB1 16272868 10385 271442DFPB2 17018004 10366 271355MHS 19200453 10360 271335

P6 2000 MPPRP-M 9722610 3415 15767MPPRP-W 8031480 3293 12678MSDFPB 8254000 3395 12524

PRP 9187302 3298 15178MPRP 9667492 3366 15503TPRP 9727760 3363 15708DFPB1 9565437 3431 15990DFPB2 9316936 3329 15590MHS 8893926 3393 14502

14 Mathematical Problems in Engineering

Table 3 Efficiency for 1000000-dimension problems

Problem Dim Method CPU-time Ni NfP1 1000000 MPPRP-M 4831098 22 69P2 1000000 MPPRP-M 5150450 22 69P3 1000000 MPPRP-M 13954404 26 81P4 1000000 MPPRP-M 8408822 19 120P5 1000000 MPPRP-M 9018623 36 154

Table 4 Average numerical efficiency and robustness of algorithms

Method (A-CT Std-CT) (A-Ni Std-Ni) (A-Nf Std-CT)MPPRP-M (11657 25775) (3307 76388) (15135 36523)MPPRP-W (25725 29258) (6894 119935) (92059 242337)MSDFPB (54010 56632) (124456 263861) ( 231436 663099)PRP (53178 55599) (124773 264115) (235830 676024)MPRP (53717 56475) (124336 265463) (235743 676492)TPRP (55309 57401) ( 124963 263694) (236334 675796)DFPB1 (58718 60331) (12759 263685) (237512 675685)DFPB2 (53586 56541) (122393 264211) (235434 676120)MHS (62549 67712) (12488 263778) (237192 675398)

where

119911 = [119906V]

119910 = 119860119879119887119888 = 1205911198902119899 + [minus119910119910 ]

119867 = [ 119860119879119860 minus119860119879119860minus119860119879119860 119860119879119860 ] 119863 = [0 1205751198641198990 0 ]

(67)

where 119906 isin 119877119899 V isin 119877119899 and 119906119894 = (119909119894)+ V119894 = (minus119909119894)+ forall 119894 = 1 2 119899 with (sdot)+ = max0 sdot and 1198902119899 is an 2119899-dimensional vector with all elements being one Clearly (66)is nonsmooth

Implement Algorithm 1 to solve the compressed sensingproblem of sparse signals with different compressed ratios Inthis experiment we consider a typical compressive sensingscenario where the goal is to reconstruct an 119899-length sparsesignal from 119898 observations We test the three algorithms(two popular algorithms CGD in [34] and SGCS in [35] andMPPRP-M) under three CS ratios (CS-R)119898119899 = 0125 02505 corresponding to different numbers of measurementswith 119899 = 2048 respectively The original sparse signal con-tains 32(64) randomly nonzero elements The measurement119887 is disturbed by noise ie 119887 = 119860119909 + 120596 where 120596 is theGaussian noise distributed as 119873(0 1205902119868) with 1205902 = 10minus4 119860is the Gaussian matrix generated by commend randn(119898 119899)

inMatlab The quality of restoration is measured by the meanof squared error (MSE) to the original signal 119909 that is

MSE = 1119899 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (68)

where 119909lowast is the restored signal The iterative process starts atthe measurement image ie 1199090 = 119860119879119887 and terminates whenthe relative change between successive iterates falls below10minus5 It says that 1003817100381710038171003817119891119896 minus 119891119896minus110038171003817100381710038171003817100381710038171003817119891119896minus11003817100381710038171003817 lt 10minus5 (69)

where 119891119896 = 1205911199091198961 + (12)119860119909119896 minus 11988722 and 120591 = 0005119860119879119887infinis chosen as suggested in [36] The other parameters in thisexperiment are same as those in the numerical experimentsconducted in Section 4 Numerical efficiency is shown inTable 6

Clearly from the results in Table 6 it follows that forany CS ratio MPPRP-M can recover the sparse signals moreefficiently without reduction of recovery quality (see theitalicized results in Table 6) If the sparsity level 32 is replacedby 64 in the 2048-length original signal the correspondingresults are also presented in Table 6 denoted by (sdot)6 Conclusions

In this paper we have presented a modified spectral PRPconjugate gradient derivative-free projection-based methodfor solving the large-scale nonlinear monotonic equationswhere the search direction is proved to be sufficiently descentfor any line search rule and the step length is chosen by aline search which can overcome the difficulty of choosing anappropriate weight

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

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Mathematical Problems in Engineering 3

Very recently Wan et al [13] proposed a modified spectralconjugate gradient projection method for solving nonlinearmonotone symmetric equations where 119889119896 was chosen by

119889119896 = minus119865119896 if 119896 = 0minus120579119896119865119896 + 120573119896119889119896minus1 if 119896 gt 0 (9)

and 120573119896 and 120579119896 are computed by

120579119896 = 119889119879119896minus1 (119910119896minus1 minus 119865119896119865119879119896 119904119896minus1 100381710038171003817100381711986511989610038171003817100381710038172)119889119879119896minus1

(119868 minus 119865 (119909119896) 119865 (119909119896)119879 1003817100381710038171003817119865 (119909119896)10038171003817100381710038172) 119910119896minus1 120573119896 = 119865119879119896 (119910119896minus1 minus 119904119896minus1)119889119879

119896minus1(119868 minus 119865 (119909119896) 119865 (119909119896)119879 1003817100381710038171003817119865 (119909119896)10038171003817100381710038172) 119910119896minus1

(10)

respectively 119904119896minus1 = 119909119896 minus 119909119896minus1 and 119910119896minus1 = 119865(119909119896) minus 119865(119909119896minus1) Itwas proved in [13] that 119889119896 given by (9) and (10) is sufficientlydescent and satisfies 119865119879119896 119889119896 = minus1198651198962

Note that a modified spectral PRP conjugate gradientmethod was proposed for solving unconstrained optimiza-tion problems in [30] Similar to the idea in [13] we canextend the method in [30] to solution of problem (1)Specifically we compute 120573119896 and 120579119896 in (9) by

120579119896 = 119889119879119896minus1119910119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 minus 119889119879119896minus1119865119896119865119879119896 119865119896minus1100381710038171003817100381711986511989610038171003817100381710038172 1003817100381710038171003817119865119896minus110038171003817100381710038172 120573119896 = 119865119879119896 119910119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172

(11)

respectively Although (11) gives different choices of 120573119896 and 120579119896from (10) we can also prove the following result

Proposition 1 Let 119889119896 be given by (9) and (11) en for any119896 ge 0 the following equality holds119865119879119896 119889119896 = minus 100381710038171003817100381711986511989610038171003817100381710038172 (12)

Proof For 119896 = 0 we have1198651198790 1198890 = 1198651198790 (minus1198650) = minus 1003817100381710038171003817119865010038171003817100381710038172 (13)

For 119896 = 1 we have1198651198791 1198891 = 1198651198791 (minus12057911198651 + 12057311198890)= minus1198891198790 (1198651 minus 1198650)1003817100381710038171003817119865010038171003817100381710038172 1198651198791 1198651 + 119889119879011986511198651198791 11986501003817100381710038171003817119865110038171003817100381710038172 10038171003817100381710038171198650100381710038171003817100381721198651198791 1198651

+ 1198651198791 (1198651 minus 1198650)1003817100381710038171003817119865010038171003817100381710038172 11988911987901198651 = 1198891198790119865010038171003817100381710038171198650100381710038171003817100381721198651198791 1198651= 10038171003817100381710038171198651100381710038171003817100381721003817100381710038171003817119865010038171003817100381710038172 (minus 1003817100381710038171003817119865010038171003817100381710038172) = minus 1003817100381710038171003817119865110038171003817100381710038172

(14)

We now prove that if119865119879119896minus1119889119896minus1 = minus 1003817100381710038171003817119865119896minus110038171003817100381710038172 (15)

holds for 119896 minus 1 (119896 gt 1) then (12) also holds for 119896

Actually119865119879119896 119889119896 = 119865119879119896 (minus120579119896119865119896 + 120573119896119889119896minus1)= minus119889119879119896minus1 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119865119879119896 119865119896

+ 119889119879119896minus1119865119896119865119879119896 119865119896minus1100381710038171003817100381711986511989610038171003817100381710038172 1003817100381710038171003817119865119896minus110038171003817100381710038172 119865119879119896 119865119896+ 119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119889119879119896minus1119865119896 = 119889119879119896minus1119865119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 119865119879119896 119865119896

= minus 1003817100381710038171003817119865119896minus1100381710038171003817100381721003817100381710038171003817119865119896minus110038171003817100381710038172 100381710038171003817100381711986511989610038171003817100381710038172 = minus 100381710038171003817100381711986511989610038171003817100381710038172

(16)

where the forth equality follows condition (15) Consequentlyby mathematical induction (12) holds for any 119896

Proposition 1 ensures that the idea of projection methodcan be incorporated into design of iteration schemes to solve(1) as the search direction is determined by (9) and (11)

23 Modified Line Search Rule Since it is critical to choosean appropriate step length to improve the performance ofthe iterate scheme (3) as well as determination of searchdirections we now present an inexact line search rule todetermine 120572119896 in (3)

Very recently Ou and Li [31] presented a line search ruleas follows find a nonnegative step length 120572119896 = max119904120588119894 119894 =0 1 2 such that the following inequality is as followsminus119865 (119909119896 + 120572119896119889119896)119879 119889119896 ge 120590120572119896120574119896 100381710038171003817100381711988911989610038171003817100381710038172 (17)

where 119889119896 is a fixed search direction 119904 gt 0 is a given initialstep length 120588 isin (0 1) and 120590 gt 0 are two given constants and120574119896 is specified by120574119896 = 120582119896 + (1 minus 120582119896) 1003817100381710038171003817119865 (119909119896 + 120572119896119889119896)1003817100381710038171003817 (18)

In [31] it was required that 120582119896 in (18) satisfies 120582119896 isin[120582119898119894119899 120582119898119886119909] sube (0 1] Clearly 120582119896 and (1 minus 120582119896) are the weightsfor the values 1 and 119865(119909119896 + 120572119896119889119896) respectively In thepractical implementation it is may be difficult to choose anappropriate 120582119896 To overcome this difficulty we choose 120574119896 in(18) by 120574119896 = min 1 1003817100381710038171003817119865 (119909119896 + 120572119896119889119896)1003817100381710038171003817 (19)

It is sure that for the newmethod as a combination of (9) (11)(17) and (19) we need to establish its convergence theory andto further test its numerical performance

Remark 2 In [31] to ensure that well-defined the line search(17) is well-defined it is assumed that 119889119896 satisfies119865 (119909119896)119879 119889119896 le minus120591 1003817100381710038171003817119865 (119909119896)10038171003817100381710038172 (20)

where 120591 gt 0 is a given constant By Proposition 1 it is clearthat 119889119896 chosen by (9) and (11) satisfies (20) as 120591 = 1

4 Mathematical Problems in Engineering

InputAn initial point 1199090 isin 119877119899 positive constants 119896119898119886119909 120590 120576 119904 and 120588 isin (0 1)Begin119896 larr997888 01198650 larr997888 119865(1199090)While (119865119896 ge 120576 and 119896 lt 119896119898119886119909)Step 1 (Search direction)Compute 119889119896 by (9) and (11)Step 2 (Step length)120572 larr997888 119904Compute 119865119896 larr997888 119865(119909119896 + 120572119889119896)While (minus119865119879119896 119889119896 lt 1205901205721205741198961198891198962)120572 larr997888 120588120572Compute 119865119896 larr997888 119865(119909119896 + 120572119889119896)EndWhile120572119896 larr997888 120572Step 3 (Projection and update)119911119896 larr997888 119909119896 + 120572119896119889119896If 119865(119911119896) le 120576119909119896+1 larr997888 119911119896BreakEnd IfCompute 119909119896+1 by (7)119865119896+1 larr997888 119865(119909119896+1)119896 larr997888 119896 + 1EndWhileEnd

Algorithm 1 Modified spectral PRP derivative-free projection-based algorithm (MPPRP)

24 Development of New Projection-Based Algorithm Withthe above preparation we are in a position to develop analgorithm to solve problem (1) by combining the projectiontechnique and the new methods to determine a searchdirection and a step length

We now present the computer procedure of Algorithm 1

Remark 3 Since Algorithm 1 does not involve computing theJacobian matrix of 119865 or its approximation both informationstorage and computational cost of the algorithm are lower Invirtue of this advantage Algorithm 1 is helpful to solution oflarge-scale problems In next section we will prove that Algo-rithm 1 is applicable even if 119865 is nonsmooth Our numericaltests in Section 4will further show that Algorithm 1 can find asingular solution of problem (1) (see problem 5 and Table 5)

Remark 4 Compared with the algorithm developed in [13]problem (1) is not assumed to be a symmetric system ofequations

3 Convergence

In this section we are going to study global convergence ofAlgorithm 1

Apart from different choices of search direction and steplength Algorithm 1 can be treated as a variant of the projec-tion algorithm in [29] So similar to some critical points ofestablishing global convergence in [29] we attempt to prove

that Algorithm 1 is globally convergent Very recently locallylinear convergence was proved in [32] for some PRP-typeprojection methods

We first state the following mild assumptions

Assumption 5 The function 119865 is monotone on 119877119899Assumption 6 The solution set of problem (1) is nonempty

Assumption 7 The function 119865 is Lipschitz continuous on 119877119899namely there exists a positive constant 119871 such that for all119909 119910 isin 119877119899 1003817100381710038171003817119865 (119909) minus 119865 (119910)1003817100381710038171003817 le 119871 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 (21)

Under these assumptions we can prove that Algorithm 1has the following nice properties

Lemma 8 Let 119909119896 be a sequence generated by Algorithm 1 IfAssumptions 5 6 and 7 hold then

(1) for any 119909lowast such that 119865(119909lowast) = 01003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172 le 1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 (22)

(2) e sequence 119909119896 is bounded(3) If 119909119896 is a finite sequence then the last iterate point is

a solution of problem (1) otherwise119899sum119894=1

1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 lt infin (23)

Mathematical Problems in Engineering 5

and

lim119896997888rarrinfin

1003817100381710038171003817119909119896+1 minus 1199091198961003817100381710038171003817 = 0 (24)

(4) e sequence 119865(119909119896) is bounded Hence there existsa constant 119872119891 such that 119865(119909119896) le 119872119891Proof (1) Let 119909lowast be any point such that 119865(119909lowast) = 0 Then bymonotonicity of 119865 we have⟨119865 (119911119896) 119909lowast minus 119911119896⟩ le 0 (25)

From (7) it is also easy to verify that 119909119896+1 is the projection of119909119896 onto the halfspace119909 isin 119877119899 | ⟨119865 (119911119896) 119909 minus 119911119896⟩ le 0 (26)

Thus it follows from (25) that 119909lowast belongs to this halfspaceFrom the basic properties of projection operator [33] weknow that ⟨119909119896 minus 119909119896+1 119909119896+1 minus 119909lowast⟩ ge 0 (27)

Consequently1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 = 1003817100381710038171003817119909119896 minus 119909119896+110038171003817100381710038172 + 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172+ 2 ⟨119909119896 minus 119909119896+1 119909119896+1 minus 119909lowast⟩ge 1003817100381710038171003817119909119896 minus 119909119896+110038171003817100381710038172 + 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172 (28)

The desired result (22) is directly obtained from (28)(2) From (28) it is clear that the sequence 119909119896 minus 119909lowast is

nonnegative and decreasing Thus 119909119896minus119909lowast is a convergentsequence It is concluded that 119909119896 is bounded

(3) From (28) we know1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 le 1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172 (29)

Thus

119899sum119896=1

1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 le 119899sum119896=1

(1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172)= 10038171003817100381710038171199091 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119899+1 minus 119909lowast10038171003817100381710038172 (30)

Since the sequence 119909119896 is bounded the series sum119899119896=1 119909119896+1 minus1199091198962 is convergent Consequentlylim119896997888rarrinfin

1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 = 0 (31)

The third result has been proved(4) For any 119909119896 by Lipschitz continuity we have1003817100381710038171003817119865 (119909119896)1003817100381710038171003817 = 1003817100381710038171003817119865 (119909119896) minus 119865 (119909lowast)1003817100381710038171003817 le 119871 1003817100381710038171003817119909119896 minus 119909lowast1003817100381710038171003817 (32)

Since 119909119896 minus119909lowast is convergent we conclude that 119865(119909119896) isbounded Consequently for all 119896 isin 119873 there exists a constant119872119891 such that 119865(119909119896) le 119872119891

Lemma 8 indicates that for the sequence 119909119896 generatedby Algorithm 1 the sequence 119909119896 minus 119909lowast is decreasing andconvergent and the sequence 119909119896 is bounded where 119909lowast isany solution of problem (1)

Lemma 9 Suppose that Assumptions 5 6 and 7 hold Let119889119896 be a sequence generated by Algorithm 1 If there exists aconstant 1205760 gt 0 such that for any positive integer 11989610038171003817100381710038171198651198961003817100381710038171003817 ge 1205760 (33)

en the sequence of directions 119889119896 is bounded ie thereexists a constant 119872 gt 0 such that for any positive integer 11989610038171003817100381710038171198891198961003817100381710038171003817 le 119872 (34)

Proof From (9) (11) (12) and the results of Lemma 8 itfollows that10038171003817100381710038171198891198961003817100381710038171003817 = 1003817100381710038171003817minus120579119896119865119896 + 120573119896119889119896minus11003817100381710038171003817 = 1003817100381710038171003817100381710038171003817100381710038171003817minus119889119879119896minus1 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119865119896

+ 119889119879119896minus1119865119896119865119879119896 119865119896minus1100381710038171003817100381711986511989610038171003817100381710038172 1003817100381710038171003817119865119896minus110038171003817100381710038172 119865119896 + 119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119889119896minus11003817100381710038171003817100381710038171003817100381710038171003817 = 1003817100381710038171003817100381710038171003817100381710038171003817minus119865119896minus 119889119879119896minus1119865119896119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 100381710038171003817100381711986511989610038171003817100381710038172 119865119896 + 119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119889119896minus11003817100381710038171003817100381710038171003817100381710038171003817le 2 100381710038171003817100381711986511989610038171003817100381710038171003817100381710038171003817119865119896minus110038171003817100381710038172 1003817100381710038171003817119889119896minus11003817100381710038171003817 1003817100381710038171003817119910119896minus11003817100381710038171003817 + 10038171003817100381710038171198651198961003817100381710038171003817 le 211987111987211989112057620 1003817100381710038171003817119909119896minus 119909119896minus11003817100381710038171003817 1003817100381710038171003817119889119896minus11003817100381710038171003817 + 119872119891

(35)

From (24) we know that there exist a positive integer 1198961 anda positive number 1205761 (0 lt 1205761 lt 1) such that for all 119896 ge 1198961211987111987211989112057620 1003817100381710038171003817119909119896 minus 119909119896minus11003817100381710038171003817 lt 1205761 (36)

Hence10038171003817100381710038171198891198961003817100381710038171003817 le 1205761 1003817100381710038171003817119889119896minus11003817100381710038171003817 + 119872119891 le 12057621 1003817100381710038171003817119889119896minus21003817100381710038171003817 + 1205761119872119891 + 119872119891le 12057611989611 10038171003817100381710038171003817119889119896110038171003817100381710038171003817 + 1205761198961minus11 119872119891 + sdot sdot sdot + 1205761119872119891 + 119872119891 le 10038171003817100381710038171003817119889119896110038171003817100381710038171003817+ 119872119891 1 minus 120576119896111 minus 1205761 le 10038171003817100381710038171003817119889119896110038171003817100381710038171003817 + 1198721198911 minus 1205761

(37)

Let 1198721 = 1198891198961 + 119872119891(1 minus 1205761) Take119872 = max 100381710038171003817100381711988901003817100381710038171003817 100381710038171003817100381711988911003817100381710038171003817 10038171003817100381710038171003817119889119896110038171003817100381710038171003817 1198721 (38)

Then 119889119896 le 119872 holds for any positive integer 119896Lemma 10 Suppose that Assumptions 5 6 and 7 hold Let119909119896 and 119889119896 be two sequences generated by Algorithm 1enthe line search rule (17) of Step 2 in Algorithm 1 is well-defined

6 Mathematical Problems in Engineering

Proof Our aim is to show that the line search rule (17) termi-nates finitely with a positive step length 120572119896 By contradictionsuppose that for some iterate indexes such as 119896lowast condition(17) does not hold As a result for all 119898 isin 119873minus119865 (119909119896lowast + 119904120588119898119889119896lowast)119879 119889119896lowast lt 120590119904120588119898120574119896lowast 1003817100381710038171003817119889119896lowast10038171003817100381710038172 (39)

From (18) and the termination condition of Algorithm 1 itfollows that for all 119898 isin 1198730 le 120576 le 120574119896lowast le 1 (40)

By taking the limit as 119898 997888rarr infin in both sides of (39) we have

minus119865 (119909119896lowast)119879 119889119896lowast le 0 (41)

Equations (41) contradicts the fact that minus119865119879119896 119889119896 = minus1198651198962 gt 0for all 119896 That is to say the line search rule terminates finitelywith a positive step length 120572119896 ie the line search step ofAlgorithm 1 is well-defined

With the above preparation we are now state the conver-gence result of Algorithm 1

Theorem 11 Suppose that Assumptions 5 6 and 7 hold Let119909119896 be a sequence generated by Algorithm 1 en

lim inf119896997888rarrinfin

10038171003817100381710038171198651198961003817100381710038171003817 = 0 (42)

Proof For the sake of contradiction we suppose that theconclusion is not trueThen by the definition of inferior limitthere exists a constant 1205760 gt 0 such that for any 119896 isin 11987310038171003817100381710038171198651198961003817100381710038171003817 ge 1205760 (43)

Consequently from100381710038171003817100381711986511989610038171003817100381710038172 = minus119865119879119896 119889119896 le 10038171003817100381710038171198651198961003817100381710038171003817 10038171003817100381710038171198891198961003817100381710038171003817 (44)

it follows that 119889119896 ge 1205760 gt 0 for any 119896 isin 119873From (7) (17) and (40) we get

1003817100381710038171003817119909119896+1 minus 1199091198961003817100381710038171003817 = 100381610038161003816100381610038161003816119865 (119911119896)119879 (119909119896 minus 119911119896)1003816100381610038161003816100381610038161003817100381710038171003817119865 (119911119896)1003817100381710038171003817 = 100381610038161003816100381610038161003816120572119896119865 (119911119896)119879 1198891198961003816100381610038161003816100381610038161003817100381710038171003817119865 (119911119896)1003817100381710038171003817ge 1205901205722119896120574119896 1003817100381710038171003817119889119896100381710038171003817100381721003817100381710038171003817119865 (119911119896)1003817100381710038171003817 ge 1205901205761205722119896 100381710038171003817100381711988911989610038171003817100381710038172119872119891 ge 0 (45)

Combining (24) and (45) we obtain

lim119896997888rarrinfin

120572119896 10038171003817100381710038171198891198961003817100381710038171003817 = 0 (46)

Since 119889119896 ge 1205760 gt 0 for any 119896 isin 119873 we have

lim119896997888rarrinfin

120572119896 = 0 (47)

Clearly 120572lowast119896 = 120588minus1120572119896 does not satisfy in (17) It says that

minus119865 (119909119896 + 120572lowast119896119889119896)119879 119889119896 lt 120590120572lowast119896120574119896 100381710038171003817100381711988911989610038171003817100381710038172 (48)

By Lemmas 8 and 9 we know that the two sequences 119909119896and 119889119896 are bounded Without loss of generality we choosea subset 119870 isin 119873 such that

lim119896997888rarrinfin119896isin119870

119909119896 = 119909lowastlim119896997888rarrinfin119896isin119870

119889119896 = 119889lowast (49)

Taking the limit in the two sides of (48) as 119896 997888rarr infin (119896 isin 119870)it holds that

119865 (119909lowast)119879 119889lowast ge 0 (50)

On the other hand from (43) we know that

119865 (119909119896)119879 119889119896 = minus 100381710038171003817100381711986511989610038171003817100381710038172 le minus12057620 lt 0 (51)

By taking the limit 119896 997888rarr infin in the two sides of (51) for 119896 isin 119870we get

119865 (119909lowast)119879 119889lowast le minus1205760 lt 0 (52)

It contradicts (50) Thus the proof of Theorem 11 has beencompleted

Remark 12 (only with 120574119896 being generated by (19)) As 120574119896 isdetermined by (18) the proofs are similar

Since the proof of Theorem 11 does not involve differen-tiability of 119865 let alone nonsingularity of its Jacobian matrixwe know that the following result holds

Corollary 13 For any nonsmooth or singular function 119865 let119909119896 be a sequence generated as Algorithm 1 is used to solveproblem (1) Under Assumptions 5 6 and 7 it holds that

lim inf119896997888rarrinfin

10038171003817100381710038171198651198961003817100381710038171003817 = 0 (53)

It should be pointed out that the global convergenceof Algorithm 1 depends on assumption on monotonicity of119865 For nonmonotonic function 119865 Algorithm 1 may be notapplicable

4 Numerical Experiments

In this section by numerical experiments we are going tostudy the effectiveness and robustness of Algorithm 1 forsolving large-scale system of equations

We first collect some benchmark test problems availablein the literature

Problem 14 (see [5]) The elements of 119865(119909) are given by119865119894 (119909) = 2119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (54)

Problem 15 (see [5]) The elements of 119865(119909) are given by119865119894 (119909) = 2119909119894 minus 1003816100381610038161003816sin (119909119894)1003816100381610038161003816 119894 = 1 2 119899 minus 1 (55)

Mathematical Problems in Engineering 7

Problem 16 (see [15]) The elements of 119865(119909) are given by

1198651 (119909) = 1199091 minus 119890cos((1199091+1199092)(119899+1))119865119894 (119909) = 119909119894 minus 119890cos((119909119894minus1+119909119894+119909119894+1)(119899+1)) 119894 = 2 119899 minus 1119865119899 (119909) = 119909119899 minus 119890cos((119909119899minus1+119909119899)(119899+1)) (56)

Problem 17 (see [28]) The elements of 119865(119909) are given by

119865 (119909) = 119860119909 + 119887119887 = (minus1 minus1 minus1 minus1)119879

119860 = ((((((

52 11 52 1d d d

d d 11 52

))))))

(57)

Clearly problem 4 is a linear system of equations which isused to test Algorithm 1 in this special case

Problem 18 (see [15]) The elements of 119865(119909) are given by

119865119894 (119909) = 119909119894 minus 11198991199092119894 + 1119899 119899sum119894=1

119909119894 + 119894 119894 = 2 119899 minus 1 (58)

Problem 19 (see [15]) The elements of 119865(119909) are given by

1198651 (119909) = 1311990931 + 1211990922119865119894 (119909) = minus121199092119894 + 11989431199093119894 + 121199092119894+1 119894 = 2 119899 minus 1119865119899 (119909) = minus121199092119899 + 11989931199093119899

(59)

Clearly apart from problem 4 all the others are nonlinearsystem of equations Problem 15 is nonsmooth at point(0 0 0) The size of all the test problems is variable Ifthis size is larger than 1000 the problem can be regarded tobe large-scale We solve all the test problems with differentsizes by Algorithm 1 especially in comparison with theexisting seven similar algorithms such as those developedvery recently in [10 13ndash15 28]

All the algorithms are coded inMATLABR2010b and runon a personal computer with a 22GHZ CPU processor 8GB

memory and Windows 10 operation system The relevantparameters of algorithms are specified by119896119898119886119909 = 100000120590 = 03119904 = 05120588 = 07

(60)

120582119896 = 12058202 if 119896 = 1120582119896minus1 + 120582119896minus22 if 119896 gt 1 (61)

where 1205820 = 1 The termination condition is 119865119896 le 10minus4Numerical performance of all the algorithms is reported

in Tables 1 2 and 3 Table 1 shows the numerical performanceof all the eight algorithms with the fixed initial points 1199090 =(minus1 minus1)119879 for problems 1 5 and 6 1199090 = (1 1)119879 forproblems 2 and 4 1199090 = (10 10)119879 for problem 3 Table 2demonstrates the numerical performance of all the eightalgorithms with initial points randomly generated byMatlabrsquosCode ldquorand(n1)rdquo Table 3 shows the numerical performanceof our algorithm with dimension of 1000000

For simplification of statement we use the followingnotations

Dim the dimension of test problems

Ni the number of iterations

Nf the number of function evaluations

MPPRP-M the developed algorithm with 120574119896 determined by(19) in this paper

MPPRP-W the developed algorithmwith 120574119896 generated by (18)in this paper

MSDFPB the modified spectral derivative-free projection-based algorithm in [13]

PRP the PRP conjugate gradient derivative-free projection-based methods in [28]

MPRP the modified PRP conjugate gradient derivative-freeprojection-based methods in [10]

TPRP the two-term PRP conjugate gradient derivative-freeprojection-based methods in [10]

DFPB1 the steepest descent derivative-free projection-basedmethods in [14] with search direction as follows

119889119896 = minus1198650 if 119896 = 0minus119865119896 + 120573119896119889119896minus1 + 120579119896 if 119896 gt 0 (62)

where 120573119896 = 120573119875119877119875119896 = 119865119879119896 119910119896minus1119865119896minus12 120579119896 = (119865119879119896 119910119896minus1)120596119896minus12119865119896minus14 119910119896minus1 = 119865119896 minus 119865119896minus1 and 120596119896minus1 = 120572119896minus1119889119896minus1

8 Mathematical Problems in Engineering

Table 1 Numerical performance with fixed initial points

Problem Dim Method CPU-time Ni NfP1 10000 MPPRP-M 0049106 19 60

MPPRP-W 0083986 28 132MSDFPB 0207396 49 357

PRP 0210513 57 373MPRP 0225433 49 357TPRP 0205193 49 357DFPB1 0208191 52 360DFPB2 0180871 41 333MHS 0201207 49 357

P1 20000 MPPRP-M 0097751 20 63MPPRP-W 0202214 34 187MSDFPB 0538910 63 522

PRP 0601340 72 544MPRP 0578528 63 522TPRP 0567042 63 522DFPB1 0564317 67 534DFPB2 0538403 56 503MHS 0568248 63 522

P1 50000 MPPRP-M 0189582 20 63MPPRP-W 0776513 47 317MSDFPB 2349394 93 913

PRP 2426823 101 926MPRP 2373229 93 913TPRP 232399 93 913DFPB1 2421556 97 925DFPB2 2364173 86 894MHS 2601235 93 913

P1 100000 MPPRP-M 0519087 21 66MPPRP-W 2762991 61 478MSDFPB 9002493 127 1384

PRP 9108308 135 1402MPRP 9448788 127 1384TPRP 8986777 127 1384DFPB1 9100524 130 1391DFPB2 8819755 119 1360MHS 9419715 127 1384

P2 10000 MPPRP-M 0054535 19 60MPPRP-W 0088811 28 132MSDFPB 0217590 49 357

PRP 0218645 57 373MPRP 0226537 49 357TPRP 0226429 49 357DFPB1 0210470 52 360DFPB2 0204578 41 333MHS 0234442 49 357

P2 20000 MPPRP-M 0089917 20 63MPPRP-W 0198822 34 187MSDFPB 0563358 63 522

PRP 0604772 72 544

Mathematical Problems in Engineering 9

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0670294 63 522TPRP 0677819 63 522DFPB1 0679165 67 534DFPB2 0607357 56 503MHS 0589872 63 522

P2 50000 MPPRP-M 0191132 20 63MPPRP-W 0720108 47 317MSDFPB 2248779 93 913

PRP 2464597 101 926MPRP 2353173 93 913TPRP 2342777 93 913DFPB1 2363392 97 925DFPB2 2279782 86 894MHS 2384060 93 913

P2 100000 MPPRP-M 0516081 21 66MPPRP-W 2734834 61 478MSDFPB 9004299 127 1384

PRP 8763470 135 1402MPRP 8647855 127 1384TPRP 8626593 127 1384DFPB1 8626593 130 1391DFPB2 8476836 119 1360MHS 8905454 127 1384

P3 5000 MPPRP-M 0077881 22 69MPPRP-W 0955814 127 1360MSDFPB 2155000 242 3141

PRP 2346950 251 3167MPRP 2200878 242 3141TPRP 2204338 242 3141DFPB1 2246450 245 3148DFPB2 2209059 234 3117MHS 2236389 242 3141

P3 10000 MPPRP-M 0145481 23 72MPPRP-W 2590862 174 2059MSDFPB 6749566 337 4751

PRP 6892930 346 4767MPRP 6843273 337 4751TPRP 6885266 337 4751DFPB1 7222119 341 4763DFPB2 6881919 330 4732MHS 6992060 337 4751

P3 15000 MPPRP-M 0190225 23 72MPPRP-W 4894470 212 2656MSDFPB 12242514 411 6059

PRP 13494894 420 6078MPRP 13547676 411 6059TPRP 12255626 411 6059DFPB1 12473808 415 6071

10 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfDFPB2 12734751 404 6040MHS 12425026 411 6059

P3 20000 MPPRP-M 0239615 23 72MPPRP-W 7505229 242 3126MSDFPB 18305460 472 7144

PRP 19195911 481 7169MPRP 19482789 472 7144TPRP 19011112 472 7144DFPB1 19165800 475 7151DFPB2 19428512 464 7120MHS 18928398 472 7144

P4 10000 MPPRP-M 0061549 16 102MPPRP-W 0113153 19 131MSDFPB 0289696 68 555

PRP 0187397 39 391MPRP 0233670 48 462TPRP 0358863 89 658DFPB1 0657537 185 1142DFPB2 0213256 40 422MHS 0414446 62 803

P4 20000 MPPRP-M 0125760 17 108MPPRP-W 0256772 31 203MSDFPB 0703061 79 730

PRP 0558050 52 565MPRP 0663444 59 635TPRP 0822848 100 833DFPB1 1442003 203 1352DFPB2 0565771 51 596MHS 0674281 60 724

P4 50000 MPPRP-M 0295365 17 108MPPRP-W 0961757 38 273MSDFPB 2972641 102 1109

PRP 2655601 73 943MPRP 2738925 81 1007TPRP 3209823 122 1207DFPB1 4783178 234 1771DFPB2 2644189 75 981MHS 6106110 158 2717

P4 100000 MPPRP-M 0768978 17 108MPPRP-W 3113887 45 357MSDFPB 10575505 129 1568

PRP 9164701 101 1406MPRP 9756561 108 1465TPRP 11151034 146 1651DFPB1 16812871 275 2301DFPB2 9292510 99 1423MHS 21792651 179 3305

P5 100 MPPRP-M 0012253 18 77MPPRP-W 0018540 62 555MSDFPB 0049702 129 1522

PRP 0033290 144 1558

Mathematical Problems in Engineering 11

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0033017 139 1560TPRP 0034758 128 1518DFPB1 0026313 150 1601DFPB2 0029466 131 1538MHS 0040165 133 1539

P5 500 MPPRP-M 0016447 22 93MPPRP-W 0231399 544 9066MSDFPB 0772128 1313 25232

PRP 0727796 1332 25274MPRP 0585951 1372 25394TPRP 0577368 1315 25238DFPB1 0578794 1337 25315DFPB2 0589438 1317 25247MHS 0595316 1315 25241

P5 1000 MPPRP-M 0017438 23 97MPPRP-W 1072332 1505 29740MSDFPB 3218997 3680 81797

PRP 2889154 3696 81825MPRP 2963704 3735 81946TPRP 2948045 3682 81802DFPB1 3040253 3704 81880DFPB2 3045603 3683 81803MHS 3220028 3678 81791

P5 2000 MPPRP-M 0021613 25 106MPPRP-W 5944706 4206 95732MSDFPB 17227153 10341 260347

PRP 16073363 10355 260379MPRP 16960973 10394 260490TPRP 16826907 10343 260352DFPB1 16257104 10364 260424DFPB2 16739385 10345 260356MHS 17503252 10339 260342

P6 100 MPPRP-M 0217224 1182 3717MPPRP-W 0193153 1148 3605MSDFPB 0237793 1186 3706

PRP 0220389 1207 3766MPRP 0224211 1186 3658TPRP 0248104 1195 3722DFPB1 0236804 1201 3750DFPB2 0235470 1170 3620MHS 0236250 1198 3720

P6 500 MPPRP-M 0944828 1348 5050MPPRP-W 0794555 1278 4482MSDFPB 0797832 1185 3968

PRP 0911934 1322 4267MPRP 0845336 1258 3983TPRP 0912343 1326 4335DFPB1 0903852 1321 4300DFPB2 1020228 1299 4143MHS 0918611 1307 4147

P6 1000 MPPRP-M 2337405 1495 6667MPPRP-W 1680275 1381 4951MSDFPB 1536904 1331 4320

12 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfPRP 1851282 1389 4553MPRP 1647356 1277 4040TPRP 1686418 1273 4109DFPB1 1792886 1360 4409DFPB2 1812678 1343 4330MHS 1775351 1341 4283

P6 2000 MPPRP-M 6738143 1737 11016MPPRP-W 3257538 1440 4936MSDFPB 3065903 1381 4524

PRP 3574894 1390 4522MPRP 3381889 1316 4247TPRP 3565741 1378 4510DFPB1 3609619 1377 4511DFPB2 3647662 984 3232MHS 3608315 1362 4397

DFPB2 the steepest descent derivative-free projection-basedmethods in [14] with 120579119896 in (62) being replaced by

120579119896 = 119865119879119896 120596119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 + (119865119879119896 119910119896minus1) 1003817100381710038171003817119910119896minus1100381710038171003817100381721003817100381710038171003817119865119896minus110038171003817100381710038174 (63)

MHS the MHS-PRP conjugate gradient derivative-freeprojection-based methods in [15]

From the results in Tables 1 2 and 3 it follows that ouralgorithm (MPPRP) outperforms the other seven algorithmsno matter how to choose the initial points (see the italicizedresults) Especially it seems to more efficiently solve large-scale test problems Actually Table 3 shows that MPPRP-Mcan solve the first five Problems with dimension of 1000000in less time compared with the other algorithms

In order to further measure the efficiency difference ofall the eight algorithms we calculate the average numberof iteration the average consumed CPU time and theirstandard deviations respectively In Table 4 A-Ni and Std-Ni stand for the average number of iteration and its stan-dard deviation respectively A-CT and Std-CT represent theaverage consumed CPU time and its standard deviationrespectively The average number of function evaluation andits standard deviation are denoted by A-Nf and Std-Nfrespectively Clearly Std-Ni Std-Nf and Std-CT can showrobustness of all the algorithms

The results in Table 4 demonstrate that both ofMPPRP-Mand MPPRP-W outperform the other seven algorithms

In the end of this section we use our algorithm to solvea singular system of equations The next test problem is amodified version of problem 1

Problem 20 The elements of 119865(119909) are given by119865119894 (119909) = 119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (64)

The initial point is fixed as 1199090 = (1 1)119879

Note that problem 7 is singular since zero is its solutionand the Jacobian matrix 1198651015840(0) is singular We implementAlgorithm 1 to solve problem 7 with different dimen-sions to test whether it can find the singular solution ornot

The results in Table 5 indicate that Algorithm 1 is alsoefficient to solve the singular system of equations

5 Preliminary Applicationin Compressed Sensing

In this section we will apply our algorithm to solve anengineering problem originated from compressed sensing ofsparse signals

Let 119860 isin 119877119898times119899 (119898 ≪ 119899) be a linear operator let 119909 isin 119877119899be a sparse or a nearly sparse original signal and let 119887 isin119877119898 be an observed value which satisfies the following linearequations

119887 = 119860119909 (65)

The original signal 119909 is desirable to be reconstructedfrom the linear equations 119860119909 = 119887 Unfortunately itis often that this linear equation is underdetermined orill-conditioned in practice and has infinitely many solu-tions From the fundamental principles of compressed sens-ing it is a reasonable approach to seek the sparsest oneamong all the solutions which contains the least nonzerocomponents

In [2] it was shown that the compressed sensing of sparsesignals can be formulated the following nonlinear system ofequations

min 119911 (119867 + 119863) 119911 + 119888 = 0 (66)

Mathematical Problems in Engineering 13

Table 2 Efficiency with random initial points

Problem Dim Method CPU-time Ni NfP1 100000 MPPRP-M 0489058 20 63

MPPRP-W 2005818 42 268MSDFPB 4565133 79 717

PRP 4757480 87 825MPRP 4561151 79 796TPRP 4631983 79 796DFPB1 4763311 82 806DFPB2 4610199 74 776MHS 4879031 79 796

P2 100000 MPPRP-M 0483632 20 63MPPRP-W 2251160 42 268MSDFPB 4375745 79 717

PRP 4561437 87 825MPRP 4830143 79 796TPRP 4560123 79 796DFPB1 4427558 82 806DFPB2 4550071 74 776MHS 4428999 79 796

P3 100000 MPPRP-M 1187410 20 63MPPRP-W 5215137 46 306MSDFPB 10136405 79 710

PRP 10808303 87 816MPRP 10124779 79 789TPRP 10068535 79 789DFPB1 10372113 82 799DFPB2 9458892 71 757MHS 10137711 79 789

P4 100000 MPPRP-M 9152258 253 1306MPPRP-W 11371204 260 1423MSDFPB 13374780 293 1837

PRP 8352463 170 1352MPRP 9223706 184 1474TPRP 14279675 302 2184DFPB1 15329979 336 2386DFPB2 11244725 231 1770MHS 17738408 216 2902

P5 2000 MPPRP-M 0019793 25 106MPPRP-W 7148874 4207 95776MSDFPB 16293374 10362 260980

PRP 16689685 10375 271376MPRP 16112906 10415 271538TPRP 16006655 10364 271349DFPB1 16272868 10385 271442DFPB2 17018004 10366 271355MHS 19200453 10360 271335

P6 2000 MPPRP-M 9722610 3415 15767MPPRP-W 8031480 3293 12678MSDFPB 8254000 3395 12524

PRP 9187302 3298 15178MPRP 9667492 3366 15503TPRP 9727760 3363 15708DFPB1 9565437 3431 15990DFPB2 9316936 3329 15590MHS 8893926 3393 14502

14 Mathematical Problems in Engineering

Table 3 Efficiency for 1000000-dimension problems

Problem Dim Method CPU-time Ni NfP1 1000000 MPPRP-M 4831098 22 69P2 1000000 MPPRP-M 5150450 22 69P3 1000000 MPPRP-M 13954404 26 81P4 1000000 MPPRP-M 8408822 19 120P5 1000000 MPPRP-M 9018623 36 154

Table 4 Average numerical efficiency and robustness of algorithms

Method (A-CT Std-CT) (A-Ni Std-Ni) (A-Nf Std-CT)MPPRP-M (11657 25775) (3307 76388) (15135 36523)MPPRP-W (25725 29258) (6894 119935) (92059 242337)MSDFPB (54010 56632) (124456 263861) ( 231436 663099)PRP (53178 55599) (124773 264115) (235830 676024)MPRP (53717 56475) (124336 265463) (235743 676492)TPRP (55309 57401) ( 124963 263694) (236334 675796)DFPB1 (58718 60331) (12759 263685) (237512 675685)DFPB2 (53586 56541) (122393 264211) (235434 676120)MHS (62549 67712) (12488 263778) (237192 675398)

where

119911 = [119906V]

119910 = 119860119879119887119888 = 1205911198902119899 + [minus119910119910 ]

119867 = [ 119860119879119860 minus119860119879119860minus119860119879119860 119860119879119860 ] 119863 = [0 1205751198641198990 0 ]

(67)

where 119906 isin 119877119899 V isin 119877119899 and 119906119894 = (119909119894)+ V119894 = (minus119909119894)+ forall 119894 = 1 2 119899 with (sdot)+ = max0 sdot and 1198902119899 is an 2119899-dimensional vector with all elements being one Clearly (66)is nonsmooth

Implement Algorithm 1 to solve the compressed sensingproblem of sparse signals with different compressed ratios Inthis experiment we consider a typical compressive sensingscenario where the goal is to reconstruct an 119899-length sparsesignal from 119898 observations We test the three algorithms(two popular algorithms CGD in [34] and SGCS in [35] andMPPRP-M) under three CS ratios (CS-R)119898119899 = 0125 02505 corresponding to different numbers of measurementswith 119899 = 2048 respectively The original sparse signal con-tains 32(64) randomly nonzero elements The measurement119887 is disturbed by noise ie 119887 = 119860119909 + 120596 where 120596 is theGaussian noise distributed as 119873(0 1205902119868) with 1205902 = 10minus4 119860is the Gaussian matrix generated by commend randn(119898 119899)

inMatlab The quality of restoration is measured by the meanof squared error (MSE) to the original signal 119909 that is

MSE = 1119899 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (68)

where 119909lowast is the restored signal The iterative process starts atthe measurement image ie 1199090 = 119860119879119887 and terminates whenthe relative change between successive iterates falls below10minus5 It says that 1003817100381710038171003817119891119896 minus 119891119896minus110038171003817100381710038171003817100381710038171003817119891119896minus11003817100381710038171003817 lt 10minus5 (69)

where 119891119896 = 1205911199091198961 + (12)119860119909119896 minus 11988722 and 120591 = 0005119860119879119887infinis chosen as suggested in [36] The other parameters in thisexperiment are same as those in the numerical experimentsconducted in Section 4 Numerical efficiency is shown inTable 6

Clearly from the results in Table 6 it follows that forany CS ratio MPPRP-M can recover the sparse signals moreefficiently without reduction of recovery quality (see theitalicized results in Table 6) If the sparsity level 32 is replacedby 64 in the 2048-length original signal the correspondingresults are also presented in Table 6 denoted by (sdot)6 Conclusions

In this paper we have presented a modified spectral PRPconjugate gradient derivative-free projection-based methodfor solving the large-scale nonlinear monotonic equationswhere the search direction is proved to be sufficiently descentfor any line search rule and the step length is chosen by aline search which can overcome the difficulty of choosing anappropriate weight

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

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4 Mathematical Problems in Engineering

InputAn initial point 1199090 isin 119877119899 positive constants 119896119898119886119909 120590 120576 119904 and 120588 isin (0 1)Begin119896 larr997888 01198650 larr997888 119865(1199090)While (119865119896 ge 120576 and 119896 lt 119896119898119886119909)Step 1 (Search direction)Compute 119889119896 by (9) and (11)Step 2 (Step length)120572 larr997888 119904Compute 119865119896 larr997888 119865(119909119896 + 120572119889119896)While (minus119865119879119896 119889119896 lt 1205901205721205741198961198891198962)120572 larr997888 120588120572Compute 119865119896 larr997888 119865(119909119896 + 120572119889119896)EndWhile120572119896 larr997888 120572Step 3 (Projection and update)119911119896 larr997888 119909119896 + 120572119896119889119896If 119865(119911119896) le 120576119909119896+1 larr997888 119911119896BreakEnd IfCompute 119909119896+1 by (7)119865119896+1 larr997888 119865(119909119896+1)119896 larr997888 119896 + 1EndWhileEnd

Algorithm 1 Modified spectral PRP derivative-free projection-based algorithm (MPPRP)

24 Development of New Projection-Based Algorithm Withthe above preparation we are in a position to develop analgorithm to solve problem (1) by combining the projectiontechnique and the new methods to determine a searchdirection and a step length

We now present the computer procedure of Algorithm 1

Remark 3 Since Algorithm 1 does not involve computing theJacobian matrix of 119865 or its approximation both informationstorage and computational cost of the algorithm are lower Invirtue of this advantage Algorithm 1 is helpful to solution oflarge-scale problems In next section we will prove that Algo-rithm 1 is applicable even if 119865 is nonsmooth Our numericaltests in Section 4will further show that Algorithm 1 can find asingular solution of problem (1) (see problem 5 and Table 5)

Remark 4 Compared with the algorithm developed in [13]problem (1) is not assumed to be a symmetric system ofequations

3 Convergence

In this section we are going to study global convergence ofAlgorithm 1

Apart from different choices of search direction and steplength Algorithm 1 can be treated as a variant of the projec-tion algorithm in [29] So similar to some critical points ofestablishing global convergence in [29] we attempt to prove

that Algorithm 1 is globally convergent Very recently locallylinear convergence was proved in [32] for some PRP-typeprojection methods

We first state the following mild assumptions

Assumption 5 The function 119865 is monotone on 119877119899Assumption 6 The solution set of problem (1) is nonempty

Assumption 7 The function 119865 is Lipschitz continuous on 119877119899namely there exists a positive constant 119871 such that for all119909 119910 isin 119877119899 1003817100381710038171003817119865 (119909) minus 119865 (119910)1003817100381710038171003817 le 119871 1003817100381710038171003817119909 minus 1199101003817100381710038171003817 (21)

Under these assumptions we can prove that Algorithm 1has the following nice properties

Lemma 8 Let 119909119896 be a sequence generated by Algorithm 1 IfAssumptions 5 6 and 7 hold then

(1) for any 119909lowast such that 119865(119909lowast) = 01003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172 le 1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 (22)

(2) e sequence 119909119896 is bounded(3) If 119909119896 is a finite sequence then the last iterate point is

a solution of problem (1) otherwise119899sum119894=1

1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 lt infin (23)

Mathematical Problems in Engineering 5

and

lim119896997888rarrinfin

1003817100381710038171003817119909119896+1 minus 1199091198961003817100381710038171003817 = 0 (24)

(4) e sequence 119865(119909119896) is bounded Hence there existsa constant 119872119891 such that 119865(119909119896) le 119872119891Proof (1) Let 119909lowast be any point such that 119865(119909lowast) = 0 Then bymonotonicity of 119865 we have⟨119865 (119911119896) 119909lowast minus 119911119896⟩ le 0 (25)

From (7) it is also easy to verify that 119909119896+1 is the projection of119909119896 onto the halfspace119909 isin 119877119899 | ⟨119865 (119911119896) 119909 minus 119911119896⟩ le 0 (26)

Thus it follows from (25) that 119909lowast belongs to this halfspaceFrom the basic properties of projection operator [33] weknow that ⟨119909119896 minus 119909119896+1 119909119896+1 minus 119909lowast⟩ ge 0 (27)

Consequently1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 = 1003817100381710038171003817119909119896 minus 119909119896+110038171003817100381710038172 + 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172+ 2 ⟨119909119896 minus 119909119896+1 119909119896+1 minus 119909lowast⟩ge 1003817100381710038171003817119909119896 minus 119909119896+110038171003817100381710038172 + 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172 (28)

The desired result (22) is directly obtained from (28)(2) From (28) it is clear that the sequence 119909119896 minus 119909lowast is

nonnegative and decreasing Thus 119909119896minus119909lowast is a convergentsequence It is concluded that 119909119896 is bounded

(3) From (28) we know1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 le 1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172 (29)

Thus

119899sum119896=1

1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 le 119899sum119896=1

(1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172)= 10038171003817100381710038171199091 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119899+1 minus 119909lowast10038171003817100381710038172 (30)

Since the sequence 119909119896 is bounded the series sum119899119896=1 119909119896+1 minus1199091198962 is convergent Consequentlylim119896997888rarrinfin

1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 = 0 (31)

The third result has been proved(4) For any 119909119896 by Lipschitz continuity we have1003817100381710038171003817119865 (119909119896)1003817100381710038171003817 = 1003817100381710038171003817119865 (119909119896) minus 119865 (119909lowast)1003817100381710038171003817 le 119871 1003817100381710038171003817119909119896 minus 119909lowast1003817100381710038171003817 (32)

Since 119909119896 minus119909lowast is convergent we conclude that 119865(119909119896) isbounded Consequently for all 119896 isin 119873 there exists a constant119872119891 such that 119865(119909119896) le 119872119891

Lemma 8 indicates that for the sequence 119909119896 generatedby Algorithm 1 the sequence 119909119896 minus 119909lowast is decreasing andconvergent and the sequence 119909119896 is bounded where 119909lowast isany solution of problem (1)

Lemma 9 Suppose that Assumptions 5 6 and 7 hold Let119889119896 be a sequence generated by Algorithm 1 If there exists aconstant 1205760 gt 0 such that for any positive integer 11989610038171003817100381710038171198651198961003817100381710038171003817 ge 1205760 (33)

en the sequence of directions 119889119896 is bounded ie thereexists a constant 119872 gt 0 such that for any positive integer 11989610038171003817100381710038171198891198961003817100381710038171003817 le 119872 (34)

Proof From (9) (11) (12) and the results of Lemma 8 itfollows that10038171003817100381710038171198891198961003817100381710038171003817 = 1003817100381710038171003817minus120579119896119865119896 + 120573119896119889119896minus11003817100381710038171003817 = 1003817100381710038171003817100381710038171003817100381710038171003817minus119889119879119896minus1 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119865119896

+ 119889119879119896minus1119865119896119865119879119896 119865119896minus1100381710038171003817100381711986511989610038171003817100381710038172 1003817100381710038171003817119865119896minus110038171003817100381710038172 119865119896 + 119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119889119896minus11003817100381710038171003817100381710038171003817100381710038171003817 = 1003817100381710038171003817100381710038171003817100381710038171003817minus119865119896minus 119889119879119896minus1119865119896119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 100381710038171003817100381711986511989610038171003817100381710038172 119865119896 + 119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119889119896minus11003817100381710038171003817100381710038171003817100381710038171003817le 2 100381710038171003817100381711986511989610038171003817100381710038171003817100381710038171003817119865119896minus110038171003817100381710038172 1003817100381710038171003817119889119896minus11003817100381710038171003817 1003817100381710038171003817119910119896minus11003817100381710038171003817 + 10038171003817100381710038171198651198961003817100381710038171003817 le 211987111987211989112057620 1003817100381710038171003817119909119896minus 119909119896minus11003817100381710038171003817 1003817100381710038171003817119889119896minus11003817100381710038171003817 + 119872119891

(35)

From (24) we know that there exist a positive integer 1198961 anda positive number 1205761 (0 lt 1205761 lt 1) such that for all 119896 ge 1198961211987111987211989112057620 1003817100381710038171003817119909119896 minus 119909119896minus11003817100381710038171003817 lt 1205761 (36)

Hence10038171003817100381710038171198891198961003817100381710038171003817 le 1205761 1003817100381710038171003817119889119896minus11003817100381710038171003817 + 119872119891 le 12057621 1003817100381710038171003817119889119896minus21003817100381710038171003817 + 1205761119872119891 + 119872119891le 12057611989611 10038171003817100381710038171003817119889119896110038171003817100381710038171003817 + 1205761198961minus11 119872119891 + sdot sdot sdot + 1205761119872119891 + 119872119891 le 10038171003817100381710038171003817119889119896110038171003817100381710038171003817+ 119872119891 1 minus 120576119896111 minus 1205761 le 10038171003817100381710038171003817119889119896110038171003817100381710038171003817 + 1198721198911 minus 1205761

(37)

Let 1198721 = 1198891198961 + 119872119891(1 minus 1205761) Take119872 = max 100381710038171003817100381711988901003817100381710038171003817 100381710038171003817100381711988911003817100381710038171003817 10038171003817100381710038171003817119889119896110038171003817100381710038171003817 1198721 (38)

Then 119889119896 le 119872 holds for any positive integer 119896Lemma 10 Suppose that Assumptions 5 6 and 7 hold Let119909119896 and 119889119896 be two sequences generated by Algorithm 1enthe line search rule (17) of Step 2 in Algorithm 1 is well-defined

6 Mathematical Problems in Engineering

Proof Our aim is to show that the line search rule (17) termi-nates finitely with a positive step length 120572119896 By contradictionsuppose that for some iterate indexes such as 119896lowast condition(17) does not hold As a result for all 119898 isin 119873minus119865 (119909119896lowast + 119904120588119898119889119896lowast)119879 119889119896lowast lt 120590119904120588119898120574119896lowast 1003817100381710038171003817119889119896lowast10038171003817100381710038172 (39)

From (18) and the termination condition of Algorithm 1 itfollows that for all 119898 isin 1198730 le 120576 le 120574119896lowast le 1 (40)

By taking the limit as 119898 997888rarr infin in both sides of (39) we have

minus119865 (119909119896lowast)119879 119889119896lowast le 0 (41)

Equations (41) contradicts the fact that minus119865119879119896 119889119896 = minus1198651198962 gt 0for all 119896 That is to say the line search rule terminates finitelywith a positive step length 120572119896 ie the line search step ofAlgorithm 1 is well-defined

With the above preparation we are now state the conver-gence result of Algorithm 1

Theorem 11 Suppose that Assumptions 5 6 and 7 hold Let119909119896 be a sequence generated by Algorithm 1 en

lim inf119896997888rarrinfin

10038171003817100381710038171198651198961003817100381710038171003817 = 0 (42)

Proof For the sake of contradiction we suppose that theconclusion is not trueThen by the definition of inferior limitthere exists a constant 1205760 gt 0 such that for any 119896 isin 11987310038171003817100381710038171198651198961003817100381710038171003817 ge 1205760 (43)

Consequently from100381710038171003817100381711986511989610038171003817100381710038172 = minus119865119879119896 119889119896 le 10038171003817100381710038171198651198961003817100381710038171003817 10038171003817100381710038171198891198961003817100381710038171003817 (44)

it follows that 119889119896 ge 1205760 gt 0 for any 119896 isin 119873From (7) (17) and (40) we get

1003817100381710038171003817119909119896+1 minus 1199091198961003817100381710038171003817 = 100381610038161003816100381610038161003816119865 (119911119896)119879 (119909119896 minus 119911119896)1003816100381610038161003816100381610038161003817100381710038171003817119865 (119911119896)1003817100381710038171003817 = 100381610038161003816100381610038161003816120572119896119865 (119911119896)119879 1198891198961003816100381610038161003816100381610038161003817100381710038171003817119865 (119911119896)1003817100381710038171003817ge 1205901205722119896120574119896 1003817100381710038171003817119889119896100381710038171003817100381721003817100381710038171003817119865 (119911119896)1003817100381710038171003817 ge 1205901205761205722119896 100381710038171003817100381711988911989610038171003817100381710038172119872119891 ge 0 (45)

Combining (24) and (45) we obtain

lim119896997888rarrinfin

120572119896 10038171003817100381710038171198891198961003817100381710038171003817 = 0 (46)

Since 119889119896 ge 1205760 gt 0 for any 119896 isin 119873 we have

lim119896997888rarrinfin

120572119896 = 0 (47)

Clearly 120572lowast119896 = 120588minus1120572119896 does not satisfy in (17) It says that

minus119865 (119909119896 + 120572lowast119896119889119896)119879 119889119896 lt 120590120572lowast119896120574119896 100381710038171003817100381711988911989610038171003817100381710038172 (48)

By Lemmas 8 and 9 we know that the two sequences 119909119896and 119889119896 are bounded Without loss of generality we choosea subset 119870 isin 119873 such that

lim119896997888rarrinfin119896isin119870

119909119896 = 119909lowastlim119896997888rarrinfin119896isin119870

119889119896 = 119889lowast (49)

Taking the limit in the two sides of (48) as 119896 997888rarr infin (119896 isin 119870)it holds that

119865 (119909lowast)119879 119889lowast ge 0 (50)

On the other hand from (43) we know that

119865 (119909119896)119879 119889119896 = minus 100381710038171003817100381711986511989610038171003817100381710038172 le minus12057620 lt 0 (51)

By taking the limit 119896 997888rarr infin in the two sides of (51) for 119896 isin 119870we get

119865 (119909lowast)119879 119889lowast le minus1205760 lt 0 (52)

It contradicts (50) Thus the proof of Theorem 11 has beencompleted

Remark 12 (only with 120574119896 being generated by (19)) As 120574119896 isdetermined by (18) the proofs are similar

Since the proof of Theorem 11 does not involve differen-tiability of 119865 let alone nonsingularity of its Jacobian matrixwe know that the following result holds

Corollary 13 For any nonsmooth or singular function 119865 let119909119896 be a sequence generated as Algorithm 1 is used to solveproblem (1) Under Assumptions 5 6 and 7 it holds that

lim inf119896997888rarrinfin

10038171003817100381710038171198651198961003817100381710038171003817 = 0 (53)

It should be pointed out that the global convergenceof Algorithm 1 depends on assumption on monotonicity of119865 For nonmonotonic function 119865 Algorithm 1 may be notapplicable

4 Numerical Experiments

In this section by numerical experiments we are going tostudy the effectiveness and robustness of Algorithm 1 forsolving large-scale system of equations

We first collect some benchmark test problems availablein the literature

Problem 14 (see [5]) The elements of 119865(119909) are given by119865119894 (119909) = 2119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (54)

Problem 15 (see [5]) The elements of 119865(119909) are given by119865119894 (119909) = 2119909119894 minus 1003816100381610038161003816sin (119909119894)1003816100381610038161003816 119894 = 1 2 119899 minus 1 (55)

Mathematical Problems in Engineering 7

Problem 16 (see [15]) The elements of 119865(119909) are given by

1198651 (119909) = 1199091 minus 119890cos((1199091+1199092)(119899+1))119865119894 (119909) = 119909119894 minus 119890cos((119909119894minus1+119909119894+119909119894+1)(119899+1)) 119894 = 2 119899 minus 1119865119899 (119909) = 119909119899 minus 119890cos((119909119899minus1+119909119899)(119899+1)) (56)

Problem 17 (see [28]) The elements of 119865(119909) are given by

119865 (119909) = 119860119909 + 119887119887 = (minus1 minus1 minus1 minus1)119879

119860 = ((((((

52 11 52 1d d d

d d 11 52

))))))

(57)

Clearly problem 4 is a linear system of equations which isused to test Algorithm 1 in this special case

Problem 18 (see [15]) The elements of 119865(119909) are given by

119865119894 (119909) = 119909119894 minus 11198991199092119894 + 1119899 119899sum119894=1

119909119894 + 119894 119894 = 2 119899 minus 1 (58)

Problem 19 (see [15]) The elements of 119865(119909) are given by

1198651 (119909) = 1311990931 + 1211990922119865119894 (119909) = minus121199092119894 + 11989431199093119894 + 121199092119894+1 119894 = 2 119899 minus 1119865119899 (119909) = minus121199092119899 + 11989931199093119899

(59)

Clearly apart from problem 4 all the others are nonlinearsystem of equations Problem 15 is nonsmooth at point(0 0 0) The size of all the test problems is variable Ifthis size is larger than 1000 the problem can be regarded tobe large-scale We solve all the test problems with differentsizes by Algorithm 1 especially in comparison with theexisting seven similar algorithms such as those developedvery recently in [10 13ndash15 28]

All the algorithms are coded inMATLABR2010b and runon a personal computer with a 22GHZ CPU processor 8GB

memory and Windows 10 operation system The relevantparameters of algorithms are specified by119896119898119886119909 = 100000120590 = 03119904 = 05120588 = 07

(60)

120582119896 = 12058202 if 119896 = 1120582119896minus1 + 120582119896minus22 if 119896 gt 1 (61)

where 1205820 = 1 The termination condition is 119865119896 le 10minus4Numerical performance of all the algorithms is reported

in Tables 1 2 and 3 Table 1 shows the numerical performanceof all the eight algorithms with the fixed initial points 1199090 =(minus1 minus1)119879 for problems 1 5 and 6 1199090 = (1 1)119879 forproblems 2 and 4 1199090 = (10 10)119879 for problem 3 Table 2demonstrates the numerical performance of all the eightalgorithms with initial points randomly generated byMatlabrsquosCode ldquorand(n1)rdquo Table 3 shows the numerical performanceof our algorithm with dimension of 1000000

For simplification of statement we use the followingnotations

Dim the dimension of test problems

Ni the number of iterations

Nf the number of function evaluations

MPPRP-M the developed algorithm with 120574119896 determined by(19) in this paper

MPPRP-W the developed algorithmwith 120574119896 generated by (18)in this paper

MSDFPB the modified spectral derivative-free projection-based algorithm in [13]

PRP the PRP conjugate gradient derivative-free projection-based methods in [28]

MPRP the modified PRP conjugate gradient derivative-freeprojection-based methods in [10]

TPRP the two-term PRP conjugate gradient derivative-freeprojection-based methods in [10]

DFPB1 the steepest descent derivative-free projection-basedmethods in [14] with search direction as follows

119889119896 = minus1198650 if 119896 = 0minus119865119896 + 120573119896119889119896minus1 + 120579119896 if 119896 gt 0 (62)

where 120573119896 = 120573119875119877119875119896 = 119865119879119896 119910119896minus1119865119896minus12 120579119896 = (119865119879119896 119910119896minus1)120596119896minus12119865119896minus14 119910119896minus1 = 119865119896 minus 119865119896minus1 and 120596119896minus1 = 120572119896minus1119889119896minus1

8 Mathematical Problems in Engineering

Table 1 Numerical performance with fixed initial points

Problem Dim Method CPU-time Ni NfP1 10000 MPPRP-M 0049106 19 60

MPPRP-W 0083986 28 132MSDFPB 0207396 49 357

PRP 0210513 57 373MPRP 0225433 49 357TPRP 0205193 49 357DFPB1 0208191 52 360DFPB2 0180871 41 333MHS 0201207 49 357

P1 20000 MPPRP-M 0097751 20 63MPPRP-W 0202214 34 187MSDFPB 0538910 63 522

PRP 0601340 72 544MPRP 0578528 63 522TPRP 0567042 63 522DFPB1 0564317 67 534DFPB2 0538403 56 503MHS 0568248 63 522

P1 50000 MPPRP-M 0189582 20 63MPPRP-W 0776513 47 317MSDFPB 2349394 93 913

PRP 2426823 101 926MPRP 2373229 93 913TPRP 232399 93 913DFPB1 2421556 97 925DFPB2 2364173 86 894MHS 2601235 93 913

P1 100000 MPPRP-M 0519087 21 66MPPRP-W 2762991 61 478MSDFPB 9002493 127 1384

PRP 9108308 135 1402MPRP 9448788 127 1384TPRP 8986777 127 1384DFPB1 9100524 130 1391DFPB2 8819755 119 1360MHS 9419715 127 1384

P2 10000 MPPRP-M 0054535 19 60MPPRP-W 0088811 28 132MSDFPB 0217590 49 357

PRP 0218645 57 373MPRP 0226537 49 357TPRP 0226429 49 357DFPB1 0210470 52 360DFPB2 0204578 41 333MHS 0234442 49 357

P2 20000 MPPRP-M 0089917 20 63MPPRP-W 0198822 34 187MSDFPB 0563358 63 522

PRP 0604772 72 544

Mathematical Problems in Engineering 9

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0670294 63 522TPRP 0677819 63 522DFPB1 0679165 67 534DFPB2 0607357 56 503MHS 0589872 63 522

P2 50000 MPPRP-M 0191132 20 63MPPRP-W 0720108 47 317MSDFPB 2248779 93 913

PRP 2464597 101 926MPRP 2353173 93 913TPRP 2342777 93 913DFPB1 2363392 97 925DFPB2 2279782 86 894MHS 2384060 93 913

P2 100000 MPPRP-M 0516081 21 66MPPRP-W 2734834 61 478MSDFPB 9004299 127 1384

PRP 8763470 135 1402MPRP 8647855 127 1384TPRP 8626593 127 1384DFPB1 8626593 130 1391DFPB2 8476836 119 1360MHS 8905454 127 1384

P3 5000 MPPRP-M 0077881 22 69MPPRP-W 0955814 127 1360MSDFPB 2155000 242 3141

PRP 2346950 251 3167MPRP 2200878 242 3141TPRP 2204338 242 3141DFPB1 2246450 245 3148DFPB2 2209059 234 3117MHS 2236389 242 3141

P3 10000 MPPRP-M 0145481 23 72MPPRP-W 2590862 174 2059MSDFPB 6749566 337 4751

PRP 6892930 346 4767MPRP 6843273 337 4751TPRP 6885266 337 4751DFPB1 7222119 341 4763DFPB2 6881919 330 4732MHS 6992060 337 4751

P3 15000 MPPRP-M 0190225 23 72MPPRP-W 4894470 212 2656MSDFPB 12242514 411 6059

PRP 13494894 420 6078MPRP 13547676 411 6059TPRP 12255626 411 6059DFPB1 12473808 415 6071

10 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfDFPB2 12734751 404 6040MHS 12425026 411 6059

P3 20000 MPPRP-M 0239615 23 72MPPRP-W 7505229 242 3126MSDFPB 18305460 472 7144

PRP 19195911 481 7169MPRP 19482789 472 7144TPRP 19011112 472 7144DFPB1 19165800 475 7151DFPB2 19428512 464 7120MHS 18928398 472 7144

P4 10000 MPPRP-M 0061549 16 102MPPRP-W 0113153 19 131MSDFPB 0289696 68 555

PRP 0187397 39 391MPRP 0233670 48 462TPRP 0358863 89 658DFPB1 0657537 185 1142DFPB2 0213256 40 422MHS 0414446 62 803

P4 20000 MPPRP-M 0125760 17 108MPPRP-W 0256772 31 203MSDFPB 0703061 79 730

PRP 0558050 52 565MPRP 0663444 59 635TPRP 0822848 100 833DFPB1 1442003 203 1352DFPB2 0565771 51 596MHS 0674281 60 724

P4 50000 MPPRP-M 0295365 17 108MPPRP-W 0961757 38 273MSDFPB 2972641 102 1109

PRP 2655601 73 943MPRP 2738925 81 1007TPRP 3209823 122 1207DFPB1 4783178 234 1771DFPB2 2644189 75 981MHS 6106110 158 2717

P4 100000 MPPRP-M 0768978 17 108MPPRP-W 3113887 45 357MSDFPB 10575505 129 1568

PRP 9164701 101 1406MPRP 9756561 108 1465TPRP 11151034 146 1651DFPB1 16812871 275 2301DFPB2 9292510 99 1423MHS 21792651 179 3305

P5 100 MPPRP-M 0012253 18 77MPPRP-W 0018540 62 555MSDFPB 0049702 129 1522

PRP 0033290 144 1558

Mathematical Problems in Engineering 11

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0033017 139 1560TPRP 0034758 128 1518DFPB1 0026313 150 1601DFPB2 0029466 131 1538MHS 0040165 133 1539

P5 500 MPPRP-M 0016447 22 93MPPRP-W 0231399 544 9066MSDFPB 0772128 1313 25232

PRP 0727796 1332 25274MPRP 0585951 1372 25394TPRP 0577368 1315 25238DFPB1 0578794 1337 25315DFPB2 0589438 1317 25247MHS 0595316 1315 25241

P5 1000 MPPRP-M 0017438 23 97MPPRP-W 1072332 1505 29740MSDFPB 3218997 3680 81797

PRP 2889154 3696 81825MPRP 2963704 3735 81946TPRP 2948045 3682 81802DFPB1 3040253 3704 81880DFPB2 3045603 3683 81803MHS 3220028 3678 81791

P5 2000 MPPRP-M 0021613 25 106MPPRP-W 5944706 4206 95732MSDFPB 17227153 10341 260347

PRP 16073363 10355 260379MPRP 16960973 10394 260490TPRP 16826907 10343 260352DFPB1 16257104 10364 260424DFPB2 16739385 10345 260356MHS 17503252 10339 260342

P6 100 MPPRP-M 0217224 1182 3717MPPRP-W 0193153 1148 3605MSDFPB 0237793 1186 3706

PRP 0220389 1207 3766MPRP 0224211 1186 3658TPRP 0248104 1195 3722DFPB1 0236804 1201 3750DFPB2 0235470 1170 3620MHS 0236250 1198 3720

P6 500 MPPRP-M 0944828 1348 5050MPPRP-W 0794555 1278 4482MSDFPB 0797832 1185 3968

PRP 0911934 1322 4267MPRP 0845336 1258 3983TPRP 0912343 1326 4335DFPB1 0903852 1321 4300DFPB2 1020228 1299 4143MHS 0918611 1307 4147

P6 1000 MPPRP-M 2337405 1495 6667MPPRP-W 1680275 1381 4951MSDFPB 1536904 1331 4320

12 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfPRP 1851282 1389 4553MPRP 1647356 1277 4040TPRP 1686418 1273 4109DFPB1 1792886 1360 4409DFPB2 1812678 1343 4330MHS 1775351 1341 4283

P6 2000 MPPRP-M 6738143 1737 11016MPPRP-W 3257538 1440 4936MSDFPB 3065903 1381 4524

PRP 3574894 1390 4522MPRP 3381889 1316 4247TPRP 3565741 1378 4510DFPB1 3609619 1377 4511DFPB2 3647662 984 3232MHS 3608315 1362 4397

DFPB2 the steepest descent derivative-free projection-basedmethods in [14] with 120579119896 in (62) being replaced by

120579119896 = 119865119879119896 120596119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 + (119865119879119896 119910119896minus1) 1003817100381710038171003817119910119896minus1100381710038171003817100381721003817100381710038171003817119865119896minus110038171003817100381710038174 (63)

MHS the MHS-PRP conjugate gradient derivative-freeprojection-based methods in [15]

From the results in Tables 1 2 and 3 it follows that ouralgorithm (MPPRP) outperforms the other seven algorithmsno matter how to choose the initial points (see the italicizedresults) Especially it seems to more efficiently solve large-scale test problems Actually Table 3 shows that MPPRP-Mcan solve the first five Problems with dimension of 1000000in less time compared with the other algorithms

In order to further measure the efficiency difference ofall the eight algorithms we calculate the average numberof iteration the average consumed CPU time and theirstandard deviations respectively In Table 4 A-Ni and Std-Ni stand for the average number of iteration and its stan-dard deviation respectively A-CT and Std-CT represent theaverage consumed CPU time and its standard deviationrespectively The average number of function evaluation andits standard deviation are denoted by A-Nf and Std-Nfrespectively Clearly Std-Ni Std-Nf and Std-CT can showrobustness of all the algorithms

The results in Table 4 demonstrate that both ofMPPRP-Mand MPPRP-W outperform the other seven algorithms

In the end of this section we use our algorithm to solvea singular system of equations The next test problem is amodified version of problem 1

Problem 20 The elements of 119865(119909) are given by119865119894 (119909) = 119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (64)

The initial point is fixed as 1199090 = (1 1)119879

Note that problem 7 is singular since zero is its solutionand the Jacobian matrix 1198651015840(0) is singular We implementAlgorithm 1 to solve problem 7 with different dimen-sions to test whether it can find the singular solution ornot

The results in Table 5 indicate that Algorithm 1 is alsoefficient to solve the singular system of equations

5 Preliminary Applicationin Compressed Sensing

In this section we will apply our algorithm to solve anengineering problem originated from compressed sensing ofsparse signals

Let 119860 isin 119877119898times119899 (119898 ≪ 119899) be a linear operator let 119909 isin 119877119899be a sparse or a nearly sparse original signal and let 119887 isin119877119898 be an observed value which satisfies the following linearequations

119887 = 119860119909 (65)

The original signal 119909 is desirable to be reconstructedfrom the linear equations 119860119909 = 119887 Unfortunately itis often that this linear equation is underdetermined orill-conditioned in practice and has infinitely many solu-tions From the fundamental principles of compressed sens-ing it is a reasonable approach to seek the sparsest oneamong all the solutions which contains the least nonzerocomponents

In [2] it was shown that the compressed sensing of sparsesignals can be formulated the following nonlinear system ofequations

min 119911 (119867 + 119863) 119911 + 119888 = 0 (66)

Mathematical Problems in Engineering 13

Table 2 Efficiency with random initial points

Problem Dim Method CPU-time Ni NfP1 100000 MPPRP-M 0489058 20 63

MPPRP-W 2005818 42 268MSDFPB 4565133 79 717

PRP 4757480 87 825MPRP 4561151 79 796TPRP 4631983 79 796DFPB1 4763311 82 806DFPB2 4610199 74 776MHS 4879031 79 796

P2 100000 MPPRP-M 0483632 20 63MPPRP-W 2251160 42 268MSDFPB 4375745 79 717

PRP 4561437 87 825MPRP 4830143 79 796TPRP 4560123 79 796DFPB1 4427558 82 806DFPB2 4550071 74 776MHS 4428999 79 796

P3 100000 MPPRP-M 1187410 20 63MPPRP-W 5215137 46 306MSDFPB 10136405 79 710

PRP 10808303 87 816MPRP 10124779 79 789TPRP 10068535 79 789DFPB1 10372113 82 799DFPB2 9458892 71 757MHS 10137711 79 789

P4 100000 MPPRP-M 9152258 253 1306MPPRP-W 11371204 260 1423MSDFPB 13374780 293 1837

PRP 8352463 170 1352MPRP 9223706 184 1474TPRP 14279675 302 2184DFPB1 15329979 336 2386DFPB2 11244725 231 1770MHS 17738408 216 2902

P5 2000 MPPRP-M 0019793 25 106MPPRP-W 7148874 4207 95776MSDFPB 16293374 10362 260980

PRP 16689685 10375 271376MPRP 16112906 10415 271538TPRP 16006655 10364 271349DFPB1 16272868 10385 271442DFPB2 17018004 10366 271355MHS 19200453 10360 271335

P6 2000 MPPRP-M 9722610 3415 15767MPPRP-W 8031480 3293 12678MSDFPB 8254000 3395 12524

PRP 9187302 3298 15178MPRP 9667492 3366 15503TPRP 9727760 3363 15708DFPB1 9565437 3431 15990DFPB2 9316936 3329 15590MHS 8893926 3393 14502

14 Mathematical Problems in Engineering

Table 3 Efficiency for 1000000-dimension problems

Problem Dim Method CPU-time Ni NfP1 1000000 MPPRP-M 4831098 22 69P2 1000000 MPPRP-M 5150450 22 69P3 1000000 MPPRP-M 13954404 26 81P4 1000000 MPPRP-M 8408822 19 120P5 1000000 MPPRP-M 9018623 36 154

Table 4 Average numerical efficiency and robustness of algorithms

Method (A-CT Std-CT) (A-Ni Std-Ni) (A-Nf Std-CT)MPPRP-M (11657 25775) (3307 76388) (15135 36523)MPPRP-W (25725 29258) (6894 119935) (92059 242337)MSDFPB (54010 56632) (124456 263861) ( 231436 663099)PRP (53178 55599) (124773 264115) (235830 676024)MPRP (53717 56475) (124336 265463) (235743 676492)TPRP (55309 57401) ( 124963 263694) (236334 675796)DFPB1 (58718 60331) (12759 263685) (237512 675685)DFPB2 (53586 56541) (122393 264211) (235434 676120)MHS (62549 67712) (12488 263778) (237192 675398)

where

119911 = [119906V]

119910 = 119860119879119887119888 = 1205911198902119899 + [minus119910119910 ]

119867 = [ 119860119879119860 minus119860119879119860minus119860119879119860 119860119879119860 ] 119863 = [0 1205751198641198990 0 ]

(67)

where 119906 isin 119877119899 V isin 119877119899 and 119906119894 = (119909119894)+ V119894 = (minus119909119894)+ forall 119894 = 1 2 119899 with (sdot)+ = max0 sdot and 1198902119899 is an 2119899-dimensional vector with all elements being one Clearly (66)is nonsmooth

Implement Algorithm 1 to solve the compressed sensingproblem of sparse signals with different compressed ratios Inthis experiment we consider a typical compressive sensingscenario where the goal is to reconstruct an 119899-length sparsesignal from 119898 observations We test the three algorithms(two popular algorithms CGD in [34] and SGCS in [35] andMPPRP-M) under three CS ratios (CS-R)119898119899 = 0125 02505 corresponding to different numbers of measurementswith 119899 = 2048 respectively The original sparse signal con-tains 32(64) randomly nonzero elements The measurement119887 is disturbed by noise ie 119887 = 119860119909 + 120596 where 120596 is theGaussian noise distributed as 119873(0 1205902119868) with 1205902 = 10minus4 119860is the Gaussian matrix generated by commend randn(119898 119899)

inMatlab The quality of restoration is measured by the meanof squared error (MSE) to the original signal 119909 that is

MSE = 1119899 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (68)

where 119909lowast is the restored signal The iterative process starts atthe measurement image ie 1199090 = 119860119879119887 and terminates whenthe relative change between successive iterates falls below10minus5 It says that 1003817100381710038171003817119891119896 minus 119891119896minus110038171003817100381710038171003817100381710038171003817119891119896minus11003817100381710038171003817 lt 10minus5 (69)

where 119891119896 = 1205911199091198961 + (12)119860119909119896 minus 11988722 and 120591 = 0005119860119879119887infinis chosen as suggested in [36] The other parameters in thisexperiment are same as those in the numerical experimentsconducted in Section 4 Numerical efficiency is shown inTable 6

Clearly from the results in Table 6 it follows that forany CS ratio MPPRP-M can recover the sparse signals moreefficiently without reduction of recovery quality (see theitalicized results in Table 6) If the sparsity level 32 is replacedby 64 in the 2048-length original signal the correspondingresults are also presented in Table 6 denoted by (sdot)6 Conclusions

In this paper we have presented a modified spectral PRPconjugate gradient derivative-free projection-based methodfor solving the large-scale nonlinear monotonic equationswhere the search direction is proved to be sufficiently descentfor any line search rule and the step length is chosen by aline search which can overcome the difficulty of choosing anappropriate weight

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

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Mathematical Problems in Engineering 5

and

lim119896997888rarrinfin

1003817100381710038171003817119909119896+1 minus 1199091198961003817100381710038171003817 = 0 (24)

(4) e sequence 119865(119909119896) is bounded Hence there existsa constant 119872119891 such that 119865(119909119896) le 119872119891Proof (1) Let 119909lowast be any point such that 119865(119909lowast) = 0 Then bymonotonicity of 119865 we have⟨119865 (119911119896) 119909lowast minus 119911119896⟩ le 0 (25)

From (7) it is also easy to verify that 119909119896+1 is the projection of119909119896 onto the halfspace119909 isin 119877119899 | ⟨119865 (119911119896) 119909 minus 119911119896⟩ le 0 (26)

Thus it follows from (25) that 119909lowast belongs to this halfspaceFrom the basic properties of projection operator [33] weknow that ⟨119909119896 minus 119909119896+1 119909119896+1 minus 119909lowast⟩ ge 0 (27)

Consequently1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 = 1003817100381710038171003817119909119896 minus 119909119896+110038171003817100381710038172 + 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172+ 2 ⟨119909119896 minus 119909119896+1 119909119896+1 minus 119909lowast⟩ge 1003817100381710038171003817119909119896 minus 119909119896+110038171003817100381710038172 + 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172 (28)

The desired result (22) is directly obtained from (28)(2) From (28) it is clear that the sequence 119909119896 minus 119909lowast is

nonnegative and decreasing Thus 119909119896minus119909lowast is a convergentsequence It is concluded that 119909119896 is bounded

(3) From (28) we know1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 le 1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172 (29)

Thus

119899sum119896=1

1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 le 119899sum119896=1

(1003817100381710038171003817119909119896 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119896+1 minus 119909lowast10038171003817100381710038172)= 10038171003817100381710038171199091 minus 119909lowast10038171003817100381710038172 minus 1003817100381710038171003817119909119899+1 minus 119909lowast10038171003817100381710038172 (30)

Since the sequence 119909119896 is bounded the series sum119899119896=1 119909119896+1 minus1199091198962 is convergent Consequentlylim119896997888rarrinfin

1003817100381710038171003817119909119896+1 minus 11990911989610038171003817100381710038172 = 0 (31)

The third result has been proved(4) For any 119909119896 by Lipschitz continuity we have1003817100381710038171003817119865 (119909119896)1003817100381710038171003817 = 1003817100381710038171003817119865 (119909119896) minus 119865 (119909lowast)1003817100381710038171003817 le 119871 1003817100381710038171003817119909119896 minus 119909lowast1003817100381710038171003817 (32)

Since 119909119896 minus119909lowast is convergent we conclude that 119865(119909119896) isbounded Consequently for all 119896 isin 119873 there exists a constant119872119891 such that 119865(119909119896) le 119872119891

Lemma 8 indicates that for the sequence 119909119896 generatedby Algorithm 1 the sequence 119909119896 minus 119909lowast is decreasing andconvergent and the sequence 119909119896 is bounded where 119909lowast isany solution of problem (1)

Lemma 9 Suppose that Assumptions 5 6 and 7 hold Let119889119896 be a sequence generated by Algorithm 1 If there exists aconstant 1205760 gt 0 such that for any positive integer 11989610038171003817100381710038171198651198961003817100381710038171003817 ge 1205760 (33)

en the sequence of directions 119889119896 is bounded ie thereexists a constant 119872 gt 0 such that for any positive integer 11989610038171003817100381710038171198891198961003817100381710038171003817 le 119872 (34)

Proof From (9) (11) (12) and the results of Lemma 8 itfollows that10038171003817100381710038171198891198961003817100381710038171003817 = 1003817100381710038171003817minus120579119896119865119896 + 120573119896119889119896minus11003817100381710038171003817 = 1003817100381710038171003817100381710038171003817100381710038171003817minus119889119879119896minus1 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119865119896

+ 119889119879119896minus1119865119896119865119879119896 119865119896minus1100381710038171003817100381711986511989610038171003817100381710038172 1003817100381710038171003817119865119896minus110038171003817100381710038172 119865119896 + 119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119889119896minus11003817100381710038171003817100381710038171003817100381710038171003817 = 1003817100381710038171003817100381710038171003817100381710038171003817minus119865119896minus 119889119879119896minus1119865119896119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 100381710038171003817100381711986511989610038171003817100381710038172 119865119896 + 119865119879119896 (119865119896 minus 119865119896minus1)1003817100381710038171003817119865119896minus110038171003817100381710038172 119889119896minus11003817100381710038171003817100381710038171003817100381710038171003817le 2 100381710038171003817100381711986511989610038171003817100381710038171003817100381710038171003817119865119896minus110038171003817100381710038172 1003817100381710038171003817119889119896minus11003817100381710038171003817 1003817100381710038171003817119910119896minus11003817100381710038171003817 + 10038171003817100381710038171198651198961003817100381710038171003817 le 211987111987211989112057620 1003817100381710038171003817119909119896minus 119909119896minus11003817100381710038171003817 1003817100381710038171003817119889119896minus11003817100381710038171003817 + 119872119891

(35)

From (24) we know that there exist a positive integer 1198961 anda positive number 1205761 (0 lt 1205761 lt 1) such that for all 119896 ge 1198961211987111987211989112057620 1003817100381710038171003817119909119896 minus 119909119896minus11003817100381710038171003817 lt 1205761 (36)

Hence10038171003817100381710038171198891198961003817100381710038171003817 le 1205761 1003817100381710038171003817119889119896minus11003817100381710038171003817 + 119872119891 le 12057621 1003817100381710038171003817119889119896minus21003817100381710038171003817 + 1205761119872119891 + 119872119891le 12057611989611 10038171003817100381710038171003817119889119896110038171003817100381710038171003817 + 1205761198961minus11 119872119891 + sdot sdot sdot + 1205761119872119891 + 119872119891 le 10038171003817100381710038171003817119889119896110038171003817100381710038171003817+ 119872119891 1 minus 120576119896111 minus 1205761 le 10038171003817100381710038171003817119889119896110038171003817100381710038171003817 + 1198721198911 minus 1205761

(37)

Let 1198721 = 1198891198961 + 119872119891(1 minus 1205761) Take119872 = max 100381710038171003817100381711988901003817100381710038171003817 100381710038171003817100381711988911003817100381710038171003817 10038171003817100381710038171003817119889119896110038171003817100381710038171003817 1198721 (38)

Then 119889119896 le 119872 holds for any positive integer 119896Lemma 10 Suppose that Assumptions 5 6 and 7 hold Let119909119896 and 119889119896 be two sequences generated by Algorithm 1enthe line search rule (17) of Step 2 in Algorithm 1 is well-defined

6 Mathematical Problems in Engineering

Proof Our aim is to show that the line search rule (17) termi-nates finitely with a positive step length 120572119896 By contradictionsuppose that for some iterate indexes such as 119896lowast condition(17) does not hold As a result for all 119898 isin 119873minus119865 (119909119896lowast + 119904120588119898119889119896lowast)119879 119889119896lowast lt 120590119904120588119898120574119896lowast 1003817100381710038171003817119889119896lowast10038171003817100381710038172 (39)

From (18) and the termination condition of Algorithm 1 itfollows that for all 119898 isin 1198730 le 120576 le 120574119896lowast le 1 (40)

By taking the limit as 119898 997888rarr infin in both sides of (39) we have

minus119865 (119909119896lowast)119879 119889119896lowast le 0 (41)

Equations (41) contradicts the fact that minus119865119879119896 119889119896 = minus1198651198962 gt 0for all 119896 That is to say the line search rule terminates finitelywith a positive step length 120572119896 ie the line search step ofAlgorithm 1 is well-defined

With the above preparation we are now state the conver-gence result of Algorithm 1

Theorem 11 Suppose that Assumptions 5 6 and 7 hold Let119909119896 be a sequence generated by Algorithm 1 en

lim inf119896997888rarrinfin

10038171003817100381710038171198651198961003817100381710038171003817 = 0 (42)

Proof For the sake of contradiction we suppose that theconclusion is not trueThen by the definition of inferior limitthere exists a constant 1205760 gt 0 such that for any 119896 isin 11987310038171003817100381710038171198651198961003817100381710038171003817 ge 1205760 (43)

Consequently from100381710038171003817100381711986511989610038171003817100381710038172 = minus119865119879119896 119889119896 le 10038171003817100381710038171198651198961003817100381710038171003817 10038171003817100381710038171198891198961003817100381710038171003817 (44)

it follows that 119889119896 ge 1205760 gt 0 for any 119896 isin 119873From (7) (17) and (40) we get

1003817100381710038171003817119909119896+1 minus 1199091198961003817100381710038171003817 = 100381610038161003816100381610038161003816119865 (119911119896)119879 (119909119896 minus 119911119896)1003816100381610038161003816100381610038161003817100381710038171003817119865 (119911119896)1003817100381710038171003817 = 100381610038161003816100381610038161003816120572119896119865 (119911119896)119879 1198891198961003816100381610038161003816100381610038161003817100381710038171003817119865 (119911119896)1003817100381710038171003817ge 1205901205722119896120574119896 1003817100381710038171003817119889119896100381710038171003817100381721003817100381710038171003817119865 (119911119896)1003817100381710038171003817 ge 1205901205761205722119896 100381710038171003817100381711988911989610038171003817100381710038172119872119891 ge 0 (45)

Combining (24) and (45) we obtain

lim119896997888rarrinfin

120572119896 10038171003817100381710038171198891198961003817100381710038171003817 = 0 (46)

Since 119889119896 ge 1205760 gt 0 for any 119896 isin 119873 we have

lim119896997888rarrinfin

120572119896 = 0 (47)

Clearly 120572lowast119896 = 120588minus1120572119896 does not satisfy in (17) It says that

minus119865 (119909119896 + 120572lowast119896119889119896)119879 119889119896 lt 120590120572lowast119896120574119896 100381710038171003817100381711988911989610038171003817100381710038172 (48)

By Lemmas 8 and 9 we know that the two sequences 119909119896and 119889119896 are bounded Without loss of generality we choosea subset 119870 isin 119873 such that

lim119896997888rarrinfin119896isin119870

119909119896 = 119909lowastlim119896997888rarrinfin119896isin119870

119889119896 = 119889lowast (49)

Taking the limit in the two sides of (48) as 119896 997888rarr infin (119896 isin 119870)it holds that

119865 (119909lowast)119879 119889lowast ge 0 (50)

On the other hand from (43) we know that

119865 (119909119896)119879 119889119896 = minus 100381710038171003817100381711986511989610038171003817100381710038172 le minus12057620 lt 0 (51)

By taking the limit 119896 997888rarr infin in the two sides of (51) for 119896 isin 119870we get

119865 (119909lowast)119879 119889lowast le minus1205760 lt 0 (52)

It contradicts (50) Thus the proof of Theorem 11 has beencompleted

Remark 12 (only with 120574119896 being generated by (19)) As 120574119896 isdetermined by (18) the proofs are similar

Since the proof of Theorem 11 does not involve differen-tiability of 119865 let alone nonsingularity of its Jacobian matrixwe know that the following result holds

Corollary 13 For any nonsmooth or singular function 119865 let119909119896 be a sequence generated as Algorithm 1 is used to solveproblem (1) Under Assumptions 5 6 and 7 it holds that

lim inf119896997888rarrinfin

10038171003817100381710038171198651198961003817100381710038171003817 = 0 (53)

It should be pointed out that the global convergenceof Algorithm 1 depends on assumption on monotonicity of119865 For nonmonotonic function 119865 Algorithm 1 may be notapplicable

4 Numerical Experiments

In this section by numerical experiments we are going tostudy the effectiveness and robustness of Algorithm 1 forsolving large-scale system of equations

We first collect some benchmark test problems availablein the literature

Problem 14 (see [5]) The elements of 119865(119909) are given by119865119894 (119909) = 2119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (54)

Problem 15 (see [5]) The elements of 119865(119909) are given by119865119894 (119909) = 2119909119894 minus 1003816100381610038161003816sin (119909119894)1003816100381610038161003816 119894 = 1 2 119899 minus 1 (55)

Mathematical Problems in Engineering 7

Problem 16 (see [15]) The elements of 119865(119909) are given by

1198651 (119909) = 1199091 minus 119890cos((1199091+1199092)(119899+1))119865119894 (119909) = 119909119894 minus 119890cos((119909119894minus1+119909119894+119909119894+1)(119899+1)) 119894 = 2 119899 minus 1119865119899 (119909) = 119909119899 minus 119890cos((119909119899minus1+119909119899)(119899+1)) (56)

Problem 17 (see [28]) The elements of 119865(119909) are given by

119865 (119909) = 119860119909 + 119887119887 = (minus1 minus1 minus1 minus1)119879

119860 = ((((((

52 11 52 1d d d

d d 11 52

))))))

(57)

Clearly problem 4 is a linear system of equations which isused to test Algorithm 1 in this special case

Problem 18 (see [15]) The elements of 119865(119909) are given by

119865119894 (119909) = 119909119894 minus 11198991199092119894 + 1119899 119899sum119894=1

119909119894 + 119894 119894 = 2 119899 minus 1 (58)

Problem 19 (see [15]) The elements of 119865(119909) are given by

1198651 (119909) = 1311990931 + 1211990922119865119894 (119909) = minus121199092119894 + 11989431199093119894 + 121199092119894+1 119894 = 2 119899 minus 1119865119899 (119909) = minus121199092119899 + 11989931199093119899

(59)

Clearly apart from problem 4 all the others are nonlinearsystem of equations Problem 15 is nonsmooth at point(0 0 0) The size of all the test problems is variable Ifthis size is larger than 1000 the problem can be regarded tobe large-scale We solve all the test problems with differentsizes by Algorithm 1 especially in comparison with theexisting seven similar algorithms such as those developedvery recently in [10 13ndash15 28]

All the algorithms are coded inMATLABR2010b and runon a personal computer with a 22GHZ CPU processor 8GB

memory and Windows 10 operation system The relevantparameters of algorithms are specified by119896119898119886119909 = 100000120590 = 03119904 = 05120588 = 07

(60)

120582119896 = 12058202 if 119896 = 1120582119896minus1 + 120582119896minus22 if 119896 gt 1 (61)

where 1205820 = 1 The termination condition is 119865119896 le 10minus4Numerical performance of all the algorithms is reported

in Tables 1 2 and 3 Table 1 shows the numerical performanceof all the eight algorithms with the fixed initial points 1199090 =(minus1 minus1)119879 for problems 1 5 and 6 1199090 = (1 1)119879 forproblems 2 and 4 1199090 = (10 10)119879 for problem 3 Table 2demonstrates the numerical performance of all the eightalgorithms with initial points randomly generated byMatlabrsquosCode ldquorand(n1)rdquo Table 3 shows the numerical performanceof our algorithm with dimension of 1000000

For simplification of statement we use the followingnotations

Dim the dimension of test problems

Ni the number of iterations

Nf the number of function evaluations

MPPRP-M the developed algorithm with 120574119896 determined by(19) in this paper

MPPRP-W the developed algorithmwith 120574119896 generated by (18)in this paper

MSDFPB the modified spectral derivative-free projection-based algorithm in [13]

PRP the PRP conjugate gradient derivative-free projection-based methods in [28]

MPRP the modified PRP conjugate gradient derivative-freeprojection-based methods in [10]

TPRP the two-term PRP conjugate gradient derivative-freeprojection-based methods in [10]

DFPB1 the steepest descent derivative-free projection-basedmethods in [14] with search direction as follows

119889119896 = minus1198650 if 119896 = 0minus119865119896 + 120573119896119889119896minus1 + 120579119896 if 119896 gt 0 (62)

where 120573119896 = 120573119875119877119875119896 = 119865119879119896 119910119896minus1119865119896minus12 120579119896 = (119865119879119896 119910119896minus1)120596119896minus12119865119896minus14 119910119896minus1 = 119865119896 minus 119865119896minus1 and 120596119896minus1 = 120572119896minus1119889119896minus1

8 Mathematical Problems in Engineering

Table 1 Numerical performance with fixed initial points

Problem Dim Method CPU-time Ni NfP1 10000 MPPRP-M 0049106 19 60

MPPRP-W 0083986 28 132MSDFPB 0207396 49 357

PRP 0210513 57 373MPRP 0225433 49 357TPRP 0205193 49 357DFPB1 0208191 52 360DFPB2 0180871 41 333MHS 0201207 49 357

P1 20000 MPPRP-M 0097751 20 63MPPRP-W 0202214 34 187MSDFPB 0538910 63 522

PRP 0601340 72 544MPRP 0578528 63 522TPRP 0567042 63 522DFPB1 0564317 67 534DFPB2 0538403 56 503MHS 0568248 63 522

P1 50000 MPPRP-M 0189582 20 63MPPRP-W 0776513 47 317MSDFPB 2349394 93 913

PRP 2426823 101 926MPRP 2373229 93 913TPRP 232399 93 913DFPB1 2421556 97 925DFPB2 2364173 86 894MHS 2601235 93 913

P1 100000 MPPRP-M 0519087 21 66MPPRP-W 2762991 61 478MSDFPB 9002493 127 1384

PRP 9108308 135 1402MPRP 9448788 127 1384TPRP 8986777 127 1384DFPB1 9100524 130 1391DFPB2 8819755 119 1360MHS 9419715 127 1384

P2 10000 MPPRP-M 0054535 19 60MPPRP-W 0088811 28 132MSDFPB 0217590 49 357

PRP 0218645 57 373MPRP 0226537 49 357TPRP 0226429 49 357DFPB1 0210470 52 360DFPB2 0204578 41 333MHS 0234442 49 357

P2 20000 MPPRP-M 0089917 20 63MPPRP-W 0198822 34 187MSDFPB 0563358 63 522

PRP 0604772 72 544

Mathematical Problems in Engineering 9

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0670294 63 522TPRP 0677819 63 522DFPB1 0679165 67 534DFPB2 0607357 56 503MHS 0589872 63 522

P2 50000 MPPRP-M 0191132 20 63MPPRP-W 0720108 47 317MSDFPB 2248779 93 913

PRP 2464597 101 926MPRP 2353173 93 913TPRP 2342777 93 913DFPB1 2363392 97 925DFPB2 2279782 86 894MHS 2384060 93 913

P2 100000 MPPRP-M 0516081 21 66MPPRP-W 2734834 61 478MSDFPB 9004299 127 1384

PRP 8763470 135 1402MPRP 8647855 127 1384TPRP 8626593 127 1384DFPB1 8626593 130 1391DFPB2 8476836 119 1360MHS 8905454 127 1384

P3 5000 MPPRP-M 0077881 22 69MPPRP-W 0955814 127 1360MSDFPB 2155000 242 3141

PRP 2346950 251 3167MPRP 2200878 242 3141TPRP 2204338 242 3141DFPB1 2246450 245 3148DFPB2 2209059 234 3117MHS 2236389 242 3141

P3 10000 MPPRP-M 0145481 23 72MPPRP-W 2590862 174 2059MSDFPB 6749566 337 4751

PRP 6892930 346 4767MPRP 6843273 337 4751TPRP 6885266 337 4751DFPB1 7222119 341 4763DFPB2 6881919 330 4732MHS 6992060 337 4751

P3 15000 MPPRP-M 0190225 23 72MPPRP-W 4894470 212 2656MSDFPB 12242514 411 6059

PRP 13494894 420 6078MPRP 13547676 411 6059TPRP 12255626 411 6059DFPB1 12473808 415 6071

10 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfDFPB2 12734751 404 6040MHS 12425026 411 6059

P3 20000 MPPRP-M 0239615 23 72MPPRP-W 7505229 242 3126MSDFPB 18305460 472 7144

PRP 19195911 481 7169MPRP 19482789 472 7144TPRP 19011112 472 7144DFPB1 19165800 475 7151DFPB2 19428512 464 7120MHS 18928398 472 7144

P4 10000 MPPRP-M 0061549 16 102MPPRP-W 0113153 19 131MSDFPB 0289696 68 555

PRP 0187397 39 391MPRP 0233670 48 462TPRP 0358863 89 658DFPB1 0657537 185 1142DFPB2 0213256 40 422MHS 0414446 62 803

P4 20000 MPPRP-M 0125760 17 108MPPRP-W 0256772 31 203MSDFPB 0703061 79 730

PRP 0558050 52 565MPRP 0663444 59 635TPRP 0822848 100 833DFPB1 1442003 203 1352DFPB2 0565771 51 596MHS 0674281 60 724

P4 50000 MPPRP-M 0295365 17 108MPPRP-W 0961757 38 273MSDFPB 2972641 102 1109

PRP 2655601 73 943MPRP 2738925 81 1007TPRP 3209823 122 1207DFPB1 4783178 234 1771DFPB2 2644189 75 981MHS 6106110 158 2717

P4 100000 MPPRP-M 0768978 17 108MPPRP-W 3113887 45 357MSDFPB 10575505 129 1568

PRP 9164701 101 1406MPRP 9756561 108 1465TPRP 11151034 146 1651DFPB1 16812871 275 2301DFPB2 9292510 99 1423MHS 21792651 179 3305

P5 100 MPPRP-M 0012253 18 77MPPRP-W 0018540 62 555MSDFPB 0049702 129 1522

PRP 0033290 144 1558

Mathematical Problems in Engineering 11

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0033017 139 1560TPRP 0034758 128 1518DFPB1 0026313 150 1601DFPB2 0029466 131 1538MHS 0040165 133 1539

P5 500 MPPRP-M 0016447 22 93MPPRP-W 0231399 544 9066MSDFPB 0772128 1313 25232

PRP 0727796 1332 25274MPRP 0585951 1372 25394TPRP 0577368 1315 25238DFPB1 0578794 1337 25315DFPB2 0589438 1317 25247MHS 0595316 1315 25241

P5 1000 MPPRP-M 0017438 23 97MPPRP-W 1072332 1505 29740MSDFPB 3218997 3680 81797

PRP 2889154 3696 81825MPRP 2963704 3735 81946TPRP 2948045 3682 81802DFPB1 3040253 3704 81880DFPB2 3045603 3683 81803MHS 3220028 3678 81791

P5 2000 MPPRP-M 0021613 25 106MPPRP-W 5944706 4206 95732MSDFPB 17227153 10341 260347

PRP 16073363 10355 260379MPRP 16960973 10394 260490TPRP 16826907 10343 260352DFPB1 16257104 10364 260424DFPB2 16739385 10345 260356MHS 17503252 10339 260342

P6 100 MPPRP-M 0217224 1182 3717MPPRP-W 0193153 1148 3605MSDFPB 0237793 1186 3706

PRP 0220389 1207 3766MPRP 0224211 1186 3658TPRP 0248104 1195 3722DFPB1 0236804 1201 3750DFPB2 0235470 1170 3620MHS 0236250 1198 3720

P6 500 MPPRP-M 0944828 1348 5050MPPRP-W 0794555 1278 4482MSDFPB 0797832 1185 3968

PRP 0911934 1322 4267MPRP 0845336 1258 3983TPRP 0912343 1326 4335DFPB1 0903852 1321 4300DFPB2 1020228 1299 4143MHS 0918611 1307 4147

P6 1000 MPPRP-M 2337405 1495 6667MPPRP-W 1680275 1381 4951MSDFPB 1536904 1331 4320

12 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfPRP 1851282 1389 4553MPRP 1647356 1277 4040TPRP 1686418 1273 4109DFPB1 1792886 1360 4409DFPB2 1812678 1343 4330MHS 1775351 1341 4283

P6 2000 MPPRP-M 6738143 1737 11016MPPRP-W 3257538 1440 4936MSDFPB 3065903 1381 4524

PRP 3574894 1390 4522MPRP 3381889 1316 4247TPRP 3565741 1378 4510DFPB1 3609619 1377 4511DFPB2 3647662 984 3232MHS 3608315 1362 4397

DFPB2 the steepest descent derivative-free projection-basedmethods in [14] with 120579119896 in (62) being replaced by

120579119896 = 119865119879119896 120596119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 + (119865119879119896 119910119896minus1) 1003817100381710038171003817119910119896minus1100381710038171003817100381721003817100381710038171003817119865119896minus110038171003817100381710038174 (63)

MHS the MHS-PRP conjugate gradient derivative-freeprojection-based methods in [15]

From the results in Tables 1 2 and 3 it follows that ouralgorithm (MPPRP) outperforms the other seven algorithmsno matter how to choose the initial points (see the italicizedresults) Especially it seems to more efficiently solve large-scale test problems Actually Table 3 shows that MPPRP-Mcan solve the first five Problems with dimension of 1000000in less time compared with the other algorithms

In order to further measure the efficiency difference ofall the eight algorithms we calculate the average numberof iteration the average consumed CPU time and theirstandard deviations respectively In Table 4 A-Ni and Std-Ni stand for the average number of iteration and its stan-dard deviation respectively A-CT and Std-CT represent theaverage consumed CPU time and its standard deviationrespectively The average number of function evaluation andits standard deviation are denoted by A-Nf and Std-Nfrespectively Clearly Std-Ni Std-Nf and Std-CT can showrobustness of all the algorithms

The results in Table 4 demonstrate that both ofMPPRP-Mand MPPRP-W outperform the other seven algorithms

In the end of this section we use our algorithm to solvea singular system of equations The next test problem is amodified version of problem 1

Problem 20 The elements of 119865(119909) are given by119865119894 (119909) = 119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (64)

The initial point is fixed as 1199090 = (1 1)119879

Note that problem 7 is singular since zero is its solutionand the Jacobian matrix 1198651015840(0) is singular We implementAlgorithm 1 to solve problem 7 with different dimen-sions to test whether it can find the singular solution ornot

The results in Table 5 indicate that Algorithm 1 is alsoefficient to solve the singular system of equations

5 Preliminary Applicationin Compressed Sensing

In this section we will apply our algorithm to solve anengineering problem originated from compressed sensing ofsparse signals

Let 119860 isin 119877119898times119899 (119898 ≪ 119899) be a linear operator let 119909 isin 119877119899be a sparse or a nearly sparse original signal and let 119887 isin119877119898 be an observed value which satisfies the following linearequations

119887 = 119860119909 (65)

The original signal 119909 is desirable to be reconstructedfrom the linear equations 119860119909 = 119887 Unfortunately itis often that this linear equation is underdetermined orill-conditioned in practice and has infinitely many solu-tions From the fundamental principles of compressed sens-ing it is a reasonable approach to seek the sparsest oneamong all the solutions which contains the least nonzerocomponents

In [2] it was shown that the compressed sensing of sparsesignals can be formulated the following nonlinear system ofequations

min 119911 (119867 + 119863) 119911 + 119888 = 0 (66)

Mathematical Problems in Engineering 13

Table 2 Efficiency with random initial points

Problem Dim Method CPU-time Ni NfP1 100000 MPPRP-M 0489058 20 63

MPPRP-W 2005818 42 268MSDFPB 4565133 79 717

PRP 4757480 87 825MPRP 4561151 79 796TPRP 4631983 79 796DFPB1 4763311 82 806DFPB2 4610199 74 776MHS 4879031 79 796

P2 100000 MPPRP-M 0483632 20 63MPPRP-W 2251160 42 268MSDFPB 4375745 79 717

PRP 4561437 87 825MPRP 4830143 79 796TPRP 4560123 79 796DFPB1 4427558 82 806DFPB2 4550071 74 776MHS 4428999 79 796

P3 100000 MPPRP-M 1187410 20 63MPPRP-W 5215137 46 306MSDFPB 10136405 79 710

PRP 10808303 87 816MPRP 10124779 79 789TPRP 10068535 79 789DFPB1 10372113 82 799DFPB2 9458892 71 757MHS 10137711 79 789

P4 100000 MPPRP-M 9152258 253 1306MPPRP-W 11371204 260 1423MSDFPB 13374780 293 1837

PRP 8352463 170 1352MPRP 9223706 184 1474TPRP 14279675 302 2184DFPB1 15329979 336 2386DFPB2 11244725 231 1770MHS 17738408 216 2902

P5 2000 MPPRP-M 0019793 25 106MPPRP-W 7148874 4207 95776MSDFPB 16293374 10362 260980

PRP 16689685 10375 271376MPRP 16112906 10415 271538TPRP 16006655 10364 271349DFPB1 16272868 10385 271442DFPB2 17018004 10366 271355MHS 19200453 10360 271335

P6 2000 MPPRP-M 9722610 3415 15767MPPRP-W 8031480 3293 12678MSDFPB 8254000 3395 12524

PRP 9187302 3298 15178MPRP 9667492 3366 15503TPRP 9727760 3363 15708DFPB1 9565437 3431 15990DFPB2 9316936 3329 15590MHS 8893926 3393 14502

14 Mathematical Problems in Engineering

Table 3 Efficiency for 1000000-dimension problems

Problem Dim Method CPU-time Ni NfP1 1000000 MPPRP-M 4831098 22 69P2 1000000 MPPRP-M 5150450 22 69P3 1000000 MPPRP-M 13954404 26 81P4 1000000 MPPRP-M 8408822 19 120P5 1000000 MPPRP-M 9018623 36 154

Table 4 Average numerical efficiency and robustness of algorithms

Method (A-CT Std-CT) (A-Ni Std-Ni) (A-Nf Std-CT)MPPRP-M (11657 25775) (3307 76388) (15135 36523)MPPRP-W (25725 29258) (6894 119935) (92059 242337)MSDFPB (54010 56632) (124456 263861) ( 231436 663099)PRP (53178 55599) (124773 264115) (235830 676024)MPRP (53717 56475) (124336 265463) (235743 676492)TPRP (55309 57401) ( 124963 263694) (236334 675796)DFPB1 (58718 60331) (12759 263685) (237512 675685)DFPB2 (53586 56541) (122393 264211) (235434 676120)MHS (62549 67712) (12488 263778) (237192 675398)

where

119911 = [119906V]

119910 = 119860119879119887119888 = 1205911198902119899 + [minus119910119910 ]

119867 = [ 119860119879119860 minus119860119879119860minus119860119879119860 119860119879119860 ] 119863 = [0 1205751198641198990 0 ]

(67)

where 119906 isin 119877119899 V isin 119877119899 and 119906119894 = (119909119894)+ V119894 = (minus119909119894)+ forall 119894 = 1 2 119899 with (sdot)+ = max0 sdot and 1198902119899 is an 2119899-dimensional vector with all elements being one Clearly (66)is nonsmooth

Implement Algorithm 1 to solve the compressed sensingproblem of sparse signals with different compressed ratios Inthis experiment we consider a typical compressive sensingscenario where the goal is to reconstruct an 119899-length sparsesignal from 119898 observations We test the three algorithms(two popular algorithms CGD in [34] and SGCS in [35] andMPPRP-M) under three CS ratios (CS-R)119898119899 = 0125 02505 corresponding to different numbers of measurementswith 119899 = 2048 respectively The original sparse signal con-tains 32(64) randomly nonzero elements The measurement119887 is disturbed by noise ie 119887 = 119860119909 + 120596 where 120596 is theGaussian noise distributed as 119873(0 1205902119868) with 1205902 = 10minus4 119860is the Gaussian matrix generated by commend randn(119898 119899)

inMatlab The quality of restoration is measured by the meanof squared error (MSE) to the original signal 119909 that is

MSE = 1119899 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (68)

where 119909lowast is the restored signal The iterative process starts atthe measurement image ie 1199090 = 119860119879119887 and terminates whenthe relative change between successive iterates falls below10minus5 It says that 1003817100381710038171003817119891119896 minus 119891119896minus110038171003817100381710038171003817100381710038171003817119891119896minus11003817100381710038171003817 lt 10minus5 (69)

where 119891119896 = 1205911199091198961 + (12)119860119909119896 minus 11988722 and 120591 = 0005119860119879119887infinis chosen as suggested in [36] The other parameters in thisexperiment are same as those in the numerical experimentsconducted in Section 4 Numerical efficiency is shown inTable 6

Clearly from the results in Table 6 it follows that forany CS ratio MPPRP-M can recover the sparse signals moreefficiently without reduction of recovery quality (see theitalicized results in Table 6) If the sparsity level 32 is replacedby 64 in the 2048-length original signal the correspondingresults are also presented in Table 6 denoted by (sdot)6 Conclusions

In this paper we have presented a modified spectral PRPconjugate gradient derivative-free projection-based methodfor solving the large-scale nonlinear monotonic equationswhere the search direction is proved to be sufficiently descentfor any line search rule and the step length is chosen by aline search which can overcome the difficulty of choosing anappropriate weight

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

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6 Mathematical Problems in Engineering

Proof Our aim is to show that the line search rule (17) termi-nates finitely with a positive step length 120572119896 By contradictionsuppose that for some iterate indexes such as 119896lowast condition(17) does not hold As a result for all 119898 isin 119873minus119865 (119909119896lowast + 119904120588119898119889119896lowast)119879 119889119896lowast lt 120590119904120588119898120574119896lowast 1003817100381710038171003817119889119896lowast10038171003817100381710038172 (39)

From (18) and the termination condition of Algorithm 1 itfollows that for all 119898 isin 1198730 le 120576 le 120574119896lowast le 1 (40)

By taking the limit as 119898 997888rarr infin in both sides of (39) we have

minus119865 (119909119896lowast)119879 119889119896lowast le 0 (41)

Equations (41) contradicts the fact that minus119865119879119896 119889119896 = minus1198651198962 gt 0for all 119896 That is to say the line search rule terminates finitelywith a positive step length 120572119896 ie the line search step ofAlgorithm 1 is well-defined

With the above preparation we are now state the conver-gence result of Algorithm 1

Theorem 11 Suppose that Assumptions 5 6 and 7 hold Let119909119896 be a sequence generated by Algorithm 1 en

lim inf119896997888rarrinfin

10038171003817100381710038171198651198961003817100381710038171003817 = 0 (42)

Proof For the sake of contradiction we suppose that theconclusion is not trueThen by the definition of inferior limitthere exists a constant 1205760 gt 0 such that for any 119896 isin 11987310038171003817100381710038171198651198961003817100381710038171003817 ge 1205760 (43)

Consequently from100381710038171003817100381711986511989610038171003817100381710038172 = minus119865119879119896 119889119896 le 10038171003817100381710038171198651198961003817100381710038171003817 10038171003817100381710038171198891198961003817100381710038171003817 (44)

it follows that 119889119896 ge 1205760 gt 0 for any 119896 isin 119873From (7) (17) and (40) we get

1003817100381710038171003817119909119896+1 minus 1199091198961003817100381710038171003817 = 100381610038161003816100381610038161003816119865 (119911119896)119879 (119909119896 minus 119911119896)1003816100381610038161003816100381610038161003817100381710038171003817119865 (119911119896)1003817100381710038171003817 = 100381610038161003816100381610038161003816120572119896119865 (119911119896)119879 1198891198961003816100381610038161003816100381610038161003817100381710038171003817119865 (119911119896)1003817100381710038171003817ge 1205901205722119896120574119896 1003817100381710038171003817119889119896100381710038171003817100381721003817100381710038171003817119865 (119911119896)1003817100381710038171003817 ge 1205901205761205722119896 100381710038171003817100381711988911989610038171003817100381710038172119872119891 ge 0 (45)

Combining (24) and (45) we obtain

lim119896997888rarrinfin

120572119896 10038171003817100381710038171198891198961003817100381710038171003817 = 0 (46)

Since 119889119896 ge 1205760 gt 0 for any 119896 isin 119873 we have

lim119896997888rarrinfin

120572119896 = 0 (47)

Clearly 120572lowast119896 = 120588minus1120572119896 does not satisfy in (17) It says that

minus119865 (119909119896 + 120572lowast119896119889119896)119879 119889119896 lt 120590120572lowast119896120574119896 100381710038171003817100381711988911989610038171003817100381710038172 (48)

By Lemmas 8 and 9 we know that the two sequences 119909119896and 119889119896 are bounded Without loss of generality we choosea subset 119870 isin 119873 such that

lim119896997888rarrinfin119896isin119870

119909119896 = 119909lowastlim119896997888rarrinfin119896isin119870

119889119896 = 119889lowast (49)

Taking the limit in the two sides of (48) as 119896 997888rarr infin (119896 isin 119870)it holds that

119865 (119909lowast)119879 119889lowast ge 0 (50)

On the other hand from (43) we know that

119865 (119909119896)119879 119889119896 = minus 100381710038171003817100381711986511989610038171003817100381710038172 le minus12057620 lt 0 (51)

By taking the limit 119896 997888rarr infin in the two sides of (51) for 119896 isin 119870we get

119865 (119909lowast)119879 119889lowast le minus1205760 lt 0 (52)

It contradicts (50) Thus the proof of Theorem 11 has beencompleted

Remark 12 (only with 120574119896 being generated by (19)) As 120574119896 isdetermined by (18) the proofs are similar

Since the proof of Theorem 11 does not involve differen-tiability of 119865 let alone nonsingularity of its Jacobian matrixwe know that the following result holds

Corollary 13 For any nonsmooth or singular function 119865 let119909119896 be a sequence generated as Algorithm 1 is used to solveproblem (1) Under Assumptions 5 6 and 7 it holds that

lim inf119896997888rarrinfin

10038171003817100381710038171198651198961003817100381710038171003817 = 0 (53)

It should be pointed out that the global convergenceof Algorithm 1 depends on assumption on monotonicity of119865 For nonmonotonic function 119865 Algorithm 1 may be notapplicable

4 Numerical Experiments

In this section by numerical experiments we are going tostudy the effectiveness and robustness of Algorithm 1 forsolving large-scale system of equations

We first collect some benchmark test problems availablein the literature

Problem 14 (see [5]) The elements of 119865(119909) are given by119865119894 (119909) = 2119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (54)

Problem 15 (see [5]) The elements of 119865(119909) are given by119865119894 (119909) = 2119909119894 minus 1003816100381610038161003816sin (119909119894)1003816100381610038161003816 119894 = 1 2 119899 minus 1 (55)

Mathematical Problems in Engineering 7

Problem 16 (see [15]) The elements of 119865(119909) are given by

1198651 (119909) = 1199091 minus 119890cos((1199091+1199092)(119899+1))119865119894 (119909) = 119909119894 minus 119890cos((119909119894minus1+119909119894+119909119894+1)(119899+1)) 119894 = 2 119899 minus 1119865119899 (119909) = 119909119899 minus 119890cos((119909119899minus1+119909119899)(119899+1)) (56)

Problem 17 (see [28]) The elements of 119865(119909) are given by

119865 (119909) = 119860119909 + 119887119887 = (minus1 minus1 minus1 minus1)119879

119860 = ((((((

52 11 52 1d d d

d d 11 52

))))))

(57)

Clearly problem 4 is a linear system of equations which isused to test Algorithm 1 in this special case

Problem 18 (see [15]) The elements of 119865(119909) are given by

119865119894 (119909) = 119909119894 minus 11198991199092119894 + 1119899 119899sum119894=1

119909119894 + 119894 119894 = 2 119899 minus 1 (58)

Problem 19 (see [15]) The elements of 119865(119909) are given by

1198651 (119909) = 1311990931 + 1211990922119865119894 (119909) = minus121199092119894 + 11989431199093119894 + 121199092119894+1 119894 = 2 119899 minus 1119865119899 (119909) = minus121199092119899 + 11989931199093119899

(59)

Clearly apart from problem 4 all the others are nonlinearsystem of equations Problem 15 is nonsmooth at point(0 0 0) The size of all the test problems is variable Ifthis size is larger than 1000 the problem can be regarded tobe large-scale We solve all the test problems with differentsizes by Algorithm 1 especially in comparison with theexisting seven similar algorithms such as those developedvery recently in [10 13ndash15 28]

All the algorithms are coded inMATLABR2010b and runon a personal computer with a 22GHZ CPU processor 8GB

memory and Windows 10 operation system The relevantparameters of algorithms are specified by119896119898119886119909 = 100000120590 = 03119904 = 05120588 = 07

(60)

120582119896 = 12058202 if 119896 = 1120582119896minus1 + 120582119896minus22 if 119896 gt 1 (61)

where 1205820 = 1 The termination condition is 119865119896 le 10minus4Numerical performance of all the algorithms is reported

in Tables 1 2 and 3 Table 1 shows the numerical performanceof all the eight algorithms with the fixed initial points 1199090 =(minus1 minus1)119879 for problems 1 5 and 6 1199090 = (1 1)119879 forproblems 2 and 4 1199090 = (10 10)119879 for problem 3 Table 2demonstrates the numerical performance of all the eightalgorithms with initial points randomly generated byMatlabrsquosCode ldquorand(n1)rdquo Table 3 shows the numerical performanceof our algorithm with dimension of 1000000

For simplification of statement we use the followingnotations

Dim the dimension of test problems

Ni the number of iterations

Nf the number of function evaluations

MPPRP-M the developed algorithm with 120574119896 determined by(19) in this paper

MPPRP-W the developed algorithmwith 120574119896 generated by (18)in this paper

MSDFPB the modified spectral derivative-free projection-based algorithm in [13]

PRP the PRP conjugate gradient derivative-free projection-based methods in [28]

MPRP the modified PRP conjugate gradient derivative-freeprojection-based methods in [10]

TPRP the two-term PRP conjugate gradient derivative-freeprojection-based methods in [10]

DFPB1 the steepest descent derivative-free projection-basedmethods in [14] with search direction as follows

119889119896 = minus1198650 if 119896 = 0minus119865119896 + 120573119896119889119896minus1 + 120579119896 if 119896 gt 0 (62)

where 120573119896 = 120573119875119877119875119896 = 119865119879119896 119910119896minus1119865119896minus12 120579119896 = (119865119879119896 119910119896minus1)120596119896minus12119865119896minus14 119910119896minus1 = 119865119896 minus 119865119896minus1 and 120596119896minus1 = 120572119896minus1119889119896minus1

8 Mathematical Problems in Engineering

Table 1 Numerical performance with fixed initial points

Problem Dim Method CPU-time Ni NfP1 10000 MPPRP-M 0049106 19 60

MPPRP-W 0083986 28 132MSDFPB 0207396 49 357

PRP 0210513 57 373MPRP 0225433 49 357TPRP 0205193 49 357DFPB1 0208191 52 360DFPB2 0180871 41 333MHS 0201207 49 357

P1 20000 MPPRP-M 0097751 20 63MPPRP-W 0202214 34 187MSDFPB 0538910 63 522

PRP 0601340 72 544MPRP 0578528 63 522TPRP 0567042 63 522DFPB1 0564317 67 534DFPB2 0538403 56 503MHS 0568248 63 522

P1 50000 MPPRP-M 0189582 20 63MPPRP-W 0776513 47 317MSDFPB 2349394 93 913

PRP 2426823 101 926MPRP 2373229 93 913TPRP 232399 93 913DFPB1 2421556 97 925DFPB2 2364173 86 894MHS 2601235 93 913

P1 100000 MPPRP-M 0519087 21 66MPPRP-W 2762991 61 478MSDFPB 9002493 127 1384

PRP 9108308 135 1402MPRP 9448788 127 1384TPRP 8986777 127 1384DFPB1 9100524 130 1391DFPB2 8819755 119 1360MHS 9419715 127 1384

P2 10000 MPPRP-M 0054535 19 60MPPRP-W 0088811 28 132MSDFPB 0217590 49 357

PRP 0218645 57 373MPRP 0226537 49 357TPRP 0226429 49 357DFPB1 0210470 52 360DFPB2 0204578 41 333MHS 0234442 49 357

P2 20000 MPPRP-M 0089917 20 63MPPRP-W 0198822 34 187MSDFPB 0563358 63 522

PRP 0604772 72 544

Mathematical Problems in Engineering 9

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0670294 63 522TPRP 0677819 63 522DFPB1 0679165 67 534DFPB2 0607357 56 503MHS 0589872 63 522

P2 50000 MPPRP-M 0191132 20 63MPPRP-W 0720108 47 317MSDFPB 2248779 93 913

PRP 2464597 101 926MPRP 2353173 93 913TPRP 2342777 93 913DFPB1 2363392 97 925DFPB2 2279782 86 894MHS 2384060 93 913

P2 100000 MPPRP-M 0516081 21 66MPPRP-W 2734834 61 478MSDFPB 9004299 127 1384

PRP 8763470 135 1402MPRP 8647855 127 1384TPRP 8626593 127 1384DFPB1 8626593 130 1391DFPB2 8476836 119 1360MHS 8905454 127 1384

P3 5000 MPPRP-M 0077881 22 69MPPRP-W 0955814 127 1360MSDFPB 2155000 242 3141

PRP 2346950 251 3167MPRP 2200878 242 3141TPRP 2204338 242 3141DFPB1 2246450 245 3148DFPB2 2209059 234 3117MHS 2236389 242 3141

P3 10000 MPPRP-M 0145481 23 72MPPRP-W 2590862 174 2059MSDFPB 6749566 337 4751

PRP 6892930 346 4767MPRP 6843273 337 4751TPRP 6885266 337 4751DFPB1 7222119 341 4763DFPB2 6881919 330 4732MHS 6992060 337 4751

P3 15000 MPPRP-M 0190225 23 72MPPRP-W 4894470 212 2656MSDFPB 12242514 411 6059

PRP 13494894 420 6078MPRP 13547676 411 6059TPRP 12255626 411 6059DFPB1 12473808 415 6071

10 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfDFPB2 12734751 404 6040MHS 12425026 411 6059

P3 20000 MPPRP-M 0239615 23 72MPPRP-W 7505229 242 3126MSDFPB 18305460 472 7144

PRP 19195911 481 7169MPRP 19482789 472 7144TPRP 19011112 472 7144DFPB1 19165800 475 7151DFPB2 19428512 464 7120MHS 18928398 472 7144

P4 10000 MPPRP-M 0061549 16 102MPPRP-W 0113153 19 131MSDFPB 0289696 68 555

PRP 0187397 39 391MPRP 0233670 48 462TPRP 0358863 89 658DFPB1 0657537 185 1142DFPB2 0213256 40 422MHS 0414446 62 803

P4 20000 MPPRP-M 0125760 17 108MPPRP-W 0256772 31 203MSDFPB 0703061 79 730

PRP 0558050 52 565MPRP 0663444 59 635TPRP 0822848 100 833DFPB1 1442003 203 1352DFPB2 0565771 51 596MHS 0674281 60 724

P4 50000 MPPRP-M 0295365 17 108MPPRP-W 0961757 38 273MSDFPB 2972641 102 1109

PRP 2655601 73 943MPRP 2738925 81 1007TPRP 3209823 122 1207DFPB1 4783178 234 1771DFPB2 2644189 75 981MHS 6106110 158 2717

P4 100000 MPPRP-M 0768978 17 108MPPRP-W 3113887 45 357MSDFPB 10575505 129 1568

PRP 9164701 101 1406MPRP 9756561 108 1465TPRP 11151034 146 1651DFPB1 16812871 275 2301DFPB2 9292510 99 1423MHS 21792651 179 3305

P5 100 MPPRP-M 0012253 18 77MPPRP-W 0018540 62 555MSDFPB 0049702 129 1522

PRP 0033290 144 1558

Mathematical Problems in Engineering 11

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0033017 139 1560TPRP 0034758 128 1518DFPB1 0026313 150 1601DFPB2 0029466 131 1538MHS 0040165 133 1539

P5 500 MPPRP-M 0016447 22 93MPPRP-W 0231399 544 9066MSDFPB 0772128 1313 25232

PRP 0727796 1332 25274MPRP 0585951 1372 25394TPRP 0577368 1315 25238DFPB1 0578794 1337 25315DFPB2 0589438 1317 25247MHS 0595316 1315 25241

P5 1000 MPPRP-M 0017438 23 97MPPRP-W 1072332 1505 29740MSDFPB 3218997 3680 81797

PRP 2889154 3696 81825MPRP 2963704 3735 81946TPRP 2948045 3682 81802DFPB1 3040253 3704 81880DFPB2 3045603 3683 81803MHS 3220028 3678 81791

P5 2000 MPPRP-M 0021613 25 106MPPRP-W 5944706 4206 95732MSDFPB 17227153 10341 260347

PRP 16073363 10355 260379MPRP 16960973 10394 260490TPRP 16826907 10343 260352DFPB1 16257104 10364 260424DFPB2 16739385 10345 260356MHS 17503252 10339 260342

P6 100 MPPRP-M 0217224 1182 3717MPPRP-W 0193153 1148 3605MSDFPB 0237793 1186 3706

PRP 0220389 1207 3766MPRP 0224211 1186 3658TPRP 0248104 1195 3722DFPB1 0236804 1201 3750DFPB2 0235470 1170 3620MHS 0236250 1198 3720

P6 500 MPPRP-M 0944828 1348 5050MPPRP-W 0794555 1278 4482MSDFPB 0797832 1185 3968

PRP 0911934 1322 4267MPRP 0845336 1258 3983TPRP 0912343 1326 4335DFPB1 0903852 1321 4300DFPB2 1020228 1299 4143MHS 0918611 1307 4147

P6 1000 MPPRP-M 2337405 1495 6667MPPRP-W 1680275 1381 4951MSDFPB 1536904 1331 4320

12 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfPRP 1851282 1389 4553MPRP 1647356 1277 4040TPRP 1686418 1273 4109DFPB1 1792886 1360 4409DFPB2 1812678 1343 4330MHS 1775351 1341 4283

P6 2000 MPPRP-M 6738143 1737 11016MPPRP-W 3257538 1440 4936MSDFPB 3065903 1381 4524

PRP 3574894 1390 4522MPRP 3381889 1316 4247TPRP 3565741 1378 4510DFPB1 3609619 1377 4511DFPB2 3647662 984 3232MHS 3608315 1362 4397

DFPB2 the steepest descent derivative-free projection-basedmethods in [14] with 120579119896 in (62) being replaced by

120579119896 = 119865119879119896 120596119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 + (119865119879119896 119910119896minus1) 1003817100381710038171003817119910119896minus1100381710038171003817100381721003817100381710038171003817119865119896minus110038171003817100381710038174 (63)

MHS the MHS-PRP conjugate gradient derivative-freeprojection-based methods in [15]

From the results in Tables 1 2 and 3 it follows that ouralgorithm (MPPRP) outperforms the other seven algorithmsno matter how to choose the initial points (see the italicizedresults) Especially it seems to more efficiently solve large-scale test problems Actually Table 3 shows that MPPRP-Mcan solve the first five Problems with dimension of 1000000in less time compared with the other algorithms

In order to further measure the efficiency difference ofall the eight algorithms we calculate the average numberof iteration the average consumed CPU time and theirstandard deviations respectively In Table 4 A-Ni and Std-Ni stand for the average number of iteration and its stan-dard deviation respectively A-CT and Std-CT represent theaverage consumed CPU time and its standard deviationrespectively The average number of function evaluation andits standard deviation are denoted by A-Nf and Std-Nfrespectively Clearly Std-Ni Std-Nf and Std-CT can showrobustness of all the algorithms

The results in Table 4 demonstrate that both ofMPPRP-Mand MPPRP-W outperform the other seven algorithms

In the end of this section we use our algorithm to solvea singular system of equations The next test problem is amodified version of problem 1

Problem 20 The elements of 119865(119909) are given by119865119894 (119909) = 119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (64)

The initial point is fixed as 1199090 = (1 1)119879

Note that problem 7 is singular since zero is its solutionand the Jacobian matrix 1198651015840(0) is singular We implementAlgorithm 1 to solve problem 7 with different dimen-sions to test whether it can find the singular solution ornot

The results in Table 5 indicate that Algorithm 1 is alsoefficient to solve the singular system of equations

5 Preliminary Applicationin Compressed Sensing

In this section we will apply our algorithm to solve anengineering problem originated from compressed sensing ofsparse signals

Let 119860 isin 119877119898times119899 (119898 ≪ 119899) be a linear operator let 119909 isin 119877119899be a sparse or a nearly sparse original signal and let 119887 isin119877119898 be an observed value which satisfies the following linearequations

119887 = 119860119909 (65)

The original signal 119909 is desirable to be reconstructedfrom the linear equations 119860119909 = 119887 Unfortunately itis often that this linear equation is underdetermined orill-conditioned in practice and has infinitely many solu-tions From the fundamental principles of compressed sens-ing it is a reasonable approach to seek the sparsest oneamong all the solutions which contains the least nonzerocomponents

In [2] it was shown that the compressed sensing of sparsesignals can be formulated the following nonlinear system ofequations

min 119911 (119867 + 119863) 119911 + 119888 = 0 (66)

Mathematical Problems in Engineering 13

Table 2 Efficiency with random initial points

Problem Dim Method CPU-time Ni NfP1 100000 MPPRP-M 0489058 20 63

MPPRP-W 2005818 42 268MSDFPB 4565133 79 717

PRP 4757480 87 825MPRP 4561151 79 796TPRP 4631983 79 796DFPB1 4763311 82 806DFPB2 4610199 74 776MHS 4879031 79 796

P2 100000 MPPRP-M 0483632 20 63MPPRP-W 2251160 42 268MSDFPB 4375745 79 717

PRP 4561437 87 825MPRP 4830143 79 796TPRP 4560123 79 796DFPB1 4427558 82 806DFPB2 4550071 74 776MHS 4428999 79 796

P3 100000 MPPRP-M 1187410 20 63MPPRP-W 5215137 46 306MSDFPB 10136405 79 710

PRP 10808303 87 816MPRP 10124779 79 789TPRP 10068535 79 789DFPB1 10372113 82 799DFPB2 9458892 71 757MHS 10137711 79 789

P4 100000 MPPRP-M 9152258 253 1306MPPRP-W 11371204 260 1423MSDFPB 13374780 293 1837

PRP 8352463 170 1352MPRP 9223706 184 1474TPRP 14279675 302 2184DFPB1 15329979 336 2386DFPB2 11244725 231 1770MHS 17738408 216 2902

P5 2000 MPPRP-M 0019793 25 106MPPRP-W 7148874 4207 95776MSDFPB 16293374 10362 260980

PRP 16689685 10375 271376MPRP 16112906 10415 271538TPRP 16006655 10364 271349DFPB1 16272868 10385 271442DFPB2 17018004 10366 271355MHS 19200453 10360 271335

P6 2000 MPPRP-M 9722610 3415 15767MPPRP-W 8031480 3293 12678MSDFPB 8254000 3395 12524

PRP 9187302 3298 15178MPRP 9667492 3366 15503TPRP 9727760 3363 15708DFPB1 9565437 3431 15990DFPB2 9316936 3329 15590MHS 8893926 3393 14502

14 Mathematical Problems in Engineering

Table 3 Efficiency for 1000000-dimension problems

Problem Dim Method CPU-time Ni NfP1 1000000 MPPRP-M 4831098 22 69P2 1000000 MPPRP-M 5150450 22 69P3 1000000 MPPRP-M 13954404 26 81P4 1000000 MPPRP-M 8408822 19 120P5 1000000 MPPRP-M 9018623 36 154

Table 4 Average numerical efficiency and robustness of algorithms

Method (A-CT Std-CT) (A-Ni Std-Ni) (A-Nf Std-CT)MPPRP-M (11657 25775) (3307 76388) (15135 36523)MPPRP-W (25725 29258) (6894 119935) (92059 242337)MSDFPB (54010 56632) (124456 263861) ( 231436 663099)PRP (53178 55599) (124773 264115) (235830 676024)MPRP (53717 56475) (124336 265463) (235743 676492)TPRP (55309 57401) ( 124963 263694) (236334 675796)DFPB1 (58718 60331) (12759 263685) (237512 675685)DFPB2 (53586 56541) (122393 264211) (235434 676120)MHS (62549 67712) (12488 263778) (237192 675398)

where

119911 = [119906V]

119910 = 119860119879119887119888 = 1205911198902119899 + [minus119910119910 ]

119867 = [ 119860119879119860 minus119860119879119860minus119860119879119860 119860119879119860 ] 119863 = [0 1205751198641198990 0 ]

(67)

where 119906 isin 119877119899 V isin 119877119899 and 119906119894 = (119909119894)+ V119894 = (minus119909119894)+ forall 119894 = 1 2 119899 with (sdot)+ = max0 sdot and 1198902119899 is an 2119899-dimensional vector with all elements being one Clearly (66)is nonsmooth

Implement Algorithm 1 to solve the compressed sensingproblem of sparse signals with different compressed ratios Inthis experiment we consider a typical compressive sensingscenario where the goal is to reconstruct an 119899-length sparsesignal from 119898 observations We test the three algorithms(two popular algorithms CGD in [34] and SGCS in [35] andMPPRP-M) under three CS ratios (CS-R)119898119899 = 0125 02505 corresponding to different numbers of measurementswith 119899 = 2048 respectively The original sparse signal con-tains 32(64) randomly nonzero elements The measurement119887 is disturbed by noise ie 119887 = 119860119909 + 120596 where 120596 is theGaussian noise distributed as 119873(0 1205902119868) with 1205902 = 10minus4 119860is the Gaussian matrix generated by commend randn(119898 119899)

inMatlab The quality of restoration is measured by the meanof squared error (MSE) to the original signal 119909 that is

MSE = 1119899 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (68)

where 119909lowast is the restored signal The iterative process starts atthe measurement image ie 1199090 = 119860119879119887 and terminates whenthe relative change between successive iterates falls below10minus5 It says that 1003817100381710038171003817119891119896 minus 119891119896minus110038171003817100381710038171003817100381710038171003817119891119896minus11003817100381710038171003817 lt 10minus5 (69)

where 119891119896 = 1205911199091198961 + (12)119860119909119896 minus 11988722 and 120591 = 0005119860119879119887infinis chosen as suggested in [36] The other parameters in thisexperiment are same as those in the numerical experimentsconducted in Section 4 Numerical efficiency is shown inTable 6

Clearly from the results in Table 6 it follows that forany CS ratio MPPRP-M can recover the sparse signals moreefficiently without reduction of recovery quality (see theitalicized results in Table 6) If the sparsity level 32 is replacedby 64 in the 2048-length original signal the correspondingresults are also presented in Table 6 denoted by (sdot)6 Conclusions

In this paper we have presented a modified spectral PRPconjugate gradient derivative-free projection-based methodfor solving the large-scale nonlinear monotonic equationswhere the search direction is proved to be sufficiently descentfor any line search rule and the step length is chosen by aline search which can overcome the difficulty of choosing anappropriate weight

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

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Mathematical Problems in Engineering 7

Problem 16 (see [15]) The elements of 119865(119909) are given by

1198651 (119909) = 1199091 minus 119890cos((1199091+1199092)(119899+1))119865119894 (119909) = 119909119894 minus 119890cos((119909119894minus1+119909119894+119909119894+1)(119899+1)) 119894 = 2 119899 minus 1119865119899 (119909) = 119909119899 minus 119890cos((119909119899minus1+119909119899)(119899+1)) (56)

Problem 17 (see [28]) The elements of 119865(119909) are given by

119865 (119909) = 119860119909 + 119887119887 = (minus1 minus1 minus1 minus1)119879

119860 = ((((((

52 11 52 1d d d

d d 11 52

))))))

(57)

Clearly problem 4 is a linear system of equations which isused to test Algorithm 1 in this special case

Problem 18 (see [15]) The elements of 119865(119909) are given by

119865119894 (119909) = 119909119894 minus 11198991199092119894 + 1119899 119899sum119894=1

119909119894 + 119894 119894 = 2 119899 minus 1 (58)

Problem 19 (see [15]) The elements of 119865(119909) are given by

1198651 (119909) = 1311990931 + 1211990922119865119894 (119909) = minus121199092119894 + 11989431199093119894 + 121199092119894+1 119894 = 2 119899 minus 1119865119899 (119909) = minus121199092119899 + 11989931199093119899

(59)

Clearly apart from problem 4 all the others are nonlinearsystem of equations Problem 15 is nonsmooth at point(0 0 0) The size of all the test problems is variable Ifthis size is larger than 1000 the problem can be regarded tobe large-scale We solve all the test problems with differentsizes by Algorithm 1 especially in comparison with theexisting seven similar algorithms such as those developedvery recently in [10 13ndash15 28]

All the algorithms are coded inMATLABR2010b and runon a personal computer with a 22GHZ CPU processor 8GB

memory and Windows 10 operation system The relevantparameters of algorithms are specified by119896119898119886119909 = 100000120590 = 03119904 = 05120588 = 07

(60)

120582119896 = 12058202 if 119896 = 1120582119896minus1 + 120582119896minus22 if 119896 gt 1 (61)

where 1205820 = 1 The termination condition is 119865119896 le 10minus4Numerical performance of all the algorithms is reported

in Tables 1 2 and 3 Table 1 shows the numerical performanceof all the eight algorithms with the fixed initial points 1199090 =(minus1 minus1)119879 for problems 1 5 and 6 1199090 = (1 1)119879 forproblems 2 and 4 1199090 = (10 10)119879 for problem 3 Table 2demonstrates the numerical performance of all the eightalgorithms with initial points randomly generated byMatlabrsquosCode ldquorand(n1)rdquo Table 3 shows the numerical performanceof our algorithm with dimension of 1000000

For simplification of statement we use the followingnotations

Dim the dimension of test problems

Ni the number of iterations

Nf the number of function evaluations

MPPRP-M the developed algorithm with 120574119896 determined by(19) in this paper

MPPRP-W the developed algorithmwith 120574119896 generated by (18)in this paper

MSDFPB the modified spectral derivative-free projection-based algorithm in [13]

PRP the PRP conjugate gradient derivative-free projection-based methods in [28]

MPRP the modified PRP conjugate gradient derivative-freeprojection-based methods in [10]

TPRP the two-term PRP conjugate gradient derivative-freeprojection-based methods in [10]

DFPB1 the steepest descent derivative-free projection-basedmethods in [14] with search direction as follows

119889119896 = minus1198650 if 119896 = 0minus119865119896 + 120573119896119889119896minus1 + 120579119896 if 119896 gt 0 (62)

where 120573119896 = 120573119875119877119875119896 = 119865119879119896 119910119896minus1119865119896minus12 120579119896 = (119865119879119896 119910119896minus1)120596119896minus12119865119896minus14 119910119896minus1 = 119865119896 minus 119865119896minus1 and 120596119896minus1 = 120572119896minus1119889119896minus1

8 Mathematical Problems in Engineering

Table 1 Numerical performance with fixed initial points

Problem Dim Method CPU-time Ni NfP1 10000 MPPRP-M 0049106 19 60

MPPRP-W 0083986 28 132MSDFPB 0207396 49 357

PRP 0210513 57 373MPRP 0225433 49 357TPRP 0205193 49 357DFPB1 0208191 52 360DFPB2 0180871 41 333MHS 0201207 49 357

P1 20000 MPPRP-M 0097751 20 63MPPRP-W 0202214 34 187MSDFPB 0538910 63 522

PRP 0601340 72 544MPRP 0578528 63 522TPRP 0567042 63 522DFPB1 0564317 67 534DFPB2 0538403 56 503MHS 0568248 63 522

P1 50000 MPPRP-M 0189582 20 63MPPRP-W 0776513 47 317MSDFPB 2349394 93 913

PRP 2426823 101 926MPRP 2373229 93 913TPRP 232399 93 913DFPB1 2421556 97 925DFPB2 2364173 86 894MHS 2601235 93 913

P1 100000 MPPRP-M 0519087 21 66MPPRP-W 2762991 61 478MSDFPB 9002493 127 1384

PRP 9108308 135 1402MPRP 9448788 127 1384TPRP 8986777 127 1384DFPB1 9100524 130 1391DFPB2 8819755 119 1360MHS 9419715 127 1384

P2 10000 MPPRP-M 0054535 19 60MPPRP-W 0088811 28 132MSDFPB 0217590 49 357

PRP 0218645 57 373MPRP 0226537 49 357TPRP 0226429 49 357DFPB1 0210470 52 360DFPB2 0204578 41 333MHS 0234442 49 357

P2 20000 MPPRP-M 0089917 20 63MPPRP-W 0198822 34 187MSDFPB 0563358 63 522

PRP 0604772 72 544

Mathematical Problems in Engineering 9

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0670294 63 522TPRP 0677819 63 522DFPB1 0679165 67 534DFPB2 0607357 56 503MHS 0589872 63 522

P2 50000 MPPRP-M 0191132 20 63MPPRP-W 0720108 47 317MSDFPB 2248779 93 913

PRP 2464597 101 926MPRP 2353173 93 913TPRP 2342777 93 913DFPB1 2363392 97 925DFPB2 2279782 86 894MHS 2384060 93 913

P2 100000 MPPRP-M 0516081 21 66MPPRP-W 2734834 61 478MSDFPB 9004299 127 1384

PRP 8763470 135 1402MPRP 8647855 127 1384TPRP 8626593 127 1384DFPB1 8626593 130 1391DFPB2 8476836 119 1360MHS 8905454 127 1384

P3 5000 MPPRP-M 0077881 22 69MPPRP-W 0955814 127 1360MSDFPB 2155000 242 3141

PRP 2346950 251 3167MPRP 2200878 242 3141TPRP 2204338 242 3141DFPB1 2246450 245 3148DFPB2 2209059 234 3117MHS 2236389 242 3141

P3 10000 MPPRP-M 0145481 23 72MPPRP-W 2590862 174 2059MSDFPB 6749566 337 4751

PRP 6892930 346 4767MPRP 6843273 337 4751TPRP 6885266 337 4751DFPB1 7222119 341 4763DFPB2 6881919 330 4732MHS 6992060 337 4751

P3 15000 MPPRP-M 0190225 23 72MPPRP-W 4894470 212 2656MSDFPB 12242514 411 6059

PRP 13494894 420 6078MPRP 13547676 411 6059TPRP 12255626 411 6059DFPB1 12473808 415 6071

10 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfDFPB2 12734751 404 6040MHS 12425026 411 6059

P3 20000 MPPRP-M 0239615 23 72MPPRP-W 7505229 242 3126MSDFPB 18305460 472 7144

PRP 19195911 481 7169MPRP 19482789 472 7144TPRP 19011112 472 7144DFPB1 19165800 475 7151DFPB2 19428512 464 7120MHS 18928398 472 7144

P4 10000 MPPRP-M 0061549 16 102MPPRP-W 0113153 19 131MSDFPB 0289696 68 555

PRP 0187397 39 391MPRP 0233670 48 462TPRP 0358863 89 658DFPB1 0657537 185 1142DFPB2 0213256 40 422MHS 0414446 62 803

P4 20000 MPPRP-M 0125760 17 108MPPRP-W 0256772 31 203MSDFPB 0703061 79 730

PRP 0558050 52 565MPRP 0663444 59 635TPRP 0822848 100 833DFPB1 1442003 203 1352DFPB2 0565771 51 596MHS 0674281 60 724

P4 50000 MPPRP-M 0295365 17 108MPPRP-W 0961757 38 273MSDFPB 2972641 102 1109

PRP 2655601 73 943MPRP 2738925 81 1007TPRP 3209823 122 1207DFPB1 4783178 234 1771DFPB2 2644189 75 981MHS 6106110 158 2717

P4 100000 MPPRP-M 0768978 17 108MPPRP-W 3113887 45 357MSDFPB 10575505 129 1568

PRP 9164701 101 1406MPRP 9756561 108 1465TPRP 11151034 146 1651DFPB1 16812871 275 2301DFPB2 9292510 99 1423MHS 21792651 179 3305

P5 100 MPPRP-M 0012253 18 77MPPRP-W 0018540 62 555MSDFPB 0049702 129 1522

PRP 0033290 144 1558

Mathematical Problems in Engineering 11

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0033017 139 1560TPRP 0034758 128 1518DFPB1 0026313 150 1601DFPB2 0029466 131 1538MHS 0040165 133 1539

P5 500 MPPRP-M 0016447 22 93MPPRP-W 0231399 544 9066MSDFPB 0772128 1313 25232

PRP 0727796 1332 25274MPRP 0585951 1372 25394TPRP 0577368 1315 25238DFPB1 0578794 1337 25315DFPB2 0589438 1317 25247MHS 0595316 1315 25241

P5 1000 MPPRP-M 0017438 23 97MPPRP-W 1072332 1505 29740MSDFPB 3218997 3680 81797

PRP 2889154 3696 81825MPRP 2963704 3735 81946TPRP 2948045 3682 81802DFPB1 3040253 3704 81880DFPB2 3045603 3683 81803MHS 3220028 3678 81791

P5 2000 MPPRP-M 0021613 25 106MPPRP-W 5944706 4206 95732MSDFPB 17227153 10341 260347

PRP 16073363 10355 260379MPRP 16960973 10394 260490TPRP 16826907 10343 260352DFPB1 16257104 10364 260424DFPB2 16739385 10345 260356MHS 17503252 10339 260342

P6 100 MPPRP-M 0217224 1182 3717MPPRP-W 0193153 1148 3605MSDFPB 0237793 1186 3706

PRP 0220389 1207 3766MPRP 0224211 1186 3658TPRP 0248104 1195 3722DFPB1 0236804 1201 3750DFPB2 0235470 1170 3620MHS 0236250 1198 3720

P6 500 MPPRP-M 0944828 1348 5050MPPRP-W 0794555 1278 4482MSDFPB 0797832 1185 3968

PRP 0911934 1322 4267MPRP 0845336 1258 3983TPRP 0912343 1326 4335DFPB1 0903852 1321 4300DFPB2 1020228 1299 4143MHS 0918611 1307 4147

P6 1000 MPPRP-M 2337405 1495 6667MPPRP-W 1680275 1381 4951MSDFPB 1536904 1331 4320

12 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfPRP 1851282 1389 4553MPRP 1647356 1277 4040TPRP 1686418 1273 4109DFPB1 1792886 1360 4409DFPB2 1812678 1343 4330MHS 1775351 1341 4283

P6 2000 MPPRP-M 6738143 1737 11016MPPRP-W 3257538 1440 4936MSDFPB 3065903 1381 4524

PRP 3574894 1390 4522MPRP 3381889 1316 4247TPRP 3565741 1378 4510DFPB1 3609619 1377 4511DFPB2 3647662 984 3232MHS 3608315 1362 4397

DFPB2 the steepest descent derivative-free projection-basedmethods in [14] with 120579119896 in (62) being replaced by

120579119896 = 119865119879119896 120596119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 + (119865119879119896 119910119896minus1) 1003817100381710038171003817119910119896minus1100381710038171003817100381721003817100381710038171003817119865119896minus110038171003817100381710038174 (63)

MHS the MHS-PRP conjugate gradient derivative-freeprojection-based methods in [15]

From the results in Tables 1 2 and 3 it follows that ouralgorithm (MPPRP) outperforms the other seven algorithmsno matter how to choose the initial points (see the italicizedresults) Especially it seems to more efficiently solve large-scale test problems Actually Table 3 shows that MPPRP-Mcan solve the first five Problems with dimension of 1000000in less time compared with the other algorithms

In order to further measure the efficiency difference ofall the eight algorithms we calculate the average numberof iteration the average consumed CPU time and theirstandard deviations respectively In Table 4 A-Ni and Std-Ni stand for the average number of iteration and its stan-dard deviation respectively A-CT and Std-CT represent theaverage consumed CPU time and its standard deviationrespectively The average number of function evaluation andits standard deviation are denoted by A-Nf and Std-Nfrespectively Clearly Std-Ni Std-Nf and Std-CT can showrobustness of all the algorithms

The results in Table 4 demonstrate that both ofMPPRP-Mand MPPRP-W outperform the other seven algorithms

In the end of this section we use our algorithm to solvea singular system of equations The next test problem is amodified version of problem 1

Problem 20 The elements of 119865(119909) are given by119865119894 (119909) = 119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (64)

The initial point is fixed as 1199090 = (1 1)119879

Note that problem 7 is singular since zero is its solutionand the Jacobian matrix 1198651015840(0) is singular We implementAlgorithm 1 to solve problem 7 with different dimen-sions to test whether it can find the singular solution ornot

The results in Table 5 indicate that Algorithm 1 is alsoefficient to solve the singular system of equations

5 Preliminary Applicationin Compressed Sensing

In this section we will apply our algorithm to solve anengineering problem originated from compressed sensing ofsparse signals

Let 119860 isin 119877119898times119899 (119898 ≪ 119899) be a linear operator let 119909 isin 119877119899be a sparse or a nearly sparse original signal and let 119887 isin119877119898 be an observed value which satisfies the following linearequations

119887 = 119860119909 (65)

The original signal 119909 is desirable to be reconstructedfrom the linear equations 119860119909 = 119887 Unfortunately itis often that this linear equation is underdetermined orill-conditioned in practice and has infinitely many solu-tions From the fundamental principles of compressed sens-ing it is a reasonable approach to seek the sparsest oneamong all the solutions which contains the least nonzerocomponents

In [2] it was shown that the compressed sensing of sparsesignals can be formulated the following nonlinear system ofequations

min 119911 (119867 + 119863) 119911 + 119888 = 0 (66)

Mathematical Problems in Engineering 13

Table 2 Efficiency with random initial points

Problem Dim Method CPU-time Ni NfP1 100000 MPPRP-M 0489058 20 63

MPPRP-W 2005818 42 268MSDFPB 4565133 79 717

PRP 4757480 87 825MPRP 4561151 79 796TPRP 4631983 79 796DFPB1 4763311 82 806DFPB2 4610199 74 776MHS 4879031 79 796

P2 100000 MPPRP-M 0483632 20 63MPPRP-W 2251160 42 268MSDFPB 4375745 79 717

PRP 4561437 87 825MPRP 4830143 79 796TPRP 4560123 79 796DFPB1 4427558 82 806DFPB2 4550071 74 776MHS 4428999 79 796

P3 100000 MPPRP-M 1187410 20 63MPPRP-W 5215137 46 306MSDFPB 10136405 79 710

PRP 10808303 87 816MPRP 10124779 79 789TPRP 10068535 79 789DFPB1 10372113 82 799DFPB2 9458892 71 757MHS 10137711 79 789

P4 100000 MPPRP-M 9152258 253 1306MPPRP-W 11371204 260 1423MSDFPB 13374780 293 1837

PRP 8352463 170 1352MPRP 9223706 184 1474TPRP 14279675 302 2184DFPB1 15329979 336 2386DFPB2 11244725 231 1770MHS 17738408 216 2902

P5 2000 MPPRP-M 0019793 25 106MPPRP-W 7148874 4207 95776MSDFPB 16293374 10362 260980

PRP 16689685 10375 271376MPRP 16112906 10415 271538TPRP 16006655 10364 271349DFPB1 16272868 10385 271442DFPB2 17018004 10366 271355MHS 19200453 10360 271335

P6 2000 MPPRP-M 9722610 3415 15767MPPRP-W 8031480 3293 12678MSDFPB 8254000 3395 12524

PRP 9187302 3298 15178MPRP 9667492 3366 15503TPRP 9727760 3363 15708DFPB1 9565437 3431 15990DFPB2 9316936 3329 15590MHS 8893926 3393 14502

14 Mathematical Problems in Engineering

Table 3 Efficiency for 1000000-dimension problems

Problem Dim Method CPU-time Ni NfP1 1000000 MPPRP-M 4831098 22 69P2 1000000 MPPRP-M 5150450 22 69P3 1000000 MPPRP-M 13954404 26 81P4 1000000 MPPRP-M 8408822 19 120P5 1000000 MPPRP-M 9018623 36 154

Table 4 Average numerical efficiency and robustness of algorithms

Method (A-CT Std-CT) (A-Ni Std-Ni) (A-Nf Std-CT)MPPRP-M (11657 25775) (3307 76388) (15135 36523)MPPRP-W (25725 29258) (6894 119935) (92059 242337)MSDFPB (54010 56632) (124456 263861) ( 231436 663099)PRP (53178 55599) (124773 264115) (235830 676024)MPRP (53717 56475) (124336 265463) (235743 676492)TPRP (55309 57401) ( 124963 263694) (236334 675796)DFPB1 (58718 60331) (12759 263685) (237512 675685)DFPB2 (53586 56541) (122393 264211) (235434 676120)MHS (62549 67712) (12488 263778) (237192 675398)

where

119911 = [119906V]

119910 = 119860119879119887119888 = 1205911198902119899 + [minus119910119910 ]

119867 = [ 119860119879119860 minus119860119879119860minus119860119879119860 119860119879119860 ] 119863 = [0 1205751198641198990 0 ]

(67)

where 119906 isin 119877119899 V isin 119877119899 and 119906119894 = (119909119894)+ V119894 = (minus119909119894)+ forall 119894 = 1 2 119899 with (sdot)+ = max0 sdot and 1198902119899 is an 2119899-dimensional vector with all elements being one Clearly (66)is nonsmooth

Implement Algorithm 1 to solve the compressed sensingproblem of sparse signals with different compressed ratios Inthis experiment we consider a typical compressive sensingscenario where the goal is to reconstruct an 119899-length sparsesignal from 119898 observations We test the three algorithms(two popular algorithms CGD in [34] and SGCS in [35] andMPPRP-M) under three CS ratios (CS-R)119898119899 = 0125 02505 corresponding to different numbers of measurementswith 119899 = 2048 respectively The original sparse signal con-tains 32(64) randomly nonzero elements The measurement119887 is disturbed by noise ie 119887 = 119860119909 + 120596 where 120596 is theGaussian noise distributed as 119873(0 1205902119868) with 1205902 = 10minus4 119860is the Gaussian matrix generated by commend randn(119898 119899)

inMatlab The quality of restoration is measured by the meanof squared error (MSE) to the original signal 119909 that is

MSE = 1119899 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (68)

where 119909lowast is the restored signal The iterative process starts atthe measurement image ie 1199090 = 119860119879119887 and terminates whenthe relative change between successive iterates falls below10minus5 It says that 1003817100381710038171003817119891119896 minus 119891119896minus110038171003817100381710038171003817100381710038171003817119891119896minus11003817100381710038171003817 lt 10minus5 (69)

where 119891119896 = 1205911199091198961 + (12)119860119909119896 minus 11988722 and 120591 = 0005119860119879119887infinis chosen as suggested in [36] The other parameters in thisexperiment are same as those in the numerical experimentsconducted in Section 4 Numerical efficiency is shown inTable 6

Clearly from the results in Table 6 it follows that forany CS ratio MPPRP-M can recover the sparse signals moreefficiently without reduction of recovery quality (see theitalicized results in Table 6) If the sparsity level 32 is replacedby 64 in the 2048-length original signal the correspondingresults are also presented in Table 6 denoted by (sdot)6 Conclusions

In this paper we have presented a modified spectral PRPconjugate gradient derivative-free projection-based methodfor solving the large-scale nonlinear monotonic equationswhere the search direction is proved to be sufficiently descentfor any line search rule and the step length is chosen by aline search which can overcome the difficulty of choosing anappropriate weight

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

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8 Mathematical Problems in Engineering

Table 1 Numerical performance with fixed initial points

Problem Dim Method CPU-time Ni NfP1 10000 MPPRP-M 0049106 19 60

MPPRP-W 0083986 28 132MSDFPB 0207396 49 357

PRP 0210513 57 373MPRP 0225433 49 357TPRP 0205193 49 357DFPB1 0208191 52 360DFPB2 0180871 41 333MHS 0201207 49 357

P1 20000 MPPRP-M 0097751 20 63MPPRP-W 0202214 34 187MSDFPB 0538910 63 522

PRP 0601340 72 544MPRP 0578528 63 522TPRP 0567042 63 522DFPB1 0564317 67 534DFPB2 0538403 56 503MHS 0568248 63 522

P1 50000 MPPRP-M 0189582 20 63MPPRP-W 0776513 47 317MSDFPB 2349394 93 913

PRP 2426823 101 926MPRP 2373229 93 913TPRP 232399 93 913DFPB1 2421556 97 925DFPB2 2364173 86 894MHS 2601235 93 913

P1 100000 MPPRP-M 0519087 21 66MPPRP-W 2762991 61 478MSDFPB 9002493 127 1384

PRP 9108308 135 1402MPRP 9448788 127 1384TPRP 8986777 127 1384DFPB1 9100524 130 1391DFPB2 8819755 119 1360MHS 9419715 127 1384

P2 10000 MPPRP-M 0054535 19 60MPPRP-W 0088811 28 132MSDFPB 0217590 49 357

PRP 0218645 57 373MPRP 0226537 49 357TPRP 0226429 49 357DFPB1 0210470 52 360DFPB2 0204578 41 333MHS 0234442 49 357

P2 20000 MPPRP-M 0089917 20 63MPPRP-W 0198822 34 187MSDFPB 0563358 63 522

PRP 0604772 72 544

Mathematical Problems in Engineering 9

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0670294 63 522TPRP 0677819 63 522DFPB1 0679165 67 534DFPB2 0607357 56 503MHS 0589872 63 522

P2 50000 MPPRP-M 0191132 20 63MPPRP-W 0720108 47 317MSDFPB 2248779 93 913

PRP 2464597 101 926MPRP 2353173 93 913TPRP 2342777 93 913DFPB1 2363392 97 925DFPB2 2279782 86 894MHS 2384060 93 913

P2 100000 MPPRP-M 0516081 21 66MPPRP-W 2734834 61 478MSDFPB 9004299 127 1384

PRP 8763470 135 1402MPRP 8647855 127 1384TPRP 8626593 127 1384DFPB1 8626593 130 1391DFPB2 8476836 119 1360MHS 8905454 127 1384

P3 5000 MPPRP-M 0077881 22 69MPPRP-W 0955814 127 1360MSDFPB 2155000 242 3141

PRP 2346950 251 3167MPRP 2200878 242 3141TPRP 2204338 242 3141DFPB1 2246450 245 3148DFPB2 2209059 234 3117MHS 2236389 242 3141

P3 10000 MPPRP-M 0145481 23 72MPPRP-W 2590862 174 2059MSDFPB 6749566 337 4751

PRP 6892930 346 4767MPRP 6843273 337 4751TPRP 6885266 337 4751DFPB1 7222119 341 4763DFPB2 6881919 330 4732MHS 6992060 337 4751

P3 15000 MPPRP-M 0190225 23 72MPPRP-W 4894470 212 2656MSDFPB 12242514 411 6059

PRP 13494894 420 6078MPRP 13547676 411 6059TPRP 12255626 411 6059DFPB1 12473808 415 6071

10 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfDFPB2 12734751 404 6040MHS 12425026 411 6059

P3 20000 MPPRP-M 0239615 23 72MPPRP-W 7505229 242 3126MSDFPB 18305460 472 7144

PRP 19195911 481 7169MPRP 19482789 472 7144TPRP 19011112 472 7144DFPB1 19165800 475 7151DFPB2 19428512 464 7120MHS 18928398 472 7144

P4 10000 MPPRP-M 0061549 16 102MPPRP-W 0113153 19 131MSDFPB 0289696 68 555

PRP 0187397 39 391MPRP 0233670 48 462TPRP 0358863 89 658DFPB1 0657537 185 1142DFPB2 0213256 40 422MHS 0414446 62 803

P4 20000 MPPRP-M 0125760 17 108MPPRP-W 0256772 31 203MSDFPB 0703061 79 730

PRP 0558050 52 565MPRP 0663444 59 635TPRP 0822848 100 833DFPB1 1442003 203 1352DFPB2 0565771 51 596MHS 0674281 60 724

P4 50000 MPPRP-M 0295365 17 108MPPRP-W 0961757 38 273MSDFPB 2972641 102 1109

PRP 2655601 73 943MPRP 2738925 81 1007TPRP 3209823 122 1207DFPB1 4783178 234 1771DFPB2 2644189 75 981MHS 6106110 158 2717

P4 100000 MPPRP-M 0768978 17 108MPPRP-W 3113887 45 357MSDFPB 10575505 129 1568

PRP 9164701 101 1406MPRP 9756561 108 1465TPRP 11151034 146 1651DFPB1 16812871 275 2301DFPB2 9292510 99 1423MHS 21792651 179 3305

P5 100 MPPRP-M 0012253 18 77MPPRP-W 0018540 62 555MSDFPB 0049702 129 1522

PRP 0033290 144 1558

Mathematical Problems in Engineering 11

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0033017 139 1560TPRP 0034758 128 1518DFPB1 0026313 150 1601DFPB2 0029466 131 1538MHS 0040165 133 1539

P5 500 MPPRP-M 0016447 22 93MPPRP-W 0231399 544 9066MSDFPB 0772128 1313 25232

PRP 0727796 1332 25274MPRP 0585951 1372 25394TPRP 0577368 1315 25238DFPB1 0578794 1337 25315DFPB2 0589438 1317 25247MHS 0595316 1315 25241

P5 1000 MPPRP-M 0017438 23 97MPPRP-W 1072332 1505 29740MSDFPB 3218997 3680 81797

PRP 2889154 3696 81825MPRP 2963704 3735 81946TPRP 2948045 3682 81802DFPB1 3040253 3704 81880DFPB2 3045603 3683 81803MHS 3220028 3678 81791

P5 2000 MPPRP-M 0021613 25 106MPPRP-W 5944706 4206 95732MSDFPB 17227153 10341 260347

PRP 16073363 10355 260379MPRP 16960973 10394 260490TPRP 16826907 10343 260352DFPB1 16257104 10364 260424DFPB2 16739385 10345 260356MHS 17503252 10339 260342

P6 100 MPPRP-M 0217224 1182 3717MPPRP-W 0193153 1148 3605MSDFPB 0237793 1186 3706

PRP 0220389 1207 3766MPRP 0224211 1186 3658TPRP 0248104 1195 3722DFPB1 0236804 1201 3750DFPB2 0235470 1170 3620MHS 0236250 1198 3720

P6 500 MPPRP-M 0944828 1348 5050MPPRP-W 0794555 1278 4482MSDFPB 0797832 1185 3968

PRP 0911934 1322 4267MPRP 0845336 1258 3983TPRP 0912343 1326 4335DFPB1 0903852 1321 4300DFPB2 1020228 1299 4143MHS 0918611 1307 4147

P6 1000 MPPRP-M 2337405 1495 6667MPPRP-W 1680275 1381 4951MSDFPB 1536904 1331 4320

12 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfPRP 1851282 1389 4553MPRP 1647356 1277 4040TPRP 1686418 1273 4109DFPB1 1792886 1360 4409DFPB2 1812678 1343 4330MHS 1775351 1341 4283

P6 2000 MPPRP-M 6738143 1737 11016MPPRP-W 3257538 1440 4936MSDFPB 3065903 1381 4524

PRP 3574894 1390 4522MPRP 3381889 1316 4247TPRP 3565741 1378 4510DFPB1 3609619 1377 4511DFPB2 3647662 984 3232MHS 3608315 1362 4397

DFPB2 the steepest descent derivative-free projection-basedmethods in [14] with 120579119896 in (62) being replaced by

120579119896 = 119865119879119896 120596119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 + (119865119879119896 119910119896minus1) 1003817100381710038171003817119910119896minus1100381710038171003817100381721003817100381710038171003817119865119896minus110038171003817100381710038174 (63)

MHS the MHS-PRP conjugate gradient derivative-freeprojection-based methods in [15]

From the results in Tables 1 2 and 3 it follows that ouralgorithm (MPPRP) outperforms the other seven algorithmsno matter how to choose the initial points (see the italicizedresults) Especially it seems to more efficiently solve large-scale test problems Actually Table 3 shows that MPPRP-Mcan solve the first five Problems with dimension of 1000000in less time compared with the other algorithms

In order to further measure the efficiency difference ofall the eight algorithms we calculate the average numberof iteration the average consumed CPU time and theirstandard deviations respectively In Table 4 A-Ni and Std-Ni stand for the average number of iteration and its stan-dard deviation respectively A-CT and Std-CT represent theaverage consumed CPU time and its standard deviationrespectively The average number of function evaluation andits standard deviation are denoted by A-Nf and Std-Nfrespectively Clearly Std-Ni Std-Nf and Std-CT can showrobustness of all the algorithms

The results in Table 4 demonstrate that both ofMPPRP-Mand MPPRP-W outperform the other seven algorithms

In the end of this section we use our algorithm to solvea singular system of equations The next test problem is amodified version of problem 1

Problem 20 The elements of 119865(119909) are given by119865119894 (119909) = 119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (64)

The initial point is fixed as 1199090 = (1 1)119879

Note that problem 7 is singular since zero is its solutionand the Jacobian matrix 1198651015840(0) is singular We implementAlgorithm 1 to solve problem 7 with different dimen-sions to test whether it can find the singular solution ornot

The results in Table 5 indicate that Algorithm 1 is alsoefficient to solve the singular system of equations

5 Preliminary Applicationin Compressed Sensing

In this section we will apply our algorithm to solve anengineering problem originated from compressed sensing ofsparse signals

Let 119860 isin 119877119898times119899 (119898 ≪ 119899) be a linear operator let 119909 isin 119877119899be a sparse or a nearly sparse original signal and let 119887 isin119877119898 be an observed value which satisfies the following linearequations

119887 = 119860119909 (65)

The original signal 119909 is desirable to be reconstructedfrom the linear equations 119860119909 = 119887 Unfortunately itis often that this linear equation is underdetermined orill-conditioned in practice and has infinitely many solu-tions From the fundamental principles of compressed sens-ing it is a reasonable approach to seek the sparsest oneamong all the solutions which contains the least nonzerocomponents

In [2] it was shown that the compressed sensing of sparsesignals can be formulated the following nonlinear system ofequations

min 119911 (119867 + 119863) 119911 + 119888 = 0 (66)

Mathematical Problems in Engineering 13

Table 2 Efficiency with random initial points

Problem Dim Method CPU-time Ni NfP1 100000 MPPRP-M 0489058 20 63

MPPRP-W 2005818 42 268MSDFPB 4565133 79 717

PRP 4757480 87 825MPRP 4561151 79 796TPRP 4631983 79 796DFPB1 4763311 82 806DFPB2 4610199 74 776MHS 4879031 79 796

P2 100000 MPPRP-M 0483632 20 63MPPRP-W 2251160 42 268MSDFPB 4375745 79 717

PRP 4561437 87 825MPRP 4830143 79 796TPRP 4560123 79 796DFPB1 4427558 82 806DFPB2 4550071 74 776MHS 4428999 79 796

P3 100000 MPPRP-M 1187410 20 63MPPRP-W 5215137 46 306MSDFPB 10136405 79 710

PRP 10808303 87 816MPRP 10124779 79 789TPRP 10068535 79 789DFPB1 10372113 82 799DFPB2 9458892 71 757MHS 10137711 79 789

P4 100000 MPPRP-M 9152258 253 1306MPPRP-W 11371204 260 1423MSDFPB 13374780 293 1837

PRP 8352463 170 1352MPRP 9223706 184 1474TPRP 14279675 302 2184DFPB1 15329979 336 2386DFPB2 11244725 231 1770MHS 17738408 216 2902

P5 2000 MPPRP-M 0019793 25 106MPPRP-W 7148874 4207 95776MSDFPB 16293374 10362 260980

PRP 16689685 10375 271376MPRP 16112906 10415 271538TPRP 16006655 10364 271349DFPB1 16272868 10385 271442DFPB2 17018004 10366 271355MHS 19200453 10360 271335

P6 2000 MPPRP-M 9722610 3415 15767MPPRP-W 8031480 3293 12678MSDFPB 8254000 3395 12524

PRP 9187302 3298 15178MPRP 9667492 3366 15503TPRP 9727760 3363 15708DFPB1 9565437 3431 15990DFPB2 9316936 3329 15590MHS 8893926 3393 14502

14 Mathematical Problems in Engineering

Table 3 Efficiency for 1000000-dimension problems

Problem Dim Method CPU-time Ni NfP1 1000000 MPPRP-M 4831098 22 69P2 1000000 MPPRP-M 5150450 22 69P3 1000000 MPPRP-M 13954404 26 81P4 1000000 MPPRP-M 8408822 19 120P5 1000000 MPPRP-M 9018623 36 154

Table 4 Average numerical efficiency and robustness of algorithms

Method (A-CT Std-CT) (A-Ni Std-Ni) (A-Nf Std-CT)MPPRP-M (11657 25775) (3307 76388) (15135 36523)MPPRP-W (25725 29258) (6894 119935) (92059 242337)MSDFPB (54010 56632) (124456 263861) ( 231436 663099)PRP (53178 55599) (124773 264115) (235830 676024)MPRP (53717 56475) (124336 265463) (235743 676492)TPRP (55309 57401) ( 124963 263694) (236334 675796)DFPB1 (58718 60331) (12759 263685) (237512 675685)DFPB2 (53586 56541) (122393 264211) (235434 676120)MHS (62549 67712) (12488 263778) (237192 675398)

where

119911 = [119906V]

119910 = 119860119879119887119888 = 1205911198902119899 + [minus119910119910 ]

119867 = [ 119860119879119860 minus119860119879119860minus119860119879119860 119860119879119860 ] 119863 = [0 1205751198641198990 0 ]

(67)

where 119906 isin 119877119899 V isin 119877119899 and 119906119894 = (119909119894)+ V119894 = (minus119909119894)+ forall 119894 = 1 2 119899 with (sdot)+ = max0 sdot and 1198902119899 is an 2119899-dimensional vector with all elements being one Clearly (66)is nonsmooth

Implement Algorithm 1 to solve the compressed sensingproblem of sparse signals with different compressed ratios Inthis experiment we consider a typical compressive sensingscenario where the goal is to reconstruct an 119899-length sparsesignal from 119898 observations We test the three algorithms(two popular algorithms CGD in [34] and SGCS in [35] andMPPRP-M) under three CS ratios (CS-R)119898119899 = 0125 02505 corresponding to different numbers of measurementswith 119899 = 2048 respectively The original sparse signal con-tains 32(64) randomly nonzero elements The measurement119887 is disturbed by noise ie 119887 = 119860119909 + 120596 where 120596 is theGaussian noise distributed as 119873(0 1205902119868) with 1205902 = 10minus4 119860is the Gaussian matrix generated by commend randn(119898 119899)

inMatlab The quality of restoration is measured by the meanof squared error (MSE) to the original signal 119909 that is

MSE = 1119899 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (68)

where 119909lowast is the restored signal The iterative process starts atthe measurement image ie 1199090 = 119860119879119887 and terminates whenthe relative change between successive iterates falls below10minus5 It says that 1003817100381710038171003817119891119896 minus 119891119896minus110038171003817100381710038171003817100381710038171003817119891119896minus11003817100381710038171003817 lt 10minus5 (69)

where 119891119896 = 1205911199091198961 + (12)119860119909119896 minus 11988722 and 120591 = 0005119860119879119887infinis chosen as suggested in [36] The other parameters in thisexperiment are same as those in the numerical experimentsconducted in Section 4 Numerical efficiency is shown inTable 6

Clearly from the results in Table 6 it follows that forany CS ratio MPPRP-M can recover the sparse signals moreefficiently without reduction of recovery quality (see theitalicized results in Table 6) If the sparsity level 32 is replacedby 64 in the 2048-length original signal the correspondingresults are also presented in Table 6 denoted by (sdot)6 Conclusions

In this paper we have presented a modified spectral PRPconjugate gradient derivative-free projection-based methodfor solving the large-scale nonlinear monotonic equationswhere the search direction is proved to be sufficiently descentfor any line search rule and the step length is chosen by aline search which can overcome the difficulty of choosing anappropriate weight

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

Hindawiwwwhindawicom Volume 2018

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Mathematical Problems in Engineering

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Mathematical Problems in Engineering 9

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0670294 63 522TPRP 0677819 63 522DFPB1 0679165 67 534DFPB2 0607357 56 503MHS 0589872 63 522

P2 50000 MPPRP-M 0191132 20 63MPPRP-W 0720108 47 317MSDFPB 2248779 93 913

PRP 2464597 101 926MPRP 2353173 93 913TPRP 2342777 93 913DFPB1 2363392 97 925DFPB2 2279782 86 894MHS 2384060 93 913

P2 100000 MPPRP-M 0516081 21 66MPPRP-W 2734834 61 478MSDFPB 9004299 127 1384

PRP 8763470 135 1402MPRP 8647855 127 1384TPRP 8626593 127 1384DFPB1 8626593 130 1391DFPB2 8476836 119 1360MHS 8905454 127 1384

P3 5000 MPPRP-M 0077881 22 69MPPRP-W 0955814 127 1360MSDFPB 2155000 242 3141

PRP 2346950 251 3167MPRP 2200878 242 3141TPRP 2204338 242 3141DFPB1 2246450 245 3148DFPB2 2209059 234 3117MHS 2236389 242 3141

P3 10000 MPPRP-M 0145481 23 72MPPRP-W 2590862 174 2059MSDFPB 6749566 337 4751

PRP 6892930 346 4767MPRP 6843273 337 4751TPRP 6885266 337 4751DFPB1 7222119 341 4763DFPB2 6881919 330 4732MHS 6992060 337 4751

P3 15000 MPPRP-M 0190225 23 72MPPRP-W 4894470 212 2656MSDFPB 12242514 411 6059

PRP 13494894 420 6078MPRP 13547676 411 6059TPRP 12255626 411 6059DFPB1 12473808 415 6071

10 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfDFPB2 12734751 404 6040MHS 12425026 411 6059

P3 20000 MPPRP-M 0239615 23 72MPPRP-W 7505229 242 3126MSDFPB 18305460 472 7144

PRP 19195911 481 7169MPRP 19482789 472 7144TPRP 19011112 472 7144DFPB1 19165800 475 7151DFPB2 19428512 464 7120MHS 18928398 472 7144

P4 10000 MPPRP-M 0061549 16 102MPPRP-W 0113153 19 131MSDFPB 0289696 68 555

PRP 0187397 39 391MPRP 0233670 48 462TPRP 0358863 89 658DFPB1 0657537 185 1142DFPB2 0213256 40 422MHS 0414446 62 803

P4 20000 MPPRP-M 0125760 17 108MPPRP-W 0256772 31 203MSDFPB 0703061 79 730

PRP 0558050 52 565MPRP 0663444 59 635TPRP 0822848 100 833DFPB1 1442003 203 1352DFPB2 0565771 51 596MHS 0674281 60 724

P4 50000 MPPRP-M 0295365 17 108MPPRP-W 0961757 38 273MSDFPB 2972641 102 1109

PRP 2655601 73 943MPRP 2738925 81 1007TPRP 3209823 122 1207DFPB1 4783178 234 1771DFPB2 2644189 75 981MHS 6106110 158 2717

P4 100000 MPPRP-M 0768978 17 108MPPRP-W 3113887 45 357MSDFPB 10575505 129 1568

PRP 9164701 101 1406MPRP 9756561 108 1465TPRP 11151034 146 1651DFPB1 16812871 275 2301DFPB2 9292510 99 1423MHS 21792651 179 3305

P5 100 MPPRP-M 0012253 18 77MPPRP-W 0018540 62 555MSDFPB 0049702 129 1522

PRP 0033290 144 1558

Mathematical Problems in Engineering 11

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0033017 139 1560TPRP 0034758 128 1518DFPB1 0026313 150 1601DFPB2 0029466 131 1538MHS 0040165 133 1539

P5 500 MPPRP-M 0016447 22 93MPPRP-W 0231399 544 9066MSDFPB 0772128 1313 25232

PRP 0727796 1332 25274MPRP 0585951 1372 25394TPRP 0577368 1315 25238DFPB1 0578794 1337 25315DFPB2 0589438 1317 25247MHS 0595316 1315 25241

P5 1000 MPPRP-M 0017438 23 97MPPRP-W 1072332 1505 29740MSDFPB 3218997 3680 81797

PRP 2889154 3696 81825MPRP 2963704 3735 81946TPRP 2948045 3682 81802DFPB1 3040253 3704 81880DFPB2 3045603 3683 81803MHS 3220028 3678 81791

P5 2000 MPPRP-M 0021613 25 106MPPRP-W 5944706 4206 95732MSDFPB 17227153 10341 260347

PRP 16073363 10355 260379MPRP 16960973 10394 260490TPRP 16826907 10343 260352DFPB1 16257104 10364 260424DFPB2 16739385 10345 260356MHS 17503252 10339 260342

P6 100 MPPRP-M 0217224 1182 3717MPPRP-W 0193153 1148 3605MSDFPB 0237793 1186 3706

PRP 0220389 1207 3766MPRP 0224211 1186 3658TPRP 0248104 1195 3722DFPB1 0236804 1201 3750DFPB2 0235470 1170 3620MHS 0236250 1198 3720

P6 500 MPPRP-M 0944828 1348 5050MPPRP-W 0794555 1278 4482MSDFPB 0797832 1185 3968

PRP 0911934 1322 4267MPRP 0845336 1258 3983TPRP 0912343 1326 4335DFPB1 0903852 1321 4300DFPB2 1020228 1299 4143MHS 0918611 1307 4147

P6 1000 MPPRP-M 2337405 1495 6667MPPRP-W 1680275 1381 4951MSDFPB 1536904 1331 4320

12 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfPRP 1851282 1389 4553MPRP 1647356 1277 4040TPRP 1686418 1273 4109DFPB1 1792886 1360 4409DFPB2 1812678 1343 4330MHS 1775351 1341 4283

P6 2000 MPPRP-M 6738143 1737 11016MPPRP-W 3257538 1440 4936MSDFPB 3065903 1381 4524

PRP 3574894 1390 4522MPRP 3381889 1316 4247TPRP 3565741 1378 4510DFPB1 3609619 1377 4511DFPB2 3647662 984 3232MHS 3608315 1362 4397

DFPB2 the steepest descent derivative-free projection-basedmethods in [14] with 120579119896 in (62) being replaced by

120579119896 = 119865119879119896 120596119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 + (119865119879119896 119910119896minus1) 1003817100381710038171003817119910119896minus1100381710038171003817100381721003817100381710038171003817119865119896minus110038171003817100381710038174 (63)

MHS the MHS-PRP conjugate gradient derivative-freeprojection-based methods in [15]

From the results in Tables 1 2 and 3 it follows that ouralgorithm (MPPRP) outperforms the other seven algorithmsno matter how to choose the initial points (see the italicizedresults) Especially it seems to more efficiently solve large-scale test problems Actually Table 3 shows that MPPRP-Mcan solve the first five Problems with dimension of 1000000in less time compared with the other algorithms

In order to further measure the efficiency difference ofall the eight algorithms we calculate the average numberof iteration the average consumed CPU time and theirstandard deviations respectively In Table 4 A-Ni and Std-Ni stand for the average number of iteration and its stan-dard deviation respectively A-CT and Std-CT represent theaverage consumed CPU time and its standard deviationrespectively The average number of function evaluation andits standard deviation are denoted by A-Nf and Std-Nfrespectively Clearly Std-Ni Std-Nf and Std-CT can showrobustness of all the algorithms

The results in Table 4 demonstrate that both ofMPPRP-Mand MPPRP-W outperform the other seven algorithms

In the end of this section we use our algorithm to solvea singular system of equations The next test problem is amodified version of problem 1

Problem 20 The elements of 119865(119909) are given by119865119894 (119909) = 119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (64)

The initial point is fixed as 1199090 = (1 1)119879

Note that problem 7 is singular since zero is its solutionand the Jacobian matrix 1198651015840(0) is singular We implementAlgorithm 1 to solve problem 7 with different dimen-sions to test whether it can find the singular solution ornot

The results in Table 5 indicate that Algorithm 1 is alsoefficient to solve the singular system of equations

5 Preliminary Applicationin Compressed Sensing

In this section we will apply our algorithm to solve anengineering problem originated from compressed sensing ofsparse signals

Let 119860 isin 119877119898times119899 (119898 ≪ 119899) be a linear operator let 119909 isin 119877119899be a sparse or a nearly sparse original signal and let 119887 isin119877119898 be an observed value which satisfies the following linearequations

119887 = 119860119909 (65)

The original signal 119909 is desirable to be reconstructedfrom the linear equations 119860119909 = 119887 Unfortunately itis often that this linear equation is underdetermined orill-conditioned in practice and has infinitely many solu-tions From the fundamental principles of compressed sens-ing it is a reasonable approach to seek the sparsest oneamong all the solutions which contains the least nonzerocomponents

In [2] it was shown that the compressed sensing of sparsesignals can be formulated the following nonlinear system ofequations

min 119911 (119867 + 119863) 119911 + 119888 = 0 (66)

Mathematical Problems in Engineering 13

Table 2 Efficiency with random initial points

Problem Dim Method CPU-time Ni NfP1 100000 MPPRP-M 0489058 20 63

MPPRP-W 2005818 42 268MSDFPB 4565133 79 717

PRP 4757480 87 825MPRP 4561151 79 796TPRP 4631983 79 796DFPB1 4763311 82 806DFPB2 4610199 74 776MHS 4879031 79 796

P2 100000 MPPRP-M 0483632 20 63MPPRP-W 2251160 42 268MSDFPB 4375745 79 717

PRP 4561437 87 825MPRP 4830143 79 796TPRP 4560123 79 796DFPB1 4427558 82 806DFPB2 4550071 74 776MHS 4428999 79 796

P3 100000 MPPRP-M 1187410 20 63MPPRP-W 5215137 46 306MSDFPB 10136405 79 710

PRP 10808303 87 816MPRP 10124779 79 789TPRP 10068535 79 789DFPB1 10372113 82 799DFPB2 9458892 71 757MHS 10137711 79 789

P4 100000 MPPRP-M 9152258 253 1306MPPRP-W 11371204 260 1423MSDFPB 13374780 293 1837

PRP 8352463 170 1352MPRP 9223706 184 1474TPRP 14279675 302 2184DFPB1 15329979 336 2386DFPB2 11244725 231 1770MHS 17738408 216 2902

P5 2000 MPPRP-M 0019793 25 106MPPRP-W 7148874 4207 95776MSDFPB 16293374 10362 260980

PRP 16689685 10375 271376MPRP 16112906 10415 271538TPRP 16006655 10364 271349DFPB1 16272868 10385 271442DFPB2 17018004 10366 271355MHS 19200453 10360 271335

P6 2000 MPPRP-M 9722610 3415 15767MPPRP-W 8031480 3293 12678MSDFPB 8254000 3395 12524

PRP 9187302 3298 15178MPRP 9667492 3366 15503TPRP 9727760 3363 15708DFPB1 9565437 3431 15990DFPB2 9316936 3329 15590MHS 8893926 3393 14502

14 Mathematical Problems in Engineering

Table 3 Efficiency for 1000000-dimension problems

Problem Dim Method CPU-time Ni NfP1 1000000 MPPRP-M 4831098 22 69P2 1000000 MPPRP-M 5150450 22 69P3 1000000 MPPRP-M 13954404 26 81P4 1000000 MPPRP-M 8408822 19 120P5 1000000 MPPRP-M 9018623 36 154

Table 4 Average numerical efficiency and robustness of algorithms

Method (A-CT Std-CT) (A-Ni Std-Ni) (A-Nf Std-CT)MPPRP-M (11657 25775) (3307 76388) (15135 36523)MPPRP-W (25725 29258) (6894 119935) (92059 242337)MSDFPB (54010 56632) (124456 263861) ( 231436 663099)PRP (53178 55599) (124773 264115) (235830 676024)MPRP (53717 56475) (124336 265463) (235743 676492)TPRP (55309 57401) ( 124963 263694) (236334 675796)DFPB1 (58718 60331) (12759 263685) (237512 675685)DFPB2 (53586 56541) (122393 264211) (235434 676120)MHS (62549 67712) (12488 263778) (237192 675398)

where

119911 = [119906V]

119910 = 119860119879119887119888 = 1205911198902119899 + [minus119910119910 ]

119867 = [ 119860119879119860 minus119860119879119860minus119860119879119860 119860119879119860 ] 119863 = [0 1205751198641198990 0 ]

(67)

where 119906 isin 119877119899 V isin 119877119899 and 119906119894 = (119909119894)+ V119894 = (minus119909119894)+ forall 119894 = 1 2 119899 with (sdot)+ = max0 sdot and 1198902119899 is an 2119899-dimensional vector with all elements being one Clearly (66)is nonsmooth

Implement Algorithm 1 to solve the compressed sensingproblem of sparse signals with different compressed ratios Inthis experiment we consider a typical compressive sensingscenario where the goal is to reconstruct an 119899-length sparsesignal from 119898 observations We test the three algorithms(two popular algorithms CGD in [34] and SGCS in [35] andMPPRP-M) under three CS ratios (CS-R)119898119899 = 0125 02505 corresponding to different numbers of measurementswith 119899 = 2048 respectively The original sparse signal con-tains 32(64) randomly nonzero elements The measurement119887 is disturbed by noise ie 119887 = 119860119909 + 120596 where 120596 is theGaussian noise distributed as 119873(0 1205902119868) with 1205902 = 10minus4 119860is the Gaussian matrix generated by commend randn(119898 119899)

inMatlab The quality of restoration is measured by the meanof squared error (MSE) to the original signal 119909 that is

MSE = 1119899 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (68)

where 119909lowast is the restored signal The iterative process starts atthe measurement image ie 1199090 = 119860119879119887 and terminates whenthe relative change between successive iterates falls below10minus5 It says that 1003817100381710038171003817119891119896 minus 119891119896minus110038171003817100381710038171003817100381710038171003817119891119896minus11003817100381710038171003817 lt 10minus5 (69)

where 119891119896 = 1205911199091198961 + (12)119860119909119896 minus 11988722 and 120591 = 0005119860119879119887infinis chosen as suggested in [36] The other parameters in thisexperiment are same as those in the numerical experimentsconducted in Section 4 Numerical efficiency is shown inTable 6

Clearly from the results in Table 6 it follows that forany CS ratio MPPRP-M can recover the sparse signals moreefficiently without reduction of recovery quality (see theitalicized results in Table 6) If the sparsity level 32 is replacedby 64 in the 2048-length original signal the correspondingresults are also presented in Table 6 denoted by (sdot)6 Conclusions

In this paper we have presented a modified spectral PRPconjugate gradient derivative-free projection-based methodfor solving the large-scale nonlinear monotonic equationswhere the search direction is proved to be sufficiently descentfor any line search rule and the step length is chosen by aline search which can overcome the difficulty of choosing anappropriate weight

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

Hindawiwwwhindawicom Volume 2018

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Mathematical Problems in Engineering

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10 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfDFPB2 12734751 404 6040MHS 12425026 411 6059

P3 20000 MPPRP-M 0239615 23 72MPPRP-W 7505229 242 3126MSDFPB 18305460 472 7144

PRP 19195911 481 7169MPRP 19482789 472 7144TPRP 19011112 472 7144DFPB1 19165800 475 7151DFPB2 19428512 464 7120MHS 18928398 472 7144

P4 10000 MPPRP-M 0061549 16 102MPPRP-W 0113153 19 131MSDFPB 0289696 68 555

PRP 0187397 39 391MPRP 0233670 48 462TPRP 0358863 89 658DFPB1 0657537 185 1142DFPB2 0213256 40 422MHS 0414446 62 803

P4 20000 MPPRP-M 0125760 17 108MPPRP-W 0256772 31 203MSDFPB 0703061 79 730

PRP 0558050 52 565MPRP 0663444 59 635TPRP 0822848 100 833DFPB1 1442003 203 1352DFPB2 0565771 51 596MHS 0674281 60 724

P4 50000 MPPRP-M 0295365 17 108MPPRP-W 0961757 38 273MSDFPB 2972641 102 1109

PRP 2655601 73 943MPRP 2738925 81 1007TPRP 3209823 122 1207DFPB1 4783178 234 1771DFPB2 2644189 75 981MHS 6106110 158 2717

P4 100000 MPPRP-M 0768978 17 108MPPRP-W 3113887 45 357MSDFPB 10575505 129 1568

PRP 9164701 101 1406MPRP 9756561 108 1465TPRP 11151034 146 1651DFPB1 16812871 275 2301DFPB2 9292510 99 1423MHS 21792651 179 3305

P5 100 MPPRP-M 0012253 18 77MPPRP-W 0018540 62 555MSDFPB 0049702 129 1522

PRP 0033290 144 1558

Mathematical Problems in Engineering 11

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0033017 139 1560TPRP 0034758 128 1518DFPB1 0026313 150 1601DFPB2 0029466 131 1538MHS 0040165 133 1539

P5 500 MPPRP-M 0016447 22 93MPPRP-W 0231399 544 9066MSDFPB 0772128 1313 25232

PRP 0727796 1332 25274MPRP 0585951 1372 25394TPRP 0577368 1315 25238DFPB1 0578794 1337 25315DFPB2 0589438 1317 25247MHS 0595316 1315 25241

P5 1000 MPPRP-M 0017438 23 97MPPRP-W 1072332 1505 29740MSDFPB 3218997 3680 81797

PRP 2889154 3696 81825MPRP 2963704 3735 81946TPRP 2948045 3682 81802DFPB1 3040253 3704 81880DFPB2 3045603 3683 81803MHS 3220028 3678 81791

P5 2000 MPPRP-M 0021613 25 106MPPRP-W 5944706 4206 95732MSDFPB 17227153 10341 260347

PRP 16073363 10355 260379MPRP 16960973 10394 260490TPRP 16826907 10343 260352DFPB1 16257104 10364 260424DFPB2 16739385 10345 260356MHS 17503252 10339 260342

P6 100 MPPRP-M 0217224 1182 3717MPPRP-W 0193153 1148 3605MSDFPB 0237793 1186 3706

PRP 0220389 1207 3766MPRP 0224211 1186 3658TPRP 0248104 1195 3722DFPB1 0236804 1201 3750DFPB2 0235470 1170 3620MHS 0236250 1198 3720

P6 500 MPPRP-M 0944828 1348 5050MPPRP-W 0794555 1278 4482MSDFPB 0797832 1185 3968

PRP 0911934 1322 4267MPRP 0845336 1258 3983TPRP 0912343 1326 4335DFPB1 0903852 1321 4300DFPB2 1020228 1299 4143MHS 0918611 1307 4147

P6 1000 MPPRP-M 2337405 1495 6667MPPRP-W 1680275 1381 4951MSDFPB 1536904 1331 4320

12 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfPRP 1851282 1389 4553MPRP 1647356 1277 4040TPRP 1686418 1273 4109DFPB1 1792886 1360 4409DFPB2 1812678 1343 4330MHS 1775351 1341 4283

P6 2000 MPPRP-M 6738143 1737 11016MPPRP-W 3257538 1440 4936MSDFPB 3065903 1381 4524

PRP 3574894 1390 4522MPRP 3381889 1316 4247TPRP 3565741 1378 4510DFPB1 3609619 1377 4511DFPB2 3647662 984 3232MHS 3608315 1362 4397

DFPB2 the steepest descent derivative-free projection-basedmethods in [14] with 120579119896 in (62) being replaced by

120579119896 = 119865119879119896 120596119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 + (119865119879119896 119910119896minus1) 1003817100381710038171003817119910119896minus1100381710038171003817100381721003817100381710038171003817119865119896minus110038171003817100381710038174 (63)

MHS the MHS-PRP conjugate gradient derivative-freeprojection-based methods in [15]

From the results in Tables 1 2 and 3 it follows that ouralgorithm (MPPRP) outperforms the other seven algorithmsno matter how to choose the initial points (see the italicizedresults) Especially it seems to more efficiently solve large-scale test problems Actually Table 3 shows that MPPRP-Mcan solve the first five Problems with dimension of 1000000in less time compared with the other algorithms

In order to further measure the efficiency difference ofall the eight algorithms we calculate the average numberof iteration the average consumed CPU time and theirstandard deviations respectively In Table 4 A-Ni and Std-Ni stand for the average number of iteration and its stan-dard deviation respectively A-CT and Std-CT represent theaverage consumed CPU time and its standard deviationrespectively The average number of function evaluation andits standard deviation are denoted by A-Nf and Std-Nfrespectively Clearly Std-Ni Std-Nf and Std-CT can showrobustness of all the algorithms

The results in Table 4 demonstrate that both ofMPPRP-Mand MPPRP-W outperform the other seven algorithms

In the end of this section we use our algorithm to solvea singular system of equations The next test problem is amodified version of problem 1

Problem 20 The elements of 119865(119909) are given by119865119894 (119909) = 119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (64)

The initial point is fixed as 1199090 = (1 1)119879

Note that problem 7 is singular since zero is its solutionand the Jacobian matrix 1198651015840(0) is singular We implementAlgorithm 1 to solve problem 7 with different dimen-sions to test whether it can find the singular solution ornot

The results in Table 5 indicate that Algorithm 1 is alsoefficient to solve the singular system of equations

5 Preliminary Applicationin Compressed Sensing

In this section we will apply our algorithm to solve anengineering problem originated from compressed sensing ofsparse signals

Let 119860 isin 119877119898times119899 (119898 ≪ 119899) be a linear operator let 119909 isin 119877119899be a sparse or a nearly sparse original signal and let 119887 isin119877119898 be an observed value which satisfies the following linearequations

119887 = 119860119909 (65)

The original signal 119909 is desirable to be reconstructedfrom the linear equations 119860119909 = 119887 Unfortunately itis often that this linear equation is underdetermined orill-conditioned in practice and has infinitely many solu-tions From the fundamental principles of compressed sens-ing it is a reasonable approach to seek the sparsest oneamong all the solutions which contains the least nonzerocomponents

In [2] it was shown that the compressed sensing of sparsesignals can be formulated the following nonlinear system ofequations

min 119911 (119867 + 119863) 119911 + 119888 = 0 (66)

Mathematical Problems in Engineering 13

Table 2 Efficiency with random initial points

Problem Dim Method CPU-time Ni NfP1 100000 MPPRP-M 0489058 20 63

MPPRP-W 2005818 42 268MSDFPB 4565133 79 717

PRP 4757480 87 825MPRP 4561151 79 796TPRP 4631983 79 796DFPB1 4763311 82 806DFPB2 4610199 74 776MHS 4879031 79 796

P2 100000 MPPRP-M 0483632 20 63MPPRP-W 2251160 42 268MSDFPB 4375745 79 717

PRP 4561437 87 825MPRP 4830143 79 796TPRP 4560123 79 796DFPB1 4427558 82 806DFPB2 4550071 74 776MHS 4428999 79 796

P3 100000 MPPRP-M 1187410 20 63MPPRP-W 5215137 46 306MSDFPB 10136405 79 710

PRP 10808303 87 816MPRP 10124779 79 789TPRP 10068535 79 789DFPB1 10372113 82 799DFPB2 9458892 71 757MHS 10137711 79 789

P4 100000 MPPRP-M 9152258 253 1306MPPRP-W 11371204 260 1423MSDFPB 13374780 293 1837

PRP 8352463 170 1352MPRP 9223706 184 1474TPRP 14279675 302 2184DFPB1 15329979 336 2386DFPB2 11244725 231 1770MHS 17738408 216 2902

P5 2000 MPPRP-M 0019793 25 106MPPRP-W 7148874 4207 95776MSDFPB 16293374 10362 260980

PRP 16689685 10375 271376MPRP 16112906 10415 271538TPRP 16006655 10364 271349DFPB1 16272868 10385 271442DFPB2 17018004 10366 271355MHS 19200453 10360 271335

P6 2000 MPPRP-M 9722610 3415 15767MPPRP-W 8031480 3293 12678MSDFPB 8254000 3395 12524

PRP 9187302 3298 15178MPRP 9667492 3366 15503TPRP 9727760 3363 15708DFPB1 9565437 3431 15990DFPB2 9316936 3329 15590MHS 8893926 3393 14502

14 Mathematical Problems in Engineering

Table 3 Efficiency for 1000000-dimension problems

Problem Dim Method CPU-time Ni NfP1 1000000 MPPRP-M 4831098 22 69P2 1000000 MPPRP-M 5150450 22 69P3 1000000 MPPRP-M 13954404 26 81P4 1000000 MPPRP-M 8408822 19 120P5 1000000 MPPRP-M 9018623 36 154

Table 4 Average numerical efficiency and robustness of algorithms

Method (A-CT Std-CT) (A-Ni Std-Ni) (A-Nf Std-CT)MPPRP-M (11657 25775) (3307 76388) (15135 36523)MPPRP-W (25725 29258) (6894 119935) (92059 242337)MSDFPB (54010 56632) (124456 263861) ( 231436 663099)PRP (53178 55599) (124773 264115) (235830 676024)MPRP (53717 56475) (124336 265463) (235743 676492)TPRP (55309 57401) ( 124963 263694) (236334 675796)DFPB1 (58718 60331) (12759 263685) (237512 675685)DFPB2 (53586 56541) (122393 264211) (235434 676120)MHS (62549 67712) (12488 263778) (237192 675398)

where

119911 = [119906V]

119910 = 119860119879119887119888 = 1205911198902119899 + [minus119910119910 ]

119867 = [ 119860119879119860 minus119860119879119860minus119860119879119860 119860119879119860 ] 119863 = [0 1205751198641198990 0 ]

(67)

where 119906 isin 119877119899 V isin 119877119899 and 119906119894 = (119909119894)+ V119894 = (minus119909119894)+ forall 119894 = 1 2 119899 with (sdot)+ = max0 sdot and 1198902119899 is an 2119899-dimensional vector with all elements being one Clearly (66)is nonsmooth

Implement Algorithm 1 to solve the compressed sensingproblem of sparse signals with different compressed ratios Inthis experiment we consider a typical compressive sensingscenario where the goal is to reconstruct an 119899-length sparsesignal from 119898 observations We test the three algorithms(two popular algorithms CGD in [34] and SGCS in [35] andMPPRP-M) under three CS ratios (CS-R)119898119899 = 0125 02505 corresponding to different numbers of measurementswith 119899 = 2048 respectively The original sparse signal con-tains 32(64) randomly nonzero elements The measurement119887 is disturbed by noise ie 119887 = 119860119909 + 120596 where 120596 is theGaussian noise distributed as 119873(0 1205902119868) with 1205902 = 10minus4 119860is the Gaussian matrix generated by commend randn(119898 119899)

inMatlab The quality of restoration is measured by the meanof squared error (MSE) to the original signal 119909 that is

MSE = 1119899 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (68)

where 119909lowast is the restored signal The iterative process starts atthe measurement image ie 1199090 = 119860119879119887 and terminates whenthe relative change between successive iterates falls below10minus5 It says that 1003817100381710038171003817119891119896 minus 119891119896minus110038171003817100381710038171003817100381710038171003817119891119896minus11003817100381710038171003817 lt 10minus5 (69)

where 119891119896 = 1205911199091198961 + (12)119860119909119896 minus 11988722 and 120591 = 0005119860119879119887infinis chosen as suggested in [36] The other parameters in thisexperiment are same as those in the numerical experimentsconducted in Section 4 Numerical efficiency is shown inTable 6

Clearly from the results in Table 6 it follows that forany CS ratio MPPRP-M can recover the sparse signals moreefficiently without reduction of recovery quality (see theitalicized results in Table 6) If the sparsity level 32 is replacedby 64 in the 2048-length original signal the correspondingresults are also presented in Table 6 denoted by (sdot)6 Conclusions

In this paper we have presented a modified spectral PRPconjugate gradient derivative-free projection-based methodfor solving the large-scale nonlinear monotonic equationswhere the search direction is proved to be sufficiently descentfor any line search rule and the step length is chosen by aline search which can overcome the difficulty of choosing anappropriate weight

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 11

Table 1 Continued

Problem Dim Method CPU-time Ni NfMPRP 0033017 139 1560TPRP 0034758 128 1518DFPB1 0026313 150 1601DFPB2 0029466 131 1538MHS 0040165 133 1539

P5 500 MPPRP-M 0016447 22 93MPPRP-W 0231399 544 9066MSDFPB 0772128 1313 25232

PRP 0727796 1332 25274MPRP 0585951 1372 25394TPRP 0577368 1315 25238DFPB1 0578794 1337 25315DFPB2 0589438 1317 25247MHS 0595316 1315 25241

P5 1000 MPPRP-M 0017438 23 97MPPRP-W 1072332 1505 29740MSDFPB 3218997 3680 81797

PRP 2889154 3696 81825MPRP 2963704 3735 81946TPRP 2948045 3682 81802DFPB1 3040253 3704 81880DFPB2 3045603 3683 81803MHS 3220028 3678 81791

P5 2000 MPPRP-M 0021613 25 106MPPRP-W 5944706 4206 95732MSDFPB 17227153 10341 260347

PRP 16073363 10355 260379MPRP 16960973 10394 260490TPRP 16826907 10343 260352DFPB1 16257104 10364 260424DFPB2 16739385 10345 260356MHS 17503252 10339 260342

P6 100 MPPRP-M 0217224 1182 3717MPPRP-W 0193153 1148 3605MSDFPB 0237793 1186 3706

PRP 0220389 1207 3766MPRP 0224211 1186 3658TPRP 0248104 1195 3722DFPB1 0236804 1201 3750DFPB2 0235470 1170 3620MHS 0236250 1198 3720

P6 500 MPPRP-M 0944828 1348 5050MPPRP-W 0794555 1278 4482MSDFPB 0797832 1185 3968

PRP 0911934 1322 4267MPRP 0845336 1258 3983TPRP 0912343 1326 4335DFPB1 0903852 1321 4300DFPB2 1020228 1299 4143MHS 0918611 1307 4147

P6 1000 MPPRP-M 2337405 1495 6667MPPRP-W 1680275 1381 4951MSDFPB 1536904 1331 4320

12 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfPRP 1851282 1389 4553MPRP 1647356 1277 4040TPRP 1686418 1273 4109DFPB1 1792886 1360 4409DFPB2 1812678 1343 4330MHS 1775351 1341 4283

P6 2000 MPPRP-M 6738143 1737 11016MPPRP-W 3257538 1440 4936MSDFPB 3065903 1381 4524

PRP 3574894 1390 4522MPRP 3381889 1316 4247TPRP 3565741 1378 4510DFPB1 3609619 1377 4511DFPB2 3647662 984 3232MHS 3608315 1362 4397

DFPB2 the steepest descent derivative-free projection-basedmethods in [14] with 120579119896 in (62) being replaced by

120579119896 = 119865119879119896 120596119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 + (119865119879119896 119910119896minus1) 1003817100381710038171003817119910119896minus1100381710038171003817100381721003817100381710038171003817119865119896minus110038171003817100381710038174 (63)

MHS the MHS-PRP conjugate gradient derivative-freeprojection-based methods in [15]

From the results in Tables 1 2 and 3 it follows that ouralgorithm (MPPRP) outperforms the other seven algorithmsno matter how to choose the initial points (see the italicizedresults) Especially it seems to more efficiently solve large-scale test problems Actually Table 3 shows that MPPRP-Mcan solve the first five Problems with dimension of 1000000in less time compared with the other algorithms

In order to further measure the efficiency difference ofall the eight algorithms we calculate the average numberof iteration the average consumed CPU time and theirstandard deviations respectively In Table 4 A-Ni and Std-Ni stand for the average number of iteration and its stan-dard deviation respectively A-CT and Std-CT represent theaverage consumed CPU time and its standard deviationrespectively The average number of function evaluation andits standard deviation are denoted by A-Nf and Std-Nfrespectively Clearly Std-Ni Std-Nf and Std-CT can showrobustness of all the algorithms

The results in Table 4 demonstrate that both ofMPPRP-Mand MPPRP-W outperform the other seven algorithms

In the end of this section we use our algorithm to solvea singular system of equations The next test problem is amodified version of problem 1

Problem 20 The elements of 119865(119909) are given by119865119894 (119909) = 119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (64)

The initial point is fixed as 1199090 = (1 1)119879

Note that problem 7 is singular since zero is its solutionand the Jacobian matrix 1198651015840(0) is singular We implementAlgorithm 1 to solve problem 7 with different dimen-sions to test whether it can find the singular solution ornot

The results in Table 5 indicate that Algorithm 1 is alsoefficient to solve the singular system of equations

5 Preliminary Applicationin Compressed Sensing

In this section we will apply our algorithm to solve anengineering problem originated from compressed sensing ofsparse signals

Let 119860 isin 119877119898times119899 (119898 ≪ 119899) be a linear operator let 119909 isin 119877119899be a sparse or a nearly sparse original signal and let 119887 isin119877119898 be an observed value which satisfies the following linearequations

119887 = 119860119909 (65)

The original signal 119909 is desirable to be reconstructedfrom the linear equations 119860119909 = 119887 Unfortunately itis often that this linear equation is underdetermined orill-conditioned in practice and has infinitely many solu-tions From the fundamental principles of compressed sens-ing it is a reasonable approach to seek the sparsest oneamong all the solutions which contains the least nonzerocomponents

In [2] it was shown that the compressed sensing of sparsesignals can be formulated the following nonlinear system ofequations

min 119911 (119867 + 119863) 119911 + 119888 = 0 (66)

Mathematical Problems in Engineering 13

Table 2 Efficiency with random initial points

Problem Dim Method CPU-time Ni NfP1 100000 MPPRP-M 0489058 20 63

MPPRP-W 2005818 42 268MSDFPB 4565133 79 717

PRP 4757480 87 825MPRP 4561151 79 796TPRP 4631983 79 796DFPB1 4763311 82 806DFPB2 4610199 74 776MHS 4879031 79 796

P2 100000 MPPRP-M 0483632 20 63MPPRP-W 2251160 42 268MSDFPB 4375745 79 717

PRP 4561437 87 825MPRP 4830143 79 796TPRP 4560123 79 796DFPB1 4427558 82 806DFPB2 4550071 74 776MHS 4428999 79 796

P3 100000 MPPRP-M 1187410 20 63MPPRP-W 5215137 46 306MSDFPB 10136405 79 710

PRP 10808303 87 816MPRP 10124779 79 789TPRP 10068535 79 789DFPB1 10372113 82 799DFPB2 9458892 71 757MHS 10137711 79 789

P4 100000 MPPRP-M 9152258 253 1306MPPRP-W 11371204 260 1423MSDFPB 13374780 293 1837

PRP 8352463 170 1352MPRP 9223706 184 1474TPRP 14279675 302 2184DFPB1 15329979 336 2386DFPB2 11244725 231 1770MHS 17738408 216 2902

P5 2000 MPPRP-M 0019793 25 106MPPRP-W 7148874 4207 95776MSDFPB 16293374 10362 260980

PRP 16689685 10375 271376MPRP 16112906 10415 271538TPRP 16006655 10364 271349DFPB1 16272868 10385 271442DFPB2 17018004 10366 271355MHS 19200453 10360 271335

P6 2000 MPPRP-M 9722610 3415 15767MPPRP-W 8031480 3293 12678MSDFPB 8254000 3395 12524

PRP 9187302 3298 15178MPRP 9667492 3366 15503TPRP 9727760 3363 15708DFPB1 9565437 3431 15990DFPB2 9316936 3329 15590MHS 8893926 3393 14502

14 Mathematical Problems in Engineering

Table 3 Efficiency for 1000000-dimension problems

Problem Dim Method CPU-time Ni NfP1 1000000 MPPRP-M 4831098 22 69P2 1000000 MPPRP-M 5150450 22 69P3 1000000 MPPRP-M 13954404 26 81P4 1000000 MPPRP-M 8408822 19 120P5 1000000 MPPRP-M 9018623 36 154

Table 4 Average numerical efficiency and robustness of algorithms

Method (A-CT Std-CT) (A-Ni Std-Ni) (A-Nf Std-CT)MPPRP-M (11657 25775) (3307 76388) (15135 36523)MPPRP-W (25725 29258) (6894 119935) (92059 242337)MSDFPB (54010 56632) (124456 263861) ( 231436 663099)PRP (53178 55599) (124773 264115) (235830 676024)MPRP (53717 56475) (124336 265463) (235743 676492)TPRP (55309 57401) ( 124963 263694) (236334 675796)DFPB1 (58718 60331) (12759 263685) (237512 675685)DFPB2 (53586 56541) (122393 264211) (235434 676120)MHS (62549 67712) (12488 263778) (237192 675398)

where

119911 = [119906V]

119910 = 119860119879119887119888 = 1205911198902119899 + [minus119910119910 ]

119867 = [ 119860119879119860 minus119860119879119860minus119860119879119860 119860119879119860 ] 119863 = [0 1205751198641198990 0 ]

(67)

where 119906 isin 119877119899 V isin 119877119899 and 119906119894 = (119909119894)+ V119894 = (minus119909119894)+ forall 119894 = 1 2 119899 with (sdot)+ = max0 sdot and 1198902119899 is an 2119899-dimensional vector with all elements being one Clearly (66)is nonsmooth

Implement Algorithm 1 to solve the compressed sensingproblem of sparse signals with different compressed ratios Inthis experiment we consider a typical compressive sensingscenario where the goal is to reconstruct an 119899-length sparsesignal from 119898 observations We test the three algorithms(two popular algorithms CGD in [34] and SGCS in [35] andMPPRP-M) under three CS ratios (CS-R)119898119899 = 0125 02505 corresponding to different numbers of measurementswith 119899 = 2048 respectively The original sparse signal con-tains 32(64) randomly nonzero elements The measurement119887 is disturbed by noise ie 119887 = 119860119909 + 120596 where 120596 is theGaussian noise distributed as 119873(0 1205902119868) with 1205902 = 10minus4 119860is the Gaussian matrix generated by commend randn(119898 119899)

inMatlab The quality of restoration is measured by the meanof squared error (MSE) to the original signal 119909 that is

MSE = 1119899 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (68)

where 119909lowast is the restored signal The iterative process starts atthe measurement image ie 1199090 = 119860119879119887 and terminates whenthe relative change between successive iterates falls below10minus5 It says that 1003817100381710038171003817119891119896 minus 119891119896minus110038171003817100381710038171003817100381710038171003817119891119896minus11003817100381710038171003817 lt 10minus5 (69)

where 119891119896 = 1205911199091198961 + (12)119860119909119896 minus 11988722 and 120591 = 0005119860119879119887infinis chosen as suggested in [36] The other parameters in thisexperiment are same as those in the numerical experimentsconducted in Section 4 Numerical efficiency is shown inTable 6

Clearly from the results in Table 6 it follows that forany CS ratio MPPRP-M can recover the sparse signals moreefficiently without reduction of recovery quality (see theitalicized results in Table 6) If the sparsity level 32 is replacedby 64 in the 2048-length original signal the correspondingresults are also presented in Table 6 denoted by (sdot)6 Conclusions

In this paper we have presented a modified spectral PRPconjugate gradient derivative-free projection-based methodfor solving the large-scale nonlinear monotonic equationswhere the search direction is proved to be sufficiently descentfor any line search rule and the step length is chosen by aline search which can overcome the difficulty of choosing anappropriate weight

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

12 Mathematical Problems in Engineering

Table 1 Continued

Problem Dim Method CPU-time Ni NfPRP 1851282 1389 4553MPRP 1647356 1277 4040TPRP 1686418 1273 4109DFPB1 1792886 1360 4409DFPB2 1812678 1343 4330MHS 1775351 1341 4283

P6 2000 MPPRP-M 6738143 1737 11016MPPRP-W 3257538 1440 4936MSDFPB 3065903 1381 4524

PRP 3574894 1390 4522MPRP 3381889 1316 4247TPRP 3565741 1378 4510DFPB1 3609619 1377 4511DFPB2 3647662 984 3232MHS 3608315 1362 4397

DFPB2 the steepest descent derivative-free projection-basedmethods in [14] with 120579119896 in (62) being replaced by

120579119896 = 119865119879119896 120596119896minus11003817100381710038171003817119865119896minus110038171003817100381710038172 + (119865119879119896 119910119896minus1) 1003817100381710038171003817119910119896minus1100381710038171003817100381721003817100381710038171003817119865119896minus110038171003817100381710038174 (63)

MHS the MHS-PRP conjugate gradient derivative-freeprojection-based methods in [15]

From the results in Tables 1 2 and 3 it follows that ouralgorithm (MPPRP) outperforms the other seven algorithmsno matter how to choose the initial points (see the italicizedresults) Especially it seems to more efficiently solve large-scale test problems Actually Table 3 shows that MPPRP-Mcan solve the first five Problems with dimension of 1000000in less time compared with the other algorithms

In order to further measure the efficiency difference ofall the eight algorithms we calculate the average numberof iteration the average consumed CPU time and theirstandard deviations respectively In Table 4 A-Ni and Std-Ni stand for the average number of iteration and its stan-dard deviation respectively A-CT and Std-CT represent theaverage consumed CPU time and its standard deviationrespectively The average number of function evaluation andits standard deviation are denoted by A-Nf and Std-Nfrespectively Clearly Std-Ni Std-Nf and Std-CT can showrobustness of all the algorithms

The results in Table 4 demonstrate that both ofMPPRP-Mand MPPRP-W outperform the other seven algorithms

In the end of this section we use our algorithm to solvea singular system of equations The next test problem is amodified version of problem 1

Problem 20 The elements of 119865(119909) are given by119865119894 (119909) = 119909119894 minus sin (119909119894) 119894 = 1 2 119899 minus 1 (64)

The initial point is fixed as 1199090 = (1 1)119879

Note that problem 7 is singular since zero is its solutionand the Jacobian matrix 1198651015840(0) is singular We implementAlgorithm 1 to solve problem 7 with different dimen-sions to test whether it can find the singular solution ornot

The results in Table 5 indicate that Algorithm 1 is alsoefficient to solve the singular system of equations

5 Preliminary Applicationin Compressed Sensing

In this section we will apply our algorithm to solve anengineering problem originated from compressed sensing ofsparse signals

Let 119860 isin 119877119898times119899 (119898 ≪ 119899) be a linear operator let 119909 isin 119877119899be a sparse or a nearly sparse original signal and let 119887 isin119877119898 be an observed value which satisfies the following linearequations

119887 = 119860119909 (65)

The original signal 119909 is desirable to be reconstructedfrom the linear equations 119860119909 = 119887 Unfortunately itis often that this linear equation is underdetermined orill-conditioned in practice and has infinitely many solu-tions From the fundamental principles of compressed sens-ing it is a reasonable approach to seek the sparsest oneamong all the solutions which contains the least nonzerocomponents

In [2] it was shown that the compressed sensing of sparsesignals can be formulated the following nonlinear system ofequations

min 119911 (119867 + 119863) 119911 + 119888 = 0 (66)

Mathematical Problems in Engineering 13

Table 2 Efficiency with random initial points

Problem Dim Method CPU-time Ni NfP1 100000 MPPRP-M 0489058 20 63

MPPRP-W 2005818 42 268MSDFPB 4565133 79 717

PRP 4757480 87 825MPRP 4561151 79 796TPRP 4631983 79 796DFPB1 4763311 82 806DFPB2 4610199 74 776MHS 4879031 79 796

P2 100000 MPPRP-M 0483632 20 63MPPRP-W 2251160 42 268MSDFPB 4375745 79 717

PRP 4561437 87 825MPRP 4830143 79 796TPRP 4560123 79 796DFPB1 4427558 82 806DFPB2 4550071 74 776MHS 4428999 79 796

P3 100000 MPPRP-M 1187410 20 63MPPRP-W 5215137 46 306MSDFPB 10136405 79 710

PRP 10808303 87 816MPRP 10124779 79 789TPRP 10068535 79 789DFPB1 10372113 82 799DFPB2 9458892 71 757MHS 10137711 79 789

P4 100000 MPPRP-M 9152258 253 1306MPPRP-W 11371204 260 1423MSDFPB 13374780 293 1837

PRP 8352463 170 1352MPRP 9223706 184 1474TPRP 14279675 302 2184DFPB1 15329979 336 2386DFPB2 11244725 231 1770MHS 17738408 216 2902

P5 2000 MPPRP-M 0019793 25 106MPPRP-W 7148874 4207 95776MSDFPB 16293374 10362 260980

PRP 16689685 10375 271376MPRP 16112906 10415 271538TPRP 16006655 10364 271349DFPB1 16272868 10385 271442DFPB2 17018004 10366 271355MHS 19200453 10360 271335

P6 2000 MPPRP-M 9722610 3415 15767MPPRP-W 8031480 3293 12678MSDFPB 8254000 3395 12524

PRP 9187302 3298 15178MPRP 9667492 3366 15503TPRP 9727760 3363 15708DFPB1 9565437 3431 15990DFPB2 9316936 3329 15590MHS 8893926 3393 14502

14 Mathematical Problems in Engineering

Table 3 Efficiency for 1000000-dimension problems

Problem Dim Method CPU-time Ni NfP1 1000000 MPPRP-M 4831098 22 69P2 1000000 MPPRP-M 5150450 22 69P3 1000000 MPPRP-M 13954404 26 81P4 1000000 MPPRP-M 8408822 19 120P5 1000000 MPPRP-M 9018623 36 154

Table 4 Average numerical efficiency and robustness of algorithms

Method (A-CT Std-CT) (A-Ni Std-Ni) (A-Nf Std-CT)MPPRP-M (11657 25775) (3307 76388) (15135 36523)MPPRP-W (25725 29258) (6894 119935) (92059 242337)MSDFPB (54010 56632) (124456 263861) ( 231436 663099)PRP (53178 55599) (124773 264115) (235830 676024)MPRP (53717 56475) (124336 265463) (235743 676492)TPRP (55309 57401) ( 124963 263694) (236334 675796)DFPB1 (58718 60331) (12759 263685) (237512 675685)DFPB2 (53586 56541) (122393 264211) (235434 676120)MHS (62549 67712) (12488 263778) (237192 675398)

where

119911 = [119906V]

119910 = 119860119879119887119888 = 1205911198902119899 + [minus119910119910 ]

119867 = [ 119860119879119860 minus119860119879119860minus119860119879119860 119860119879119860 ] 119863 = [0 1205751198641198990 0 ]

(67)

where 119906 isin 119877119899 V isin 119877119899 and 119906119894 = (119909119894)+ V119894 = (minus119909119894)+ forall 119894 = 1 2 119899 with (sdot)+ = max0 sdot and 1198902119899 is an 2119899-dimensional vector with all elements being one Clearly (66)is nonsmooth

Implement Algorithm 1 to solve the compressed sensingproblem of sparse signals with different compressed ratios Inthis experiment we consider a typical compressive sensingscenario where the goal is to reconstruct an 119899-length sparsesignal from 119898 observations We test the three algorithms(two popular algorithms CGD in [34] and SGCS in [35] andMPPRP-M) under three CS ratios (CS-R)119898119899 = 0125 02505 corresponding to different numbers of measurementswith 119899 = 2048 respectively The original sparse signal con-tains 32(64) randomly nonzero elements The measurement119887 is disturbed by noise ie 119887 = 119860119909 + 120596 where 120596 is theGaussian noise distributed as 119873(0 1205902119868) with 1205902 = 10minus4 119860is the Gaussian matrix generated by commend randn(119898 119899)

inMatlab The quality of restoration is measured by the meanof squared error (MSE) to the original signal 119909 that is

MSE = 1119899 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (68)

where 119909lowast is the restored signal The iterative process starts atthe measurement image ie 1199090 = 119860119879119887 and terminates whenthe relative change between successive iterates falls below10minus5 It says that 1003817100381710038171003817119891119896 minus 119891119896minus110038171003817100381710038171003817100381710038171003817119891119896minus11003817100381710038171003817 lt 10minus5 (69)

where 119891119896 = 1205911199091198961 + (12)119860119909119896 minus 11988722 and 120591 = 0005119860119879119887infinis chosen as suggested in [36] The other parameters in thisexperiment are same as those in the numerical experimentsconducted in Section 4 Numerical efficiency is shown inTable 6

Clearly from the results in Table 6 it follows that forany CS ratio MPPRP-M can recover the sparse signals moreefficiently without reduction of recovery quality (see theitalicized results in Table 6) If the sparsity level 32 is replacedby 64 in the 2048-length original signal the correspondingresults are also presented in Table 6 denoted by (sdot)6 Conclusions

In this paper we have presented a modified spectral PRPconjugate gradient derivative-free projection-based methodfor solving the large-scale nonlinear monotonic equationswhere the search direction is proved to be sufficiently descentfor any line search rule and the step length is chosen by aline search which can overcome the difficulty of choosing anappropriate weight

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

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Mathematical Problems in Engineering 13

Table 2 Efficiency with random initial points

Problem Dim Method CPU-time Ni NfP1 100000 MPPRP-M 0489058 20 63

MPPRP-W 2005818 42 268MSDFPB 4565133 79 717

PRP 4757480 87 825MPRP 4561151 79 796TPRP 4631983 79 796DFPB1 4763311 82 806DFPB2 4610199 74 776MHS 4879031 79 796

P2 100000 MPPRP-M 0483632 20 63MPPRP-W 2251160 42 268MSDFPB 4375745 79 717

PRP 4561437 87 825MPRP 4830143 79 796TPRP 4560123 79 796DFPB1 4427558 82 806DFPB2 4550071 74 776MHS 4428999 79 796

P3 100000 MPPRP-M 1187410 20 63MPPRP-W 5215137 46 306MSDFPB 10136405 79 710

PRP 10808303 87 816MPRP 10124779 79 789TPRP 10068535 79 789DFPB1 10372113 82 799DFPB2 9458892 71 757MHS 10137711 79 789

P4 100000 MPPRP-M 9152258 253 1306MPPRP-W 11371204 260 1423MSDFPB 13374780 293 1837

PRP 8352463 170 1352MPRP 9223706 184 1474TPRP 14279675 302 2184DFPB1 15329979 336 2386DFPB2 11244725 231 1770MHS 17738408 216 2902

P5 2000 MPPRP-M 0019793 25 106MPPRP-W 7148874 4207 95776MSDFPB 16293374 10362 260980

PRP 16689685 10375 271376MPRP 16112906 10415 271538TPRP 16006655 10364 271349DFPB1 16272868 10385 271442DFPB2 17018004 10366 271355MHS 19200453 10360 271335

P6 2000 MPPRP-M 9722610 3415 15767MPPRP-W 8031480 3293 12678MSDFPB 8254000 3395 12524

PRP 9187302 3298 15178MPRP 9667492 3366 15503TPRP 9727760 3363 15708DFPB1 9565437 3431 15990DFPB2 9316936 3329 15590MHS 8893926 3393 14502

14 Mathematical Problems in Engineering

Table 3 Efficiency for 1000000-dimension problems

Problem Dim Method CPU-time Ni NfP1 1000000 MPPRP-M 4831098 22 69P2 1000000 MPPRP-M 5150450 22 69P3 1000000 MPPRP-M 13954404 26 81P4 1000000 MPPRP-M 8408822 19 120P5 1000000 MPPRP-M 9018623 36 154

Table 4 Average numerical efficiency and robustness of algorithms

Method (A-CT Std-CT) (A-Ni Std-Ni) (A-Nf Std-CT)MPPRP-M (11657 25775) (3307 76388) (15135 36523)MPPRP-W (25725 29258) (6894 119935) (92059 242337)MSDFPB (54010 56632) (124456 263861) ( 231436 663099)PRP (53178 55599) (124773 264115) (235830 676024)MPRP (53717 56475) (124336 265463) (235743 676492)TPRP (55309 57401) ( 124963 263694) (236334 675796)DFPB1 (58718 60331) (12759 263685) (237512 675685)DFPB2 (53586 56541) (122393 264211) (235434 676120)MHS (62549 67712) (12488 263778) (237192 675398)

where

119911 = [119906V]

119910 = 119860119879119887119888 = 1205911198902119899 + [minus119910119910 ]

119867 = [ 119860119879119860 minus119860119879119860minus119860119879119860 119860119879119860 ] 119863 = [0 1205751198641198990 0 ]

(67)

where 119906 isin 119877119899 V isin 119877119899 and 119906119894 = (119909119894)+ V119894 = (minus119909119894)+ forall 119894 = 1 2 119899 with (sdot)+ = max0 sdot and 1198902119899 is an 2119899-dimensional vector with all elements being one Clearly (66)is nonsmooth

Implement Algorithm 1 to solve the compressed sensingproblem of sparse signals with different compressed ratios Inthis experiment we consider a typical compressive sensingscenario where the goal is to reconstruct an 119899-length sparsesignal from 119898 observations We test the three algorithms(two popular algorithms CGD in [34] and SGCS in [35] andMPPRP-M) under three CS ratios (CS-R)119898119899 = 0125 02505 corresponding to different numbers of measurementswith 119899 = 2048 respectively The original sparse signal con-tains 32(64) randomly nonzero elements The measurement119887 is disturbed by noise ie 119887 = 119860119909 + 120596 where 120596 is theGaussian noise distributed as 119873(0 1205902119868) with 1205902 = 10minus4 119860is the Gaussian matrix generated by commend randn(119898 119899)

inMatlab The quality of restoration is measured by the meanof squared error (MSE) to the original signal 119909 that is

MSE = 1119899 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (68)

where 119909lowast is the restored signal The iterative process starts atthe measurement image ie 1199090 = 119860119879119887 and terminates whenthe relative change between successive iterates falls below10minus5 It says that 1003817100381710038171003817119891119896 minus 119891119896minus110038171003817100381710038171003817100381710038171003817119891119896minus11003817100381710038171003817 lt 10minus5 (69)

where 119891119896 = 1205911199091198961 + (12)119860119909119896 minus 11988722 and 120591 = 0005119860119879119887infinis chosen as suggested in [36] The other parameters in thisexperiment are same as those in the numerical experimentsconducted in Section 4 Numerical efficiency is shown inTable 6

Clearly from the results in Table 6 it follows that forany CS ratio MPPRP-M can recover the sparse signals moreefficiently without reduction of recovery quality (see theitalicized results in Table 6) If the sparsity level 32 is replacedby 64 in the 2048-length original signal the correspondingresults are also presented in Table 6 denoted by (sdot)6 Conclusions

In this paper we have presented a modified spectral PRPconjugate gradient derivative-free projection-based methodfor solving the large-scale nonlinear monotonic equationswhere the search direction is proved to be sufficiently descentfor any line search rule and the step length is chosen by aline search which can overcome the difficulty of choosing anappropriate weight

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

14 Mathematical Problems in Engineering

Table 3 Efficiency for 1000000-dimension problems

Problem Dim Method CPU-time Ni NfP1 1000000 MPPRP-M 4831098 22 69P2 1000000 MPPRP-M 5150450 22 69P3 1000000 MPPRP-M 13954404 26 81P4 1000000 MPPRP-M 8408822 19 120P5 1000000 MPPRP-M 9018623 36 154

Table 4 Average numerical efficiency and robustness of algorithms

Method (A-CT Std-CT) (A-Ni Std-Ni) (A-Nf Std-CT)MPPRP-M (11657 25775) (3307 76388) (15135 36523)MPPRP-W (25725 29258) (6894 119935) (92059 242337)MSDFPB (54010 56632) (124456 263861) ( 231436 663099)PRP (53178 55599) (124773 264115) (235830 676024)MPRP (53717 56475) (124336 265463) (235743 676492)TPRP (55309 57401) ( 124963 263694) (236334 675796)DFPB1 (58718 60331) (12759 263685) (237512 675685)DFPB2 (53586 56541) (122393 264211) (235434 676120)MHS (62549 67712) (12488 263778) (237192 675398)

where

119911 = [119906V]

119910 = 119860119879119887119888 = 1205911198902119899 + [minus119910119910 ]

119867 = [ 119860119879119860 minus119860119879119860minus119860119879119860 119860119879119860 ] 119863 = [0 1205751198641198990 0 ]

(67)

where 119906 isin 119877119899 V isin 119877119899 and 119906119894 = (119909119894)+ V119894 = (minus119909119894)+ forall 119894 = 1 2 119899 with (sdot)+ = max0 sdot and 1198902119899 is an 2119899-dimensional vector with all elements being one Clearly (66)is nonsmooth

Implement Algorithm 1 to solve the compressed sensingproblem of sparse signals with different compressed ratios Inthis experiment we consider a typical compressive sensingscenario where the goal is to reconstruct an 119899-length sparsesignal from 119898 observations We test the three algorithms(two popular algorithms CGD in [34] and SGCS in [35] andMPPRP-M) under three CS ratios (CS-R)119898119899 = 0125 02505 corresponding to different numbers of measurementswith 119899 = 2048 respectively The original sparse signal con-tains 32(64) randomly nonzero elements The measurement119887 is disturbed by noise ie 119887 = 119860119909 + 120596 where 120596 is theGaussian noise distributed as 119873(0 1205902119868) with 1205902 = 10minus4 119860is the Gaussian matrix generated by commend randn(119898 119899)

inMatlab The quality of restoration is measured by the meanof squared error (MSE) to the original signal 119909 that is

MSE = 1119899 1003817100381710038171003817119909 minus 119909lowast10038171003817100381710038172 (68)

where 119909lowast is the restored signal The iterative process starts atthe measurement image ie 1199090 = 119860119879119887 and terminates whenthe relative change between successive iterates falls below10minus5 It says that 1003817100381710038171003817119891119896 minus 119891119896minus110038171003817100381710038171003817100381710038171003817119891119896minus11003817100381710038171003817 lt 10minus5 (69)

where 119891119896 = 1205911199091198961 + (12)119860119909119896 minus 11988722 and 120591 = 0005119860119879119887infinis chosen as suggested in [36] The other parameters in thisexperiment are same as those in the numerical experimentsconducted in Section 4 Numerical efficiency is shown inTable 6

Clearly from the results in Table 6 it follows that forany CS ratio MPPRP-M can recover the sparse signals moreefficiently without reduction of recovery quality (see theitalicized results in Table 6) If the sparsity level 32 is replacedby 64 in the 2048-length original signal the correspondingresults are also presented in Table 6 denoted by (sdot)6 Conclusions

In this paper we have presented a modified spectral PRPconjugate gradient derivative-free projection-based methodfor solving the large-scale nonlinear monotonic equationswhere the search direction is proved to be sufficiently descentfor any line search rule and the step length is chosen by aline search which can overcome the difficulty of choosing anappropriate weight

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 15

Table 5 Numerical performance in singular case

Problem Dim Method CPU-time Ni NfP7 500 MPPRP-M 0907980 6684 20055

MPPRP-W 0823281 6684 20055MSDFPB 1063594 6684 20055

PRP 0801892 6695 20088MPRP 0856992 6684 20055TPRP 0892188 6684 20055DFPB1 0906388 6690 20073DFPB2 0894474 6684 20055MHS 1045179 6684 20055

P7 1000 MPPRP-M 1742640 8424 25275MPPRP-W 1787419 8424 25275MSDFPB 2140070 8424 25275

PRP 1773320 8435 25308MPRP 1878275 8424 25275TPRP 1856212 8424 25275DFPB1 1855210 8430 25293DFPB2 1869769 8424 25275MHS 2085814 8424 25275

P7 2000 MPPRP-M 3712167 10616 31851MPPRP-W 4182644 10616 31851MSDFPB 4622078 10617 31855

PRP 4465109 10628 31888MPRP 4716474 10617 31855TPRP 5772379 10617 31855DFPB1 5357838 10622 31870DFPB2 4663553 10616 31852MHS 5737888 10617 31855

P7 5000 MPPRP-M 20734981 14412 43239MPPRP-W 21112604 14413 43243MSDFPB 23963617 14414 43252

PRP 22231907 14425 43283MPRP 23381419 14414 43252TPRP 23871918 14414 43252DFPB1 23222197 14420 43269DFPB2 23300951 14414 43252MHS 25432421 14414 43252

Table 6 Recovering sparse signals under different CS ratios

CS-R CGD SGCS MPPRP-MIter Time MSE Iter Time MSE Iter Time MSE

0125 1047 497s 796e-006 752 344s 440e-006 607 258s 418e-006(985) (486s) (511e-003) (1018) (419s) (429e-003) 886 (363s) (416e-003)

025 263 205s 287e-006 241 192s 275e-006 178 138s 264e-006(425) (334s) (712e-006) (359) (284s) (570e-006) (274 ) ( 220s) ( 554e-006)

05 97 133s 246e-006 135 175s 159e-006 85 116s 154e-006(95) (133s) (114e-005) (146) (203s) ( 552e-006) (93) (130s) (537e-006)

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

16 Mathematical Problems in Engineering

Under mild assumptions global convergence theory hasbeen established for the developed algorithm Since our algo-rithm does not involve computing the Jacobian matrix or itsapproximation both information storage and computationalcost of the algorithm are lower That is to say our algorithmis helpful to solution of large-scale problems In addition ithas been shown that our algorithm is also applicable to solvea nonsmooth system of equations or a singular one

Numerical tests have demonstrated that our algorithmoutperforms the others by costing less number of functionevaluations less number of iterations or less CPU timeto find a solution with the same tolerance especially incomparison with some similar algorithms available in theliterature Efficiency of our algorithm has also been shown byreconstructing sparse signals in compressed sensing

In future research due to its satisfactory numericalefficiency the proposedmethod in this paper can be extendedinto solving more large-scale nonlinear system of equationsfrommany fields of sciences and engineering

Data Availability

The data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

Wedeclare that all the authors have no any conflicts of interestabout submission and publication of this paper

Authorsrsquo Contributions

Zhong Wan conceived and designed the research plan andwrote the paper Jie Guo performed the mathematical mod-elling and numerical analysis and wrote the paper

Acknowledgments

This research is supported by the National Natural ScienceFoundation of China (Grant no 71671190) and OpeningFoundation from State Key Laboratory of DevelopmentalBiology of Freshwater Fish Hunan Normal University

References

[1] S Huang and Z Wan ldquoA new nonmonotone spectral residualmethod for nonsmooth nonlinear equationsrdquo Journal of Com-putational and Applied Mathematics vol 313 pp 82ndash101 2017

[2] ZWan J Guo J Liu andW Liu ldquoAmodified spectral conjugategradient projection method for signal recoveryrdquo Signal Imageand Video Processing vol 12 no 8 pp 1455ndash1462 2018

[3] J M Ortega andW C Rheinboldt Iterative Solution of Nonlin-ear Equations in Several Variables Siam New York NY USA1970

[4] J E Dennis and B R Schnabel ldquoNumerical methods fornonlinear equations and unconstrained optimizationrdquo Classicsin Applied Mathematics p 16 1983

[5] W Zhou andD Li ldquoLimitedmemory BFGSmethod for nonlin-ear monotone equationsrdquo Journal of Computational Mathemat-ics vol 25 no 1 pp 89ndash96 2007

[6] J M Martinez ldquoA family of quasi-Newton methods for non-linear equations with direct secant updates of matrix factoriza-tionsrdquo SIAM Journal on Numerical Analysis vol 27 no 4 pp1034ndash1049 1990

[7] G Fasano F Lampariello and M Sciandrone ldquoA truncatednonmonotone Gauss-Newton method for large-scale nonlin-ear least-squares problemsrdquo Computational Optimization andApplications vol 34 no 3 pp 343ndash358 2006

[8] D Li and M Fukushima ldquoA globally and superlinearly con-vergent Gauss-Newton-based BFGS method for symmetricnonlinear equationsrdquo SIAM Journal on Numerical Analysis vol37 no 1 pp 152ndash172 1999

[9] Z Papp and S Rapajic ldquoFR type methods for systems of large-scale nonlinear monotone equationsrdquo AppliedMathematics andComputation vol 269 pp 816ndash823 2015

[10] Q Li and D-H Li ldquoA class of derivative-free methods for large-scale nonlinearmonotone equationsrdquo IMA Journal ofNumericalAnalysis (IMAJNA) vol 31 no 4 pp 1625ndash1635 2011

[11] L Zhang W Zhou and D H Li ldquoA descent modified Polak-Ribiere-Polyak conjugate gradient method and its global con-vergencerdquo IMA Journal of Numerical Analysis (IMAJNA) vol26 no 4 pp 629ndash640 2006

[12] W La Cruz J M Martnez and M Raydan ldquoSpectral residualmethod without gradient information for solving large-scalenonlinear systems of equationsrdquo Mathematics of Computationvol 75 no 255 pp 1429ndash1448 2006

[13] Z Wan W Liu and C Wang ldquoA modified spectral conjugategradient projection method for solving nonlinear monotonesymmetric equationsrdquo Pacific Journal of Optimization An Inter-national Journal vol 12 no 3 pp 603ndash622 2016

[14] M Ahookhosh K Amini and S Bahrami ldquoTwo derivative-free projection approaches for systems of large-scale nonlinearmonotone equationsrdquo Numerical Algorithms vol 64 no 1 pp21ndash42 2013

[15] Q-R Yan X-Z Peng and D-H Li ldquoA globally conver-gent derivative-free method for solving large-scale nonlinearmonotone equationsrdquo Journal of Computational and AppliedMathematics vol 234 no 3 pp 649ndash657 2010

[16] K Amini A Kamandi and S Bahrami ldquoA double-projection-based algorithm for large-scale nonlinear systems of monotoneequationsrdquo Numerical Algorithms vol 68 no 2 pp 213ndash2282015

[17] Y Li Z Wan and J Liu ldquoBi-level programming approachto optimal strategy for vendor-managed inventory problemsunder random demandrdquo e ANZIAM Journal vol 59 no 2pp 247ndash270 2017

[18] T Li and ZWan ldquoNew adaptive Barzilar-Borwein step size andits application in solving large scale optimization problemsrdquoeANZIAM Journal vol 61 no 1 pp 76ndash98 2019

[19] X J Tong and L Qi ldquoOn the convergence of a trust-region method for solving constrained nonlinear equationswith degenerate solutionsrdquo Journal of Optimization eory andApplications vol 123 no 1 pp 187ndash211 2004

[20] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 pp 164ndash168 1944

[21] D Marquardt ldquoAn algorithm for least-squares estimation ofnonlinear parametersrdquo SIAM Journal on Applied Mathematicsvol 11 no 2 pp 431ndash441 1963

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Mathematical Problems in Engineering 17

[22] C Kanzow N Yamashita and M Fukushima ldquoLevenberg-Marquardt methods with strong local convergence propertiesfor solving nonlinear equations with convex constraintsrdquo Jour-nal of Computational and Applied Mathematics vol 173 no 2pp 321ndash343 2005

[23] J K Liu and S J Li ldquoA projection method for convexconstrained monotone nonlinear equations with applicationsrdquoComputers ampMathematics with Applications vol 70 no 10 pp2442ndash2453 2015

[24] G Yuan and M Zhang ldquoA three-terms Polak-Ribiere-Polyakconjugate gradient algorithm for large-scale nonlinear equa-tionsrdquo Journal of Computational and Applied Mathematics vol286 pp 186ndash195 2015

[25] G Yuan Z Meng and Y Li ldquoA modified Hestenes and Stiefelconjugate gradient algorithm for large-scale nonsmooth min-imizations and nonlinear equationsrdquo Journal of Optimizationeory and Applications vol 168 no 1 pp 129ndash152 2016

[26] G Yuan and W Hu ldquoA conjugate gradient algorithm forlarge-scale unconstrained optimization problems and nonlin-ear equationsrdquo Journal of Inequalities andApplications vol 2018no 1 article 113 2018

[27] L Zhang and W Zhou ldquoSpectral gradient projection methodfor solving nonlinear monotone equationsrdquo Journal of Compu-tational and Applied Mathematics vol 196 no 2 pp 478ndash4842006

[28] W Cheng ldquoA PRP type method for systems of monotoneequationsrdquo Mathematical and Computer Modelling vol 50 no1-2 pp 15ndash20 2009

[29] M V Solodov and B F Svaiter ldquoA globally convergent inex-act Newton method for systems of monotone equationsrdquo inReformulation Nonsmooth Piecewise Smooth Semismooth andSmoothing Methods pp 355ndash369 Springer Boston MA USA1998

[30] Z Wan Z Yang and Y Wang ldquoNew spectral PRP conju-gate gradient method for unconstrained optimizationrdquo AppliedMathematics Letters vol 24 no 1 pp 16ndash22 2011

[31] Y Ou and J Li ldquoA new derivative-free SCG-type projectionmethod for nonlinear monotone equations with convex con-straintsrdquoAppliedMathematics and Computation vol 56 no 1-2pp 195ndash216 2018

[32] W Zhou and D Li ldquoOn the Q-linear convergence rate of a classof methods for monotone nonlinear equationsrdquo Pacific Journalof Optimization vol 14 pp 723ndash737 2018

[33] B T Polyak Introduction to Optimization Optimization Soft-ware New York NY USA 1987

[34] Y Xiao and H Zhu ldquoA conjugate gradient method to solveconvex constrained monotone equations with applications incompressive sensingrdquo Journal of Mathematical Analysis andApplications vol 405 no 1 pp 310ndash319 2013

[35] Y Xiao Q Wang and Q Hu ldquoNon-smooth equations basedmethod for l1 problems with applications to compressed sens-ingrdquo Nonlinear Analysis eory Methods and Applications vol74 no 11 pp 3570ndash3577 2011

[36] S Kim K Koh M Lustig et al ldquoA method for large-scale 1-regularized least squares problems with applications in signalprocessing and statisticsrdquo Technical Report Dept of ElectricalEngineering Stanford University Stanford Calif USA 2007

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom