research article levenberg-marquardt method for the

Research ArticleLevenberg-Marquardt Method for the EigenvalueComplementarity Problem

Yuan-yuan Chen12 and Yan Gao1

1 School of Management University of Shanghai for Science and Technology Shanghai 200093 China2 College of Mathematics Qingdao University Qingdao 266071 China

Correspondence should be addressed to Yuan-yuan Chen usstchenyuanyuan163com

Received 22 June 2014 Revised 28 August 2014 Accepted 29 August 2014 Published 30 October 2014

Academic Editor Pu-yan Nie

Copyright copy 2014 Y-y Chen and Y Gao 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

The eigenvalue complementarity problem (EiCP) is a kind of very useful model which is widely used in the study ofmany problemsin mechanics engineering and economics The EiCP was shown to be equivalent to a special nonlinear complementarity problemor a mathematical programming problem with complementarity constraints The existing methods for solving the EiCP are allnonsmooth methods including nonsmooth or semismooth Newton type methods In this paper we reformulate the EiCP as asystem of continuously differentiable equations and give the Levenberg-Marquardtmethod to solve themUndermild assumptionsthe method is proved globally convergent Finally some numerical results and the extensions of the method are also given Thenumerical experiments highlight the efficiency of the method

1 Introduction

Eigenvalue complementarity problem (EiCP) is proposedin the study of the problems in mechanics engineeringand economics The EiCP is also called cone-constrainedeigenvalue problem in [1ndash4] The EiCP is to find a solutionincluding a scalar and a nonzero vector satisfying a com-plementarity constraint on a closed convex cone The EiCPcan be reformulated to be a special complementarity problemor a mathematical programming optimization problem withcomplementarity constraints and can use nonsmooth orsemismooth Newton type method to solve it such as [5ndash7]The Levenberg-Marquardt method is one of the widely usedmethods in solving optimization problems (see for instance[8ndash15]) Use a trust region strategy to replace the line searchthe Levenberg-Marquardt method is widely considered to bethe progenitor of the trust region method approach for gen-eral unconstrained or constrained optimization problemsThe use of a trust region avoids the weaknesses of GaussNewton method that is its behavior when the Jacobianis rank deficient or nearly so rank deficient On the otherhand we reformulate the EiCP as a system of continuouslydifferentiable equations that is one of the most interesting

themes The advantage of the reformulation is that we solvethe equations with continuously differentiable functions forwhich there are rich powerful solution methods and the-ory analysis including the powerful Levenberg-Marquardtmethod So in this paper we give the Levenberg-Marquardtmethod to solve the EiCP The EiCP which we will consideris the following problem Given the matrix 119860 isin 119877

119899times119899 and thematrix 119861 isin 119877

119899times119899 which are positive definite matrix then weconsider to find a scalar 120582 isin 119877 and a vector 119909 isin 119877

119899

0such that

(119860 minus 120582119861) 119909 ge 0

119909 ge 0

119909119879

(119860 minus 120582119861) 119909 = 0

(1)

This paper is organized as follows In Section 2 we givesome background definitions and known properties Andwe also give the Levenberg-Marquardt method for the EiCPThe global convergence analysis and some discussions of theLevenberg-Marquardt method is also given In Sections 3and 4 we give some numerical results and some extensionsof the method

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 307823 6 pageshttpdxdoiorg1011552014307823

2 The Scientific World Journal

Throughout the paper 119872119899(119877) denotes a real matrix of

order 119899 and 119872119899119898(119877) denotes a real matrix of order 119899 times 119898

0119899= (0 0) and 119890

119899= (1 1)

2 Preliminaries

In this section firstly we reformulate (1) as a system ofcontinuously differentiable equations and give some prelimi-naries used in the followingThenwe propose the Levenberg-Marquardt method for the EiCP

As we all know (1) can be rewritten as

120596 = (119909

120582)

119891 (120596) = (119868119899 0) 120596 = 119909

119892 (120596) = (119860 0) 120596 minus (0119899 1) 120596 (119861 0) 120596 = (119860 minus 120582119861) 119909

119891119879

(120596) 119892 (120596) = 0

(119890119899 0) 120596 = 1

(2)

where 119891 119877119899+1

rarr 119877119899 is a continuously differentiable

function and the matrix 119860 isin 119877119899times119899 Using the nonlinear

complementarity problem (NCP) function 120601 1198772

rarr 119877which satisfied the following basic property

120601 (119886 119887) = 0 iff 119886119887 = 0 119886 ge 0 119887 ge 0 (3)

By the property (2) can be recasted as the following systemof equations

119867(120596) = (

1198671(120596)

119867119899(120596)

) = (

120601 (1198911(120596) 119892

1(120596))

120601 (119891119899(120596) 119892

119899(120596))

) = 0

(119890119899 0) 120596 = 1

(4)

Then 120596 solves the EiCP (1) if and only if 120596 solves (4) Definea merit function for (4) as

Ψ (120596) =1

2

119899

sum

119894=1

1206012

(119891119894(120596) 119892

119894(120596)) =

1

2119867 (120596)

2

(5)

We use a special NCP function named Fischer-Burmeisterfunction defined as

120601 (119886 119887) = radic1198862 + 1198872 minus (119886 + 119887) (6)

We know that the favorable property of Ψ is that Ψ isa continuously differentiable function on the whole spacealthough 119867 is not a continuously differentiable function ingeneral (see for example [16]) Thus we give the system ofcontinuously differentiable equations for (1) as the followingequations

119865 (120596) = (Ψ (120596)

(119890119899 0) 120596 minus 1

) = 0 (7)

where 119865 119877119899+1 rarr 1198772 which is a continuously differentiable

function In what follows we will give the Levenberg-Marquardt method Denote

Φ (120596) =1

2119865 (120596)

2

(8)

The least-squares formulation of (7) is the following uncon-strained optimization problem

min120596isin119877119899+1

Φ (120596) =1

2119865(120596)

2

(9)

Now we give the Levenberg-Marquardt method forsolving (1) The global convergence result of the method isalso given

Method 1 (the Levenberg-Marquardt method for the EiCP)Given 0 lt 120572 lt 1 0 lt 120573 lt 1 119901 gt 2 120588 gt 0 0 lt 120575 le 2 119875 gt 0120598 gt 0 120596

0isin 119877119899+1 1205830= 119865(120596

0)120575 and 119896 = 0

Step 1 If nablaΦ(120596) le 120598 then stop Otherwise compute 119889119896by

(1198651015840

(120596119896)119879

1198651015840

(120596119896) + 120583119896119868) 119889 + 119865

1015840

(120596119896)119879

119865 (120596119896) = 0 (10)

Step 2 If nablaΦ(120596119896)119879

119889119896+ 120588119889

119896119875

gt 0 let 119889119896= minusnablaΦ(120596

119896)

otherwise 119889119896is computed by (10) Then find the smallest

nonnegative integer119898 such that

Φ(120596119896+ 120573119898

119889119896) le Φ (120596

119896) + 120572120573

119898

nablaΦ(120596119896)119879

119889119896 (11)

Set 120596119896+1

= 120596119896+ 120573119898

119889119896

Step 3 Let 120583119896+1

= 119865(120596119896+1)120575 and 119896 = 119896+1 and go to Step 1

Now we give the global convergence of Method 1

Theorem 1 Suppose that 120596119896 119896 = 1 2 generated by

Method 1 Then each accumulation point of the sequence is astationary point of Φ

Proof Suppose that 120596119896119870rarr 120596⋆ 120596119896119870is a subsequence of

120596119896 and 119896 = 1 2 When there are infinitely many 119896 isin 119870

such that 119889119896= minusnablaΦ(120596

119896) by Proposition 116 in [17] we get

the assertion In the following we assume that if 120596119896119870is a

convergent subsequence of 120596119896 then 119889

119896is always computed

by (10) We assume that for every convergent subsequence120596119896119870for which

lim119896isin119870119896rarrinfin

nablaΦ (120596119896) = 0 (12)

we havelim sup119896isin119870119896rarrinfin

10038171003817100381710038171198891198961003817100381710038171003817 lt infin (13)

lim sup119896isin119870119896rarrinfin

100381610038161003816100381610038161003816nablaΦ(120596

119896)119879

119889119896

100381610038161003816100381610038161003816gt 0 (14)

In the following we also assume that120596119896rarr 120596⋆ Suppose that

120596⋆ is not a stationary point ofΦ By (10) we have

1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817 =

100381710038171003817100381710038171003817(1198651015840

(120596119896)119879

1198651015840

(120596119896) + 120583119896119868) 119889119896

100381710038171003817100381710038171003817

le100381710038171003817100381710038171003817(1198651015840

(120596119896)119879

1198651015840

(120596119896) + 120583119896119868)100381710038171003817100381710038171003817

10038171003817100381710038171198891198961003817100381710038171003817

(15)

The Scientific World Journal 3

therefore we have

10038171003817100381710038171198891198961003817100381710038171003817 ge

1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817

100381710038171003817100381710038171003817(1198651015840(120596

119896)119879

1198651015840 (120596119896) + 120583119896119868)100381710038171003817100381710038171003817

(16)

Obviously the denominator in the above inequality isnonzero otherwise we have nablaΦ(120596

119896) = 0 Then the

algorithmhas stoppedOn the other handwe know that thereexists a constant 120577 gt 0 such that

1003817100381710038171003817100381710038171198651015840

(120596119896)119879

1198651015840

(120596119896) + 120583119896119868100381710038171003817100381710038171003817le 120577 (17)

moreover

10038171003817100381710038171198891198961003817100381710038171003817 ge

1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817

120577 (18)

By nablaΦ(120596119896)119879

119889119896le minus120588119889

119896119875 and the fact that the gradient

nablaΦ(120596119896) is bounded on the convergent sequence 120596

119896 we get

(13) We next prove (14) If (14) is not satisfied there exists asubsequence 120596

1198961198701015840 of 120596

119896119870


100381610038161003816100381610038161003816nablaΦ(120596

119896)119879

119889119896

100381610038161003816100381610038161003816= 0 (19)

This implies that lim119896isin1198701015840119896rarrinfin

119889119896 = 0 From (18) we know

that


1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817 = 0 (20)

which contradicts with (12) Thus (14) holds So accordingto the definition given in [17] the sequence 119889

119896 is uniformly

gradient related to 120596119896 We complete the proof

Remark 2 In Method 1 we can also use some other linesearch such as the nonmonotone line search The line searchis to find the smallest nonnegative integer119898 such that

Φ(120596119896+ 120573119898

119889119896) minus max0le119895le119898(119896)

Φ(120596119896minus119895) minus 120572120573

119898

nablaΦ(120596119896)119879

119889119896le 0

(21)

where119898(0) = 0119898(119896) = min1198720 119898(119896 minus 1) + 1 and119872

0gt 0

is a integer

Remark 3 In Method 1 we can also use the followingequation to compute 119889

119896in Step 1We can find an approximate

solution 119889119896isin 119877119899 of the equation

(1198651015840

(120596119896)119879

1198651015840

(120596119896) + 120583119896119868) 119889 + 119865

1015840

(120596119896)119879

119865 (120596119896) minus 119903119896= 0

(22)

where 119903119896is the residuals and satisfies

10038171003817100381710038171199031198961003817100381710038171003817 le 120572119896

1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817 (23)

where 120572119896le 119886 lt 1 for every 119896

Table 1

Method 1 120582 119909 SPA 120582 119909

499542 (024366 075633)119879

5 (025 075)119879

700343 (050625 049373)119879

sdot sdot sdot sdot sdot sdot

766150 (071184 028784)119879

8 (1 0)119879

3 Numerical Results

Wegive somenumerical experiments for themethod Andwecompare Method 1 with the scaling and projection algorithm(denoted by SPA in [18]) The numerical results indicatethat Method 1 works quite well in practice We considerthe eigenvalue complementarity problems which are alltaken form [3 18] All codes for the method are finished inMATLAB The parameters used in the method are chosen as120588 = 10 119875 = 3 120572 = 01 and 120598 = 10minus4

Example 4 We consider

(119860 minus 120582119861) 119909 ge 0

119909 ge 0

119909119879

(119860 minus 120582119861) 119909 = 0

(24)

where

119861 = (1 0

0 1)

119860 = (8 minus1

3 4)

(25)

By [3] we know that Example 4 has three eigenval-ues Now we consider random initial points to computeExample 4 by Method 1 Numerical results for Example 4 byMethod 1 and SPA method are presented in Table 1

Table 1 shows that Method 1 are able to detect all thesolutions for the small size matrix But the SPA method canonly detect 2 solutions from [3]


(119860 minus 120582119861) 119909 ge 0

119909 ge 0

119909119879

(119860 minus 120582119861) 119909 = 0

(26)

where

119861 = (

1 0 0

0 1 0

0 0 1

)

119860 = (

8 minus1 4

3 4 05

2 minus05 6

)

(27)

From [3] we also know that Example 5 have 9 eigen-values Now we consider random initial points to compute


Table 2

Method 1 120582 119909 SPA 120582 119909

460705 (055617 044898 013390)119879

41340 (0 1 02679)119879

418288 (008023 051776 038374)119879

5 (03333 1 0)119879

503988 (014265 042924 042811)119879

6 (0 0 1)119879

587521 (030666 027220 050890)119879

8 (1 0 0)119879

600946 (012341 036685 050975)119879


701141 (035296 030395 038277)119879


800737 (040141 035921 023934)119879


936479 (047314 0290185 023667)119879


999838 (057695 013243 029060)119879


the above Example 5 by Method 1 Numerical results forExample 5 by Method 1 and the SPA method are presentedin Table 2

Table 2 shows that Method 1 is able to detect all thesolutions But the SPA method can only detect 4 Paretoeigenvalues from the analysis of [3]


(119860 minus 120582119861) 119909 ge 0

119909 ge 0

119909119879

(119860 minus 120582119861) 119909 = 0

(28)

where

119861 = (

1 0 0

0 1 0

0 0 1

)

119860 = (

100 106 minus18 minus81


2 44 37 minus7

21 38 0 2

)

(29)

From [3] we also know that Example 6 have 23 eigen-values Now we consider random initial points to computeExample 6 by Method 1 Numerical results for Example 6 byMethod 1 and the SPA method are given in Table 3

Table 3 shows that Method 1 is able to detect all 23solutions But the SPA method can only detect 2 Paretoeigenvalues from [3] The numerical results indicate thatMethod 1 works quite well for the big size EiCP in practice

Discussion In this section we study the numerical behaviorsof Method 1 for solving the Pareto eigenvalue problem TheEiCP problem is very useful in studying the optimizationproblems arising in many areas of the applied mathematicsand mechanics By using the F-B function we reformulatethe EiCP as a system of continuously differentiable equationsThen we use the Levenberg-Marquardt method to solve itFrom the above numerical results we know that Method 1 isvery effective for small size and big size EiCP problems

4 Extensions

41 Bieigenvalue Complementarity Problems (BECP) Wealsocan use Method 1 to solve the bieigenvalue complementarityproblems (denoted by BECP) The BECP is to find (120582 120583) isin119877 times 119877 and (119909 119910) isin 119877119899 0 times 119877119898 0 such that

119909 ge 0 120582119909 minus 119860119909 minus 119861119910 ge 0 119909119879

(120582119909 minus 119860119909 minus 119861119910) = 0

119910 ge 0 120583119910 minus 119862119909 minus 119863119910 ge 0 119910119879

(120583119910 minus 119862119909 minus 119863119910) = 0

(30)

where 119860 isin 119872119899(119877) 119861 isin 119872

119899119898(119877) 119861 isin 119872

119898119899(119877) and 119863 isin

119872119898(119877)

Let 120596 = (

119909

119910

120582

120583

) 119891(120596) = 119909 119892(120596) = 119910 119875(120596) = 120582119909minus119860119909minus119861119910

and119876(120596) = 120583119910minus119862119909minus119863119910We canwrite the above bieigenvaluecomplementarity problems as 119891(120596) ge 0 119892(120596) ge 0 119875(120596) ge 0119876(120596) ge 0 119891119879(120596)119875(120596) = 0 119892119879(120596)119876(120596) = 0 (119890

119899 0119899 0 0)120596 = 1

and (0119899 119890119899 0 0)120596 = 1 By using F-B function similar to

rewriting the EiCP we can rewrite the above bieigenvaluecomplementarity problems as the following equations

1198671(120596) = (

120601 (1198911(120596) 119875

1(120596))

120601 (119891119899(120596) 119875

119899(120596))

) = 0

1198672(120596) = (

120601 (1198921(120596) 119876

1(120596))

120601 (119892119899(120596) 119876

119899(120596))

) = 0

(119890119899 0119899 0 0) 120596 minus 1 = 0

(0119899 119890119899 0 0) 120596 minus 1 = 0

(31)

Let

Φ1(120596) =

1

2

10038171003817100381710038171198671(120596)1003817100381710038171003817

2

Φ2(120596) =

1

2

10038171003817100381710038171198672(120596)1003817100381710038171003817

2

(32)


Table 3

Method 1 120582 119909 SPA 120582 119909

2628209 (032922 028109 000067 068363)119879 100 (1 0 0 0)

119879

2641489 (034225 004798 030622 031649)119879 158 (0 1 0 0)

119879

2871143 (059383 030244 001458 043984)119879


2913415 (039357 016933 031781 012821)119879


3260775 (035641 026564 017973 062135)119879


3286388 (013771 033320 002167 050745)119879


3757690 (030291 019045 021854 027608)119879


4101635 (028247 019667 023447 026953)119879


4646811 (034130 028631 018549 016868)119879


4914424 (025544 018482 028418 027187)119879


6696950 (047995 022175 000256 029497)119879


7742278 (026550 028105 030904 014384)119879


7745752 (021870 031154 024021 022741)119879


9942333 (036284 032018 016593 013579)119879


9999801 (067443 013866 000391 018299)119879


10749890 (039982 033534 017631 008845)119879


12738892 (032969 037065 018570 010872)119879


14853897 (034127 045333 001989 017409)119879


15799623 (028006 037495 014411 007701)119879


19717149 (048407 060907 015118 018058)119879


20457942 (055801 067908 017056 014984)119879


22627891 (026236 032696 008010 001135)119879


23192132 (026744 033127 004324 000457)119879


Then we can give the system of continuously differentiableequations for the bieigenvalue complementarity problems asthe following continuously differentiable equations

(

Φ1(120596)

Φ2(120596)

(119890119899 0119899 0 0) 120596 minus 1

(0119899 119890119899 0 0) 120596 minus 1

) = 0 (33)

In the following numerical test the parameters used in themethod are also chosen as 120588 = 10 119875 = 3 120572 = 01 and 120598 =10minus4

Example 7 We consider the BECP where 119860 = 0119863 = 0

119861 = (1 0

0 1)

119862 = (1 0

0 1)

(34)

We use Method 1 to solve the Example 7 Results forExample 7 with random initial point are given in Table 4

42 The Paretian Version We consider using Method 1 tocompute the following Paretian version

119909 ge 0 119872 (120582) 119909 ge 0 119909119879

119872(120582) 119909 = 0 (35)

Table 4

120582 120583 119909 119910

095803 096145 (069993 030006)119879

(069782 030217)119879

096205 097990 (069259 030740)119879

(068925 031074)119879

095574 094739 (048255 051744)119879

(047451 052548)119879

096069 093502 (037373 062626)119879

(037670 062329)119879

094582 104493 (069415 030584)119879

(069598 030401)119879

096200 103264 (046627 053372)119879

(047934 052065)119879

095517 106445 (035869 064130)119879

(037196 062803)119879

095744 105645 (039169 060830)119879

(039707 060292)119879

where 119872 standing for the pencil associated to a finitecollection 119860

0 1198601 119860

119903 of real matrices of order 119899 and

119872(120582) =

119903

sum

119896=0

120582119896

119860119896 (36)

Example 8 We consider the quadratic pencil model where

119872(120582) = 1205822

(

2 0 0

0 6 0

0 0 10

) + 120582(

7 0 0

0 30 0

0 0 20

) + (

minus2 6 0

2 16 3

0 5 0

)

(37)

We use Method 1 to compute Example 8 We present theresults of Example 8 with random initial point in Table 5


Table 5

120582 119909

025315 (097503 001611 000884)119879

minus438990 (000000 100000 000000)119879

minus374482 (095986 002679 001334)119879

minus084644 (039691 042913 017392)119879

minus5000000 (000000 100000 000000)119879

minus060280 (040400 037328 022271)119879

minus366880 (083349 010559 006090)119879

minus187686 (019023 025846 055130)119879

minus198258 (004642 006140 089217)119879

minus001062 (010208 003658 086133)119879

5 Conclusion

In this paper we reformulate the EiCP as a system ofcontinuously differentiable equations and use the Levenberg-Marquardt method to solve them The numerical experi-ments show that our method is a promising method for solv-ing the EiCP The numerical experiments of the extensionsconfirm the efficiency of our method

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work is supported by National Science Foundation ofChina (11171221 11101231) Shanghai Leading Academic Dis-cipline (XTKX 2012) and Innovation Program of ShanghaiMunicipal Education Commission (14YZ094)

References

[1] J J Judice M Raydan S S Rosa and S A Santos ldquoOn thesolution of the symmetric eigenvalue complementarity problemby the spectral projected gradient algorithmrdquo Numerical Algo-rithms vol 47 no 4 pp 391ndash407 2008

[2] J J Judice H D Sherali and I M Ribeiro ldquoThe eigenvaluecomplementarity problemrdquo Computational Optimization andApplications vol 37 no 2 pp 139ndash156 2007

[3] C F Ma ldquoThe semismooth and smoothing Newton methodsfor solving Pareto eigenvalue problemrdquo Applied MathematicalModelling vol 36 no 1 pp 279ndash287 2012

[4] S Adly and A Seeger ldquoA nonsmooth algorithm for cone-constrained eigenvalue problemsrdquoComputational Optimizationand Applications vol 49 no 2 pp 299ndash318 2011

[5] S Adly and H Rammal ldquoA new method for solving Paretoeigenvalue complementarity problemsrdquo Computational Opti-mization and Applications vol 55 no 3 pp 703ndash731 2013

[6] J J Judice H D Sherali I M Ribeiro and S S RosaldquoOn the asymmetric eigenvalue complementarity problemrdquoOptimization Methods and Software vol 24 no 4-5 pp 549ndash568 2009

[7] A Pinto da Costa and A Seeger ldquoCone-constrained eigenvalueproblems theory and algorithmsrdquoComputational Optimizationand Applications vol 45 no 1 pp 25ndash57 2010

[8] C Wang Q Liu and C Ma ldquoSmoothing SQP algorithm forsemismooth equations with box constraintsrdquo ComputationalOptimization and Applications vol 55 no 2 pp 399ndash425 2013

[9] F Facchinei and C Kanzow ldquoA nonsmooth inexact Newtonmethod for the solution of large-scale nonlinear complemen-tarity problemsrdquoMathematical Programming vol 76 no 3 pp493ndash512 1997

[10] J Y Fan and Y Y Yuan ldquoOn the quadratic convergenceof the Levenberg-Marquardt method without nonsingularityassumptionrdquo Computing Archives for Scientific Computing vol74 no 1 pp 23ndash39 2005

[11] S-Q Du and Y Gao ldquoThe Levenberg-Marquardt-typemethodsfor a kind of vertical complementarity problemrdquo Journal ofApplied Mathematics vol 2011 Article ID 161853 12 pages 2011

[12] J Y Fan ldquoA Shamanskii-like Levenberg-Marquardt method fornonlinear equationsrdquoComputational Optimization andApplica-tions vol 56 no 1 pp 63ndash80 2013

[13] R Behling and A Fischer ldquoA unified local convergence analysisof inexact constrained Levenberg-Marquardt methodsrdquo Opti-mization Letters vol 6 no 5 pp 927ndash940 2012

[14] K Ueda and N Yamashita ldquoGlobal complexity bound analysisof the Levenberg-Marquardt method for nonsmooth equationsand its application to the nonlinear complementarity problemrdquoJournal of Optimization Theory and Applications vol 152 no 2pp 450ndash467 2012

[15] F Facchinei A Fischer and M Herrich ldquoA family of Newtonmethods for nonsmooth constrained systems with nonisolatedsolutionsrdquo Mathematical Methods of Operations Research vol77 no 3 pp 433ndash443 2013

[16] H Jiang M Fukushima L Qi and D Sun ldquoA trust regionmethod for solving generalized complementarity problemsrdquoSIAM Journal on Optimization vol 8 no 1 pp 140ndash157 1998

[17] D P Bertsekas Constrained Optimization and Lagrange Multi-plier Methods Academic Press New York NY USA 1982

[18] A P da Costa and A Seeger ldquoNumerical resolution of cone-constrained eigenvalue problemsrdquo Computational amp AppliedMathematics vol 28 no 1 pp 37ndash61 2009

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Throughout the paper 119872119899(119877) denotes a real matrix of

order 119899 and 119872119899119898(119877) denotes a real matrix of order 119899 times 119898

0119899= (0 0) and 119890

119899= (1 1)

2 Preliminaries

In this section firstly we reformulate (1) as a system ofcontinuously differentiable equations and give some prelimi-naries used in the followingThenwe propose the Levenberg-Marquardt method for the EiCP

As we all know (1) can be rewritten as

120596 = (119909

120582)

119891 (120596) = (119868119899 0) 120596 = 119909

119892 (120596) = (119860 0) 120596 minus (0119899 1) 120596 (119861 0) 120596 = (119860 minus 120582119861) 119909

119891119879

(120596) 119892 (120596) = 0

(119890119899 0) 120596 = 1

(2)

where 119891 119877119899+1

rarr 119877119899 is a continuously differentiable

function and the matrix 119860 isin 119877119899times119899 Using the nonlinear

complementarity problem (NCP) function 120601 1198772

rarr 119877which satisfied the following basic property

120601 (119886 119887) = 0 iff 119886119887 = 0 119886 ge 0 119887 ge 0 (3)

By the property (2) can be recasted as the following systemof equations

119867(120596) = (

1198671(120596)

119867119899(120596)

) = (

120601 (1198911(120596) 119892

1(120596))

120601 (119891119899(120596) 119892

119899(120596))

) = 0

(119890119899 0) 120596 = 1

(4)

Then 120596 solves the EiCP (1) if and only if 120596 solves (4) Definea merit function for (4) as

Ψ (120596) =1

2

119899

sum

119894=1

1206012

(119891119894(120596) 119892

119894(120596)) =

1

2119867 (120596)

2

(5)

We use a special NCP function named Fischer-Burmeisterfunction defined as

120601 (119886 119887) = radic1198862 + 1198872 minus (119886 + 119887) (6)

We know that the favorable property of Ψ is that Ψ isa continuously differentiable function on the whole spacealthough 119867 is not a continuously differentiable function ingeneral (see for example [16]) Thus we give the system ofcontinuously differentiable equations for (1) as the followingequations

119865 (120596) = (Ψ (120596)

(119890119899 0) 120596 minus 1

) = 0 (7)

where 119865 119877119899+1 rarr 1198772 which is a continuously differentiable

function In what follows we will give the Levenberg-Marquardt method Denote

Φ (120596) =1

2119865 (120596)

2

(8)

The least-squares formulation of (7) is the following uncon-strained optimization problem

min120596isin119877119899+1

Φ (120596) =1

2119865(120596)

2

(9)

Now we give the Levenberg-Marquardt method forsolving (1) The global convergence result of the method isalso given

Method 1 (the Levenberg-Marquardt method for the EiCP)Given 0 lt 120572 lt 1 0 lt 120573 lt 1 119901 gt 2 120588 gt 0 0 lt 120575 le 2 119875 gt 0120598 gt 0 120596

0isin 119877119899+1 1205830= 119865(120596

0)120575 and 119896 = 0

Step 1 If nablaΦ(120596) le 120598 then stop Otherwise compute 119889119896by

(1198651015840

(120596119896)119879

1198651015840

(120596119896) + 120583119896119868) 119889 + 119865

1015840

(120596119896)119879

119865 (120596119896) = 0 (10)

Step 2 If nablaΦ(120596119896)119879

119889119896+ 120588119889

119896119875

gt 0 let 119889119896= minusnablaΦ(120596

119896)

otherwise 119889119896is computed by (10) Then find the smallest

nonnegative integer119898 such that

Φ(120596119896+ 120573119898

119889119896) le Φ (120596

119896) + 120572120573

119898

nablaΦ(120596119896)119879

119889119896 (11)

Set 120596119896+1

= 120596119896+ 120573119898

119889119896

Step 3 Let 120583119896+1

= 119865(120596119896+1)120575 and 119896 = 119896+1 and go to Step 1

Now we give the global convergence of Method 1

Theorem 1 Suppose that 120596119896 119896 = 1 2 generated by

Method 1 Then each accumulation point of the sequence is astationary point of Φ

Proof Suppose that 120596119896119870rarr 120596⋆ 120596119896119870is a subsequence of

120596119896 and 119896 = 1 2 When there are infinitely many 119896 isin 119870

such that 119889119896= minusnablaΦ(120596

119896) by Proposition 116 in [17] we get

the assertion In the following we assume that if 120596119896119870is a

convergent subsequence of 120596119896 then 119889

119896is always computed

by (10) We assume that for every convergent subsequence120596119896119870for which


nablaΦ (120596119896) = 0 (12)

we havelim sup119896isin119870119896rarrinfin

10038171003817100381710038171198891198961003817100381710038171003817 lt infin (13)

lim sup119896isin119870119896rarrinfin

100381610038161003816100381610038161003816nablaΦ(120596

119896)119879

119889119896

100381610038161003816100381610038161003816gt 0 (14)

In the following we also assume that120596119896rarr 120596⋆ Suppose that

120596⋆ is not a stationary point ofΦ By (10) we have

1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817 =

100381710038171003817100381710038171003817(1198651015840

(120596119896)119879

1198651015840

(120596119896) + 120583119896119868) 119889119896

100381710038171003817100381710038171003817

le100381710038171003817100381710038171003817(1198651015840

(120596119896)119879

1198651015840

(120596119896) + 120583119896119868)100381710038171003817100381710038171003817

10038171003817100381710038171198891198961003817100381710038171003817

(15)


therefore we have

10038171003817100381710038171198891198961003817100381710038171003817 ge

1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817

100381710038171003817100381710038171003817(1198651015840(120596

119896)119879

1198651015840 (120596119896) + 120583119896119868)100381710038171003817100381710038171003817

(16)


119896) = 0 Then the


1003817100381710038171003817100381710038171198651015840

(120596119896)119879

1198651015840

(120596119896) + 120583119896119868100381710038171003817100381710038171003817le 120577 (17)

moreover

10038171003817100381710038171198891198961003817100381710038171003817 ge

1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817

120577 (18)

By nablaΦ(120596119896)119879

119889119896le minus120588119889



119896 we get


1198961198701015840 of 120596

119896119870


100381610038161003816100381610038161003816nablaΦ(120596

119896)119879

119889119896

100381610038161003816100381610038161003816= 0 (19)


119889119896 = 0 From (18) we know

that


1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817 = 0 (20)


119896 is uniformly



Φ(120596119896+ 120573119898

119889119896) minus max0le119895le119898(119896)

Φ(120596119896minus119895) minus 120572120573

119898

nablaΦ(120596119896)119879

119889119896le 0

(21)


0gt 0

is a integer




(1198651015840

(120596119896)119879

1198651015840

(120596119896) + 120583119896119868) 119889 + 119865

1015840

(120596119896)119879

119865 (120596119896) minus 119903119896= 0

(22)


10038171003817100381710038171199031198961003817100381710038171003817 le 120572119896

1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817 (23)


Table 1

Method 1 120582 119909 SPA 120582 119909

499542 (024366 075633)119879

5 (025 075)119879

700343 (050625 049373)119879


766150 (071184 028784)119879

8 (1 0)119879

3 Numerical Results



(119860 minus 120582119861) 119909 ge 0

119909 ge 0

119909119879

(119860 minus 120582119861) 119909 = 0

(24)

where

119861 = (1 0

0 1)

119860 = (8 minus1

3 4)

(25)




(119860 minus 120582119861) 119909 ge 0

119909 ge 0

119909119879

(119860 minus 120582119861) 119909 = 0

(26)

where

119861 = (

1 0 0

0 1 0

0 0 1

)

119860 = (

8 minus1 4

3 4 05

2 minus05 6

)

(27)



Table 2

Method 1 120582 119909 SPA 120582 119909

460705 (055617 044898 013390)119879

41340 (0 1 02679)119879

418288 (008023 051776 038374)119879

5 (03333 1 0)119879

503988 (014265 042924 042811)119879

6 (0 0 1)119879

587521 (030666 027220 050890)119879

8 (1 0 0)119879

600946 (012341 036685 050975)119879


701141 (035296 030395 038277)119879


800737 (040141 035921 023934)119879


936479 (047314 0290185 023667)119879


999838 (057695 013243 029060)119879





(119860 minus 120582119861) 119909 ge 0

119909 ge 0

119909119879

(119860 minus 120582119861) 119909 = 0

(28)

where

119861 = (

1 0 0

0 1 0

0 0 1

)

119860 = (



2 44 37 minus7

21 38 0 2

)

(29)




4 Extensions


119909 ge 0 120582119909 minus 119860119909 minus 119861119910 ge 0 119909119879

(120582119909 minus 119860119909 minus 119861119910) = 0

119910 ge 0 120583119910 minus 119862119909 minus 119863119910 ge 0 119910119879

(120583119910 minus 119862119909 minus 119863119910) = 0

(30)

where 119860 isin 119872119899(119877) 119861 isin 119872

119899119898(119877) 119861 isin 119872

119898119899(119877) and 119863 isin

119872119898(119877)

Let 120596 = (

119909

119910

120582

120583

) 119891(120596) = 119909 119892(120596) = 119910 119875(120596) = 120582119909minus119860119909minus119861119910


119899 0119899 0 0)120596 = 1



1198671(120596) = (

120601 (1198911(120596) 119875

1(120596))

120601 (119891119899(120596) 119875

119899(120596))

) = 0

1198672(120596) = (

120601 (1198921(120596) 119876

1(120596))

120601 (119892119899(120596) 119876

119899(120596))

) = 0

(119890119899 0119899 0 0) 120596 minus 1 = 0

(0119899 119890119899 0 0) 120596 minus 1 = 0

(31)

Let

Φ1(120596) =

1

2

10038171003817100381710038171198671(120596)1003817100381710038171003817

2

Φ2(120596) =

1

2

10038171003817100381710038171198672(120596)1003817100381710038171003817

2

(32)


Table 3

Method 1 120582 119909 SPA 120582 119909

2628209 (032922 028109 000067 068363)119879 100 (1 0 0 0)

119879

2641489 (034225 004798 030622 031649)119879 158 (0 1 0 0)

119879

2871143 (059383 030244 001458 043984)119879


2913415 (039357 016933 031781 012821)119879


3260775 (035641 026564 017973 062135)119879


3286388 (013771 033320 002167 050745)119879


3757690 (030291 019045 021854 027608)119879


4101635 (028247 019667 023447 026953)119879


4646811 (034130 028631 018549 016868)119879


4914424 (025544 018482 028418 027187)119879


6696950 (047995 022175 000256 029497)119879


7742278 (026550 028105 030904 014384)119879


7745752 (021870 031154 024021 022741)119879


9942333 (036284 032018 016593 013579)119879


9999801 (067443 013866 000391 018299)119879


10749890 (039982 033534 017631 008845)119879


12738892 (032969 037065 018570 010872)119879


14853897 (034127 045333 001989 017409)119879


15799623 (028006 037495 014411 007701)119879


19717149 (048407 060907 015118 018058)119879


20457942 (055801 067908 017056 014984)119879


22627891 (026236 032696 008010 001135)119879


23192132 (026744 033127 004324 000457)119879



(

Φ1(120596)

Φ2(120596)

(119890119899 0119899 0 0) 120596 minus 1

(0119899 119890119899 0 0) 120596 minus 1

) = 0 (33)



119861 = (1 0

0 1)

119862 = (1 0

0 1)

(34)



119909 ge 0 119872 (120582) 119909 ge 0 119909119879

119872(120582) 119909 = 0 (35)

Table 4

120582 120583 119909 119910

095803 096145 (069993 030006)119879

(069782 030217)119879

096205 097990 (069259 030740)119879

(068925 031074)119879

095574 094739 (048255 051744)119879

(047451 052548)119879

096069 093502 (037373 062626)119879

(037670 062329)119879

094582 104493 (069415 030584)119879

(069598 030401)119879

096200 103264 (046627 053372)119879

(047934 052065)119879

095517 106445 (035869 064130)119879

(037196 062803)119879

095744 105645 (039169 060830)119879

(039707 060292)119879


0 1198601 119860


119872(120582) =

119903

sum

119896=0

120582119896

119860119896 (36)


119872(120582) = 1205822

(

2 0 0

0 6 0

0 0 10

) + 120582(

7 0 0

0 30 0

0 0 20

) + (

minus2 6 0

2 16 3

0 5 0

)

(37)



Table 5

120582 119909

025315 (097503 001611 000884)119879

minus438990 (000000 100000 000000)119879

minus374482 (095986 002679 001334)119879

minus084644 (039691 042913 017392)119879

minus5000000 (000000 100000 000000)119879

minus060280 (040400 037328 022271)119879

minus366880 (083349 010559 006090)119879

minus187686 (019023 025846 055130)119879

minus198258 (004642 006140 089217)119879

minus001062 (010208 003658 086133)119879

5 Conclusion




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










therefore we have

10038171003817100381710038171198891198961003817100381710038171003817 ge

1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817

100381710038171003817100381710038171003817(1198651015840(120596

119896)119879

1198651015840 (120596119896) + 120583119896119868)100381710038171003817100381710038171003817

(16)


119896) = 0 Then the


1003817100381710038171003817100381710038171198651015840

(120596119896)119879

1198651015840

(120596119896) + 120583119896119868100381710038171003817100381710038171003817le 120577 (17)

moreover

10038171003817100381710038171198891198961003817100381710038171003817 ge

1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817

120577 (18)

By nablaΦ(120596119896)119879

119889119896le minus120588119889



119896 we get


1198961198701015840 of 120596

119896119870


100381610038161003816100381610038161003816nablaΦ(120596

119896)119879

119889119896

100381610038161003816100381610038161003816= 0 (19)


119889119896 = 0 From (18) we know

that


1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817 = 0 (20)


119896 is uniformly



Φ(120596119896+ 120573119898

119889119896) minus max0le119895le119898(119896)

Φ(120596119896minus119895) minus 120572120573

119898

nablaΦ(120596119896)119879

119889119896le 0

(21)


0gt 0

is a integer




(1198651015840

(120596119896)119879

1198651015840

(120596119896) + 120583119896119868) 119889 + 119865

1015840

(120596119896)119879

119865 (120596119896) minus 119903119896= 0

(22)


10038171003817100381710038171199031198961003817100381710038171003817 le 120572119896

1003817100381710038171003817nablaΦ (120596119896)1003817100381710038171003817 (23)


Table 1

Method 1 120582 119909 SPA 120582 119909

499542 (024366 075633)119879

5 (025 075)119879

700343 (050625 049373)119879


766150 (071184 028784)119879

8 (1 0)119879

3 Numerical Results



(119860 minus 120582119861) 119909 ge 0

119909 ge 0

119909119879

(119860 minus 120582119861) 119909 = 0

(24)

where

119861 = (1 0

0 1)

119860 = (8 minus1

3 4)

(25)




(119860 minus 120582119861) 119909 ge 0

119909 ge 0

119909119879

(119860 minus 120582119861) 119909 = 0

(26)

where

119861 = (

1 0 0

0 1 0

0 0 1

)

119860 = (

8 minus1 4

3 4 05

2 minus05 6

)

(27)



Table 2

Method 1 120582 119909 SPA 120582 119909

460705 (055617 044898 013390)119879

41340 (0 1 02679)119879

418288 (008023 051776 038374)119879

5 (03333 1 0)119879

503988 (014265 042924 042811)119879

6 (0 0 1)119879

587521 (030666 027220 050890)119879

8 (1 0 0)119879

600946 (012341 036685 050975)119879


701141 (035296 030395 038277)119879


800737 (040141 035921 023934)119879


936479 (047314 0290185 023667)119879


999838 (057695 013243 029060)119879





(119860 minus 120582119861) 119909 ge 0

119909 ge 0

119909119879

(119860 minus 120582119861) 119909 = 0

(28)

where

119861 = (

1 0 0

0 1 0

0 0 1

)

119860 = (



2 44 37 minus7

21 38 0 2

)

(29)




4 Extensions


119909 ge 0 120582119909 minus 119860119909 minus 119861119910 ge 0 119909119879

(120582119909 minus 119860119909 minus 119861119910) = 0

119910 ge 0 120583119910 minus 119862119909 minus 119863119910 ge 0 119910119879

(120583119910 minus 119862119909 minus 119863119910) = 0

(30)

where 119860 isin 119872119899(119877) 119861 isin 119872

119899119898(119877) 119861 isin 119872

119898119899(119877) and 119863 isin

119872119898(119877)

Let 120596 = (

119909

119910

120582

120583

) 119891(120596) = 119909 119892(120596) = 119910 119875(120596) = 120582119909minus119860119909minus119861119910


119899 0119899 0 0)120596 = 1



1198671(120596) = (

120601 (1198911(120596) 119875

1(120596))

120601 (119891119899(120596) 119875

119899(120596))

) = 0

1198672(120596) = (

120601 (1198921(120596) 119876

1(120596))

120601 (119892119899(120596) 119876

119899(120596))

) = 0

(119890119899 0119899 0 0) 120596 minus 1 = 0

(0119899 119890119899 0 0) 120596 minus 1 = 0

(31)

Let

Φ1(120596) =

1

2

10038171003817100381710038171198671(120596)1003817100381710038171003817

2

Φ2(120596) =

1

2

10038171003817100381710038171198672(120596)1003817100381710038171003817

2

(32)


Table 3

Method 1 120582 119909 SPA 120582 119909

2628209 (032922 028109 000067 068363)119879 100 (1 0 0 0)

119879

2641489 (034225 004798 030622 031649)119879 158 (0 1 0 0)

119879

2871143 (059383 030244 001458 043984)119879


2913415 (039357 016933 031781 012821)119879


3260775 (035641 026564 017973 062135)119879


3286388 (013771 033320 002167 050745)119879


3757690 (030291 019045 021854 027608)119879


4101635 (028247 019667 023447 026953)119879


4646811 (034130 028631 018549 016868)119879


4914424 (025544 018482 028418 027187)119879


6696950 (047995 022175 000256 029497)119879


7742278 (026550 028105 030904 014384)119879


7745752 (021870 031154 024021 022741)119879


9942333 (036284 032018 016593 013579)119879


9999801 (067443 013866 000391 018299)119879


10749890 (039982 033534 017631 008845)119879


12738892 (032969 037065 018570 010872)119879


14853897 (034127 045333 001989 017409)119879


15799623 (028006 037495 014411 007701)119879


19717149 (048407 060907 015118 018058)119879


20457942 (055801 067908 017056 014984)119879


22627891 (026236 032696 008010 001135)119879


23192132 (026744 033127 004324 000457)119879



(

Φ1(120596)

Φ2(120596)

(119890119899 0119899 0 0) 120596 minus 1

(0119899 119890119899 0 0) 120596 minus 1

) = 0 (33)



119861 = (1 0

0 1)

119862 = (1 0

0 1)

(34)



119909 ge 0 119872 (120582) 119909 ge 0 119909119879

119872(120582) 119909 = 0 (35)

Table 4

120582 120583 119909 119910

095803 096145 (069993 030006)119879

(069782 030217)119879

096205 097990 (069259 030740)119879

(068925 031074)119879

095574 094739 (048255 051744)119879

(047451 052548)119879

096069 093502 (037373 062626)119879

(037670 062329)119879

094582 104493 (069415 030584)119879

(069598 030401)119879

096200 103264 (046627 053372)119879

(047934 052065)119879

095517 106445 (035869 064130)119879

(037196 062803)119879

095744 105645 (039169 060830)119879

(039707 060292)119879


0 1198601 119860


119872(120582) =

119903

sum

119896=0

120582119896

119860119896 (36)


119872(120582) = 1205822

(

2 0 0

0 6 0

0 0 10

) + 120582(

7 0 0

0 30 0

0 0 20

) + (

minus2 6 0

2 16 3

0 5 0

)

(37)



Table 5

120582 119909

025315 (097503 001611 000884)119879

minus438990 (000000 100000 000000)119879

minus374482 (095986 002679 001334)119879

minus084644 (039691 042913 017392)119879

minus5000000 (000000 100000 000000)119879

minus060280 (040400 037328 022271)119879

minus366880 (083349 010559 006090)119879

minus187686 (019023 025846 055130)119879

minus198258 (004642 006140 089217)119879

minus001062 (010208 003658 086133)119879

5 Conclusion




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 2

Method 1 120582 119909 SPA 120582 119909

460705 (055617 044898 013390)119879

41340 (0 1 02679)119879

418288 (008023 051776 038374)119879

5 (03333 1 0)119879

503988 (014265 042924 042811)119879

6 (0 0 1)119879

587521 (030666 027220 050890)119879

8 (1 0 0)119879

600946 (012341 036685 050975)119879


701141 (035296 030395 038277)119879


800737 (040141 035921 023934)119879


936479 (047314 0290185 023667)119879


999838 (057695 013243 029060)119879





(119860 minus 120582119861) 119909 ge 0

119909 ge 0

119909119879

(119860 minus 120582119861) 119909 = 0

(28)

where

119861 = (

1 0 0

0 1 0

0 0 1

)

119860 = (



2 44 37 minus7

21 38 0 2

)

(29)




4 Extensions


119909 ge 0 120582119909 minus 119860119909 minus 119861119910 ge 0 119909119879

(120582119909 minus 119860119909 minus 119861119910) = 0

119910 ge 0 120583119910 minus 119862119909 minus 119863119910 ge 0 119910119879

(120583119910 minus 119862119909 minus 119863119910) = 0

(30)

where 119860 isin 119872119899(119877) 119861 isin 119872

119899119898(119877) 119861 isin 119872

119898119899(119877) and 119863 isin

119872119898(119877)

Let 120596 = (

119909

119910

120582

120583

) 119891(120596) = 119909 119892(120596) = 119910 119875(120596) = 120582119909minus119860119909minus119861119910


119899 0119899 0 0)120596 = 1



1198671(120596) = (

120601 (1198911(120596) 119875

1(120596))

120601 (119891119899(120596) 119875

119899(120596))

) = 0

1198672(120596) = (

120601 (1198921(120596) 119876

1(120596))

120601 (119892119899(120596) 119876

119899(120596))

) = 0

(119890119899 0119899 0 0) 120596 minus 1 = 0

(0119899 119890119899 0 0) 120596 minus 1 = 0

(31)

Let

Φ1(120596) =

1

2

10038171003817100381710038171198671(120596)1003817100381710038171003817

2

Φ2(120596) =

1

2

10038171003817100381710038171198672(120596)1003817100381710038171003817

2

(32)


Table 3

Method 1 120582 119909 SPA 120582 119909

2628209 (032922 028109 000067 068363)119879 100 (1 0 0 0)

119879

2641489 (034225 004798 030622 031649)119879 158 (0 1 0 0)

119879

2871143 (059383 030244 001458 043984)119879


2913415 (039357 016933 031781 012821)119879


3260775 (035641 026564 017973 062135)119879


3286388 (013771 033320 002167 050745)119879


3757690 (030291 019045 021854 027608)119879


4101635 (028247 019667 023447 026953)119879


4646811 (034130 028631 018549 016868)119879


4914424 (025544 018482 028418 027187)119879


6696950 (047995 022175 000256 029497)119879


7742278 (026550 028105 030904 014384)119879


7745752 (021870 031154 024021 022741)119879


9942333 (036284 032018 016593 013579)119879


9999801 (067443 013866 000391 018299)119879


10749890 (039982 033534 017631 008845)119879


12738892 (032969 037065 018570 010872)119879


14853897 (034127 045333 001989 017409)119879


15799623 (028006 037495 014411 007701)119879


19717149 (048407 060907 015118 018058)119879


20457942 (055801 067908 017056 014984)119879


22627891 (026236 032696 008010 001135)119879


23192132 (026744 033127 004324 000457)119879



(

Φ1(120596)

Φ2(120596)

(119890119899 0119899 0 0) 120596 minus 1

(0119899 119890119899 0 0) 120596 minus 1

) = 0 (33)



119861 = (1 0

0 1)

119862 = (1 0

0 1)

(34)



119909 ge 0 119872 (120582) 119909 ge 0 119909119879

119872(120582) 119909 = 0 (35)

Table 4

120582 120583 119909 119910

095803 096145 (069993 030006)119879

(069782 030217)119879

096205 097990 (069259 030740)119879

(068925 031074)119879

095574 094739 (048255 051744)119879

(047451 052548)119879

096069 093502 (037373 062626)119879

(037670 062329)119879

094582 104493 (069415 030584)119879

(069598 030401)119879

096200 103264 (046627 053372)119879

(047934 052065)119879

095517 106445 (035869 064130)119879

(037196 062803)119879

095744 105645 (039169 060830)119879

(039707 060292)119879


0 1198601 119860


119872(120582) =

119903

sum

119896=0

120582119896

119860119896 (36)


119872(120582) = 1205822

(

2 0 0

0 6 0

0 0 10

) + 120582(

7 0 0

0 30 0

0 0 20

) + (

minus2 6 0

2 16 3

0 5 0

)

(37)



Table 5

120582 119909

025315 (097503 001611 000884)119879

minus438990 (000000 100000 000000)119879

minus374482 (095986 002679 001334)119879

minus084644 (039691 042913 017392)119879

minus5000000 (000000 100000 000000)119879

minus060280 (040400 037328 022271)119879

minus366880 (083349 010559 006090)119879

minus187686 (019023 025846 055130)119879

minus198258 (004642 006140 089217)119879

minus001062 (010208 003658 086133)119879

5 Conclusion




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 3

Method 1 120582 119909 SPA 120582 119909

2628209 (032922 028109 000067 068363)119879 100 (1 0 0 0)

119879

2641489 (034225 004798 030622 031649)119879 158 (0 1 0 0)

119879

2871143 (059383 030244 001458 043984)119879


2913415 (039357 016933 031781 012821)119879


3260775 (035641 026564 017973 062135)119879


3286388 (013771 033320 002167 050745)119879


3757690 (030291 019045 021854 027608)119879


4101635 (028247 019667 023447 026953)119879


4646811 (034130 028631 018549 016868)119879


4914424 (025544 018482 028418 027187)119879


6696950 (047995 022175 000256 029497)119879


7742278 (026550 028105 030904 014384)119879


7745752 (021870 031154 024021 022741)119879


9942333 (036284 032018 016593 013579)119879


9999801 (067443 013866 000391 018299)119879


10749890 (039982 033534 017631 008845)119879


12738892 (032969 037065 018570 010872)119879


14853897 (034127 045333 001989 017409)119879


15799623 (028006 037495 014411 007701)119879


19717149 (048407 060907 015118 018058)119879


20457942 (055801 067908 017056 014984)119879


22627891 (026236 032696 008010 001135)119879


23192132 (026744 033127 004324 000457)119879



(

Φ1(120596)

Φ2(120596)

(119890119899 0119899 0 0) 120596 minus 1

(0119899 119890119899 0 0) 120596 minus 1

) = 0 (33)



119861 = (1 0

0 1)

119862 = (1 0

0 1)

(34)



119909 ge 0 119872 (120582) 119909 ge 0 119909119879

119872(120582) 119909 = 0 (35)

Table 4

120582 120583 119909 119910

095803 096145 (069993 030006)119879

(069782 030217)119879

096205 097990 (069259 030740)119879

(068925 031074)119879

095574 094739 (048255 051744)119879

(047451 052548)119879

096069 093502 (037373 062626)119879

(037670 062329)119879

094582 104493 (069415 030584)119879

(069598 030401)119879

096200 103264 (046627 053372)119879

(047934 052065)119879

095517 106445 (035869 064130)119879

(037196 062803)119879

095744 105645 (039169 060830)119879

(039707 060292)119879


0 1198601 119860


119872(120582) =

119903

sum

119896=0

120582119896

119860119896 (36)


119872(120582) = 1205822

(

2 0 0

0 6 0

0 0 10

) + 120582(

7 0 0

0 30 0

0 0 20

) + (

minus2 6 0

2 16 3

0 5 0

)

(37)



Table 5

120582 119909

025315 (097503 001611 000884)119879

minus438990 (000000 100000 000000)119879

minus374482 (095986 002679 001334)119879

minus084644 (039691 042913 017392)119879

minus5000000 (000000 100000 000000)119879

minus060280 (040400 037328 022271)119879

minus366880 (083349 010559 006090)119879

minus187686 (019023 025846 055130)119879

minus198258 (004642 006140 089217)119879

minus001062 (010208 003658 086133)119879

5 Conclusion




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Table 5

120582 119909

025315 (097503 001611 000884)119879

minus438990 (000000 100000 000000)119879

minus374482 (095986 002679 001334)119879

minus084644 (039691 042913 017392)119879

minus5000000 (000000 100000 000000)119879

minus060280 (040400 037328 022271)119879

minus366880 (083349 010559 006090)119879

minus187686 (019023 025846 055130)119879

minus198258 (004642 006140 089217)119879

minus001062 (010208 003658 086133)119879

5 Conclusion




Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article levenberg-marquardt method for the

Documents