Research ArticleSequential Quadratic Programming Method for Nonlinear LeastSquares Estimation and Its Application

Zhengqing Fu 1 Goulin Liu 2 and Lanlan Guo 3

1College of Mathematics and System Sciences SDUST Qingdao Shandong 266590 China2Geomatics College SDUST Qingdao Shandong 266590 Qingdao China3College of Mechanical and Electronic Engineering SDUST Qingdao Shandong 266590 China

Correspondence should be addressed to Zhengqing Fu fzhqing163com

Received 9 October 2018 Revised 18 January 2019 Accepted 19 May 2019 Published 18 June 2019

Academic Editor Sebastian Anita

Copyright copy 2019 Zhengqing Fu et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this study we propose a direction-controlled nonlinear least squares estimation model that combines the penalty functionand sequential quadratic programming The least squares model is transformed into a sequential quadratic programming modelallowing for the iteration direction to be controlled An ill-conditionedmatrix is processed by our model the least squares estimatethe ridge estimate and the results are compared based on a combination of qualitative and quantitative analyses For comparison weuse two equality indicators estimated residual fluctuation of differentmethods and the deviation between estimated and true valuesThe root-mean-squared error and standard deviation are used for quantitative analysis The results demonstrate that our proposedmodel has a smaller error than other methods Our proposed model is thereby found to be effective and has high precision It canobtain more precise results compared with other classical unwrapping algorithms as shown by unwrapping using both simulatedand real data from the Jining area in China

1 Introduction

Most problems faced during the course of processing mea-surement data are nonlinear in nature In the classical leastsquares approach the nonlinear approximation function isexpanded at an approximate value This transformation canlinearize the data allowing the problem to be solved throughthe linear least squares approach The precondition of thismodel is that initial parameter values must reach the adjustedvalue otherwise the model error is significant

Nonlinear least squares is an optimization method usedto solve nonlinear problems [1ndash4] Sequential quadratic pro-gramming is another important and effective optimizationmethod [5ndash7] A nonlinear least squares problem can betransformed into a sequential quadratic programming modeland then solved [8ndash11]

A direction-controlled estimation model is proposed inthis paper The nonlinear least squares problem is trans-formed into a sequential quadratic programming modelThe iterative point enters the feasible region via the penalty

function after which a problemrsquos optimal solution can beobtained through sequential quadratic programming [12ndash14]In the model the penalty function is used to reduce theconstraints of the initial values [15ndash19] Sequential quadraticprogramming possesses quadratic convergence The modelis contrasted with least squares and ridge estimate andsimulated data are used to check model performance [20ndash23] The root-mean-squared error (RMSE) and standarddeviation between the estimated and true values and the 120576value representing the quality of unwrapping are taken asindices of performance [24 25] We conducted experimentsthat showed the feasibility and effectiveness of the modelUnwrapping experiments using real data from the Jining areain China show that our proposed algorithm achieves moreprecise results than the least squares unwrapping algorithmsQuantitative indexes include differences in RMSE betweenrewrapped results and the original wrapped phase compu-tation time and 120576 values [26ndash28] The sequential quadraticprogramming method achieves better results with respect tothree indices

HindawiMathematical Problems in EngineeringVolume 2019 Article ID 3087949 8 pageshttpsdoiorg10115520193087949

2 Mathematical Problems in Engineering

2 Sequential Quadratic Programming Model

Quadratic optimization with equality constraints is definedas follows

min 119891 (119909) = 12119909119879119866119909 + 119892119879119909119904119905 119860119909 = 119887119909 ge 0 (1)

where 119866 isin 119877119899times119899 is a symmetric matrix 119892 isin 119877119899times1 119860 isin119877119898times119899 119887 isin 119877119898times1The Lagrange function of (1) is119871 (119909 119906) = 12119909119879119866119909 + 119892119879119909 + V119879 (119860119909 minus 119887) (2)

where V is the Lagrange multiplier At 119909 the KuhnndashTucker(KndashT) condition of (2) is119866119909 + 119892 + 119860119879V = 0119860119909 = 119887 (3)

If 119866 is a positive definite matrix then the unique solutionof (3) is 119909 = minus119866minus1119892 + 119866minus1119860119879 (119860119866minus1119860119879) (119860119866minus1119892 + 119887) (4)

In the sequential quadratic programming model anoptimal subproblem is set up to search for the next feasiblepoint in the current iterative point 119889119896 is the solution of theoptimal subproblem that is119909(119896+1) = 119909(119896) + 120572119896119889119896 (5)

where 120572119896 isin (0 1] The process for more iterations is repeatedand the optimization solution can then be obtained

The optimal subproblem of (1) can be defined as follows

min119909

nabla119891119879 (119909119896) (119909 minus 119909119896)+ 12 (119909 minus 119909119896)119879119866 (119909119896) (119909 minus 119909119896)119904119905 119904 (119909119896) + nabla119904119879 (119909119896) (119909 minus 119909119896) = 0

(6)

where 119904(119909) = 119860119909 minus 119887 nabla119891(119909119896) =(1205971198911205971199091 1205971198911205971199092 sdot sdot sdot 120597119891120597119909119899)119879 |119909=119909119896 nabla119904(119909119896) =(1205971199041205971199091 1205971199041205971199092 sdot sdot sdot 120597119904120597119909119899)119879 |119909=119909119896119866(119909119896) is the Hessian matrix of 119891(119909)119866(119909119896) = nabla2119891 (119909119896) = 12059721198911205971199092=[[[[[[[[[[[[

1205972119891 (119909)12059711990921 1205972119891 (119909)12059711990911205971199092 sdot sdot sdot 1205972119891 (119909)12059711990911205971199091199051205972119891 (119909)12059711990911205971199092 1205972119891 (119909)12059711990922 sdot sdot sdot 1205972119891 (119909)1205971199092120597119909119905sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot sdot1205972119891 (119909)1205971199091120597119909119905 1205972119891 (119909)1205971199092120597119909119905 sdot sdot sdot 1205972119891 (119909)1205971199092119905

]]]]]]]]]]]]

100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816119909=119909119896(7)

The penalty function of model (1) is defined as follows

(119909 120587) = 12119909119879119866119909 + 119892119879119909 + 120587 119860119909 minus 119887 (8)

where 120587 gt 0 is the penalty coefficientIf the objective and constraint function satisfy certain

conditions then the penalty method is convergent ie thealgorithm converges to the unique optimal result [15 29 30]

The penalty function method is simple to compute Asthe penalty coefficient increases the initial point is iterated tothe optimal solution However when the penalty coefficientis very large computation time increases

3 Sequential Quadratic Programming forNonlinear Least Squares

Synthesizing both methods the penalty function is simpleto compute in earlier iterations and the sequential quadraticprogramming method allows for quadratic convergence Adirection-controlled nonlinear least squares estimation algo-rithm using the penalty function and sequential quadraticprogramming is proposed herein The least squares modelis transformed into the quadratic optimization model Theiterative point enters the feasible region by the penaltyfunction and the optimal solution can then be obtainedby sequential quadratic programming Efficiency in problemsolving can be improved by combining the two methods

The nonlinear model is defined as follows119871 + Δ = 119891 (119909) (9)

where119871 = (1198711 1198712 sdot sdot sdot 119871119899)119879 is an 119899times1 vector119891 is a nonlinearfunction119883 = (1198831 1198832 sdot sdot sdot 119883119905)119879 is a 119905 times 1 vector to be estimatedand Δ = (Δ 1 Δ 2 sdot sdot sdot Δ 119899)119879 is an 119899 times 1 observational errorvector The error equation of (9) is119881 (119909) = 119891 (119909) minus 119871 (10)

The least squares principle must be satisfied that is

min119865 (119909) = 119881119879 (119909) 119875119881 (119909)= [119891 (119909) minus 119871]119879 119875 [119891 (119909) minus 119871] (11)

Equation (11) can be converted to quadratic optimizationwithequality constraints as follows

min 119865 (Δ119909119896) = 119881119879119875119881119904119905 119881 = 119891 (119909119896) minus 119871 (12)

The subproblem of (12) can be obtained by linearizing theequality constraint at 119909119896 The expression is as follows

min 119865 (Δ119909119896) = 119881119879119875119881119904119905 119881 = 119861 (119909119896)Δ119909119896 + 119862 (119909119896) minus 119871 (13)

Mathematical Problems in Engineering 3

Table 1 Estimated values by sequential quadratic programming least squares and the ridge estimate

True value Least squares method Ridge method Sequential quadratic programming method

119909 1 28552 08654 122051 -13035 13872 125981 12584 08283 118961 23954 05180 090641 -19306 11375 11034

RMSE 19672 02996 01852The standard deviation 43989 06699 04141

where 119861(119909119896) = 120597119891(119909)120597119909|119909=119909119896 119862(119909119896) = 119891(119909119896) and thepositive definite matrix 119875 is the weight matrix Equation (12)shows convex programming

The penalty function of (12) is (119909 120587) = 119881119879119875119881 + 120587 1003817100381710038171003817119891 (119909) minus 1198711003817100381710038171003817 (14)The steps of the algorithm using the penalty function and

sequential quadratic programming are as follows(1)The initial values are defined as 119909(0) and 1205870 the controlerror is 119890 gt 0 120576 gt 0 and the number of iterations is 119896(119896 = 0)(2)The initial value119909(0) is substituted into (14) to calculatethe iterative value 119891(119909(119896)) minus 119871 If 119891(119909(119896)) minus 119871 lt 119890 theiteration terminates(3) If 119909(119896+1) minus 119909(119896) le 120576 the result is 119909lowast = 119909(119896+1)otherwise set 119909 = 119909(119896) and move to the next step(4) By taking 119909(0) = 119909 as the new initial value(12) is solved using sequential quadratic programming Thecorrection 119909(119896) value can then be obtained(5) If 119909(119896+1) minus 119909(119896) le 120576 the result is 119909lowast = 119909(119896+1)otherwise return to step (4) until the condition is met

In the course of running the algorithm the penaltycoefficient of (6) is as follows [31]

120587 = max(1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816 [119881119879 (Δ119909119896)119879] ( 119875 00 0 ) ( 119881Δ119909119896 )Δ119909119896 1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816 2120587minus + 120573) (15)

where 120573 ge 0 is the constant and 120587minus is the penalty coefficientof the previous iteration

4 Experimental Results and Analysis

41 Simulated Data In this case 119860119883 = 119871 where119860

=(((((((((((((((((

20000 minus50000 10000 10000 minus95000minus20000 40000 10000 minus10500 85000minus20000 10000 10000 minus10000 24000minus10000 25000 40000 minus05000 70000minus10000 32000 40000 minus05000 8400010000 10000 minus30000 04000 0490030000 70000 minus30000 15000 12700050000 minus10000 minus20000 25000 minus3000040000 20000 minus20000 20100 3000040000 30000 minus20000 20000 50000

)))))))))))))))))

(16)

solu

tion

norm

|| Ｒ

||2

residual norm || A Ｒ - ＜ ||2

L-curve Tikh corner at 145683

25

2

15

1

10minus1

100

101

102

043778

06873

107911694126597

41758

65559

102928

161596

Figure 1 Ridge parameter from the L-curve

The observations are = [minus105 1045 14 12 141 minus011 212 15 901 12]119879119883 = [1199091 1199092 1199093 1199094 1199095]119879are unknown parameters Thetrue values are 119883 = [1 1 1 1 1]119879

The white noise is Δ sim 119873(0 1205902119868) where 119868 denotes theidentity matrix and 120590 = 01 119890 = 10minus2 120576 = 10minus2

The condition number of 119860 is 41397 so this is an ill-conditioned problem Using sequential quadratic program-ming least squares and the ridge estimate the estimatedresults can be compared as shown in Table 1

In Table 1 the ridge parameter of the ridge estimation is14568 Figure 1 shows the ridge parameter from the L-curve[32]

In this ill-conditioned problem all methods obtain esti-mated values The distance between the initial and truevalues is large Sequential quadratic programming obtains theclosest solution (Table 1) The RMSE and standard deviationshow that the sequential quadratic programming model isthe best and the ridge estimate method is better than theleast squares method In sequential quadratic programmingthe iterative point enters the feasible region via the penaltyfunction and the optimal solution can then be obtainedvia sequential quadratic programming The constraints onthe initial values of sequential quadratic programming arereduced however the computational cost is increased Ridge

4 Mathematical Problems in Engineering

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

3

2

1

0

-1

-2

-3

(a) Noise-free wrapped phase

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

3

2

1

0

-1

-2

-3

(b) Wrapped phase after noise

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

25

20

15

10

5

0

-5

-10

-15

-20

(c) Least squares method

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

15

10

5

0

-5

-10

-15

(d) Sequential quadratic programming method

Figure 2 Unwrapping results obtained using the above-mentioned two methods

estimation performs much better than least squares butdetermining the ridge parameters is difficult

42 Phase Unwrapping Experiments

421 Simulated Data To verify the effectiveness of the algo-rithm simulated data were used to conduct experiments In asimulated interferogram with a size of 100 pixels times 100 pixelsa coherence coefficient of 085 and Gaussian noise with astandard deviation of 08160 radians were added to the image119890 = 120576 = 10minus3 The least squares and sequential quadraticprogrammingmethods were used to unwrap images Figure 2shows the unwrapping result of each method

Both methods obtain good unwrapping results withoutlines or islands The image obtained by sequential quadraticprogramming is smoother than that obtained by least squaresand is more similar to the noise-free image

Figure 3 shows a cross-sectional view of the unwrappingphase

As shown in Figure 3 the abscissa represents the hori-zontal position (x-coordinate) of the pixel in the 50th line ofthe image The ordinate denotes the phase The maximumvalue of the cross-sectional view of sequential quadraticprogramming is close to the maximum value of the simulatedsurface The maximum value of the cross-sectional view ofthe least squares method is far from the maximum value of

Mathematical Problems in Engineering 5

100 20 30 40 50 60 70 80 90 100

15

10

5

0

minus5

minus10

minus15

(a) Noise-free100 20 30 40 50 60 70 80 90 100

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

(b) Least squares

100 20 30 40 50 60 70 80 90 100

15

10

5

0

minus5

minus10

minus15

20

(c) Sequential quadratic programming

Figure 3 Fiftieth line cross-sectional view of the unwrapping phase using least squares and sequential quadratic programming

Table 2 Unwrapping time RMSE and 120576 valueMethod Computation time (s) RMSE (rad) 120576 (rad)Least squares 12142 07594 06214Sequential quadratic programming 11866 05393 04228

Table 3 Selected parameters of ASAR image data

Number Sensor Date Type Latitude and longitude of center Polarization Ascending Wavelength022403 ENVISAT 20050306 SLC 116314 6 35305 1 VV Ascending C022404 ENVISAT 20050130 SLC 116314 4 35305 1 VV Ascending C

the simulated surface The curve obtained for of the leastsquares method is also much rougher than that of sequentialquadratic programming

Table 2 shows the running time RMSE and 120576 valueof sequential quadratic programming and least squaresobtained by phase unwrapping Both methods result inrelatively reliable unwrapping The running time is similarindicating that the efficiency of the two algorithms is almostthe same The RMSE of sequential quadratic programming issmaller than that of least squares indicating that sequentialquadratic programming is more precise 120576 indicates the qual-ity of the wrap the smaller the 120576 the better the unwrappingquality Thus the sequential quadratic programming methodprovides better unwrapping than the least squares method

The experimentwith simulated data shows that sequentialquadratic programming is a feasible method the model iseffective and has high precision

422 Observational Data Satellite data collected by ENVI-SAT ASAR covering the Jining area of Shandong provinceChina were used to test the modelrsquos performance

Table 3 shows the basic parameters of the dataThe images are 1024 times 1024 pixels with a resolution of

30m Figure 4 shows the unwrapping result of each method119890 = 120576 = 10minus3Both unwrapping methods were run successfully In the

interferogram the stripe noise is large and the fringes areunclear The phase ripple of sequential quadratic program-ming is smaller than that of least squares demonstrating thatsequential quadratic programming experiences less interfer-ence

Table 4 shows the computation time RMSE and 120576 valueThe RMSE and 120576 values of sequential quadratic program-

ming are smaller than those of the least squares methodThis

6 Mathematical Problems in Engineering

3

2

1

0

-1

-2

-3

100

200

300

400

500

600

700

800

900

1000

200 400 600 800 1000

(a) Wrapped phase

100

200

300

400

500

600

700

800

900

1000

200 400 600 800 1000

15

10

5

0

-5

-10

-15

-20

(b) Least squares method

100

200

300

400

500

600

700

800

900

1000

100 200 300 400 500 600 700 800 900 1000

10

5

0

-5

-10

-15

-20

(c) Sequential quadratic programming method

Figure 4 Unwrapping of ASAR images using least squares and sequential quadratic programming

Table 4 Unwrapping time RMSE and 120576 valuesMethod computation time (s) RMSE (rad) 120576 (rad)Least squares 725940 17594 06214Sequential quadratic programming 1336948 05393 04228

illustrates that the difference between the rewrapped resultsand the original wrapped phase is small The unwrappingphases of the sequential quadratic programming method aresmoother and more reliable However the computation timeof sequential quadratic programming is greater than that ofthe least squares method This is due to high levels of noiseand the penalty function increasing the computation time ofsequential quadratic programming

5 Conclusions

A direction-controlled nonlinear least squares estima-tion algorithm using the sequential quadratic programmingmethod is proposed herein The least squares model is

transformed into the nonlinear programming model andthe iteration direction can then be controlled by sequentialquadratic programming The ill-conditioned matrix andphase unwrapping are used to verify the method Theproposed algorithm achieves better results than alternativeapproaches in terms of RMSE and 120576 values and its effec-tiveness is demonstrated using observational satellite data Infuture research the role change if the inequality constrainsare taken into account will be considered

Data Availability

The 20050130slc and 20050306slc data used to support thefindings of this study are available from the correspondingauthor upon request

Mathematical Problems in Engineering 7

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grants nos 41774002 and 41274007)and the Scientific Research Fund Project for the Introductionof Talents of Shandong University of Science and Technology(Grants nos 2014RCJJ047 and 2015RCJJ022)

References

[1] C Han F Zheng T Guo and G He ldquoParallel algorithms forlarge-scale linearly constrained minimization problemrdquo ActaMathematicae Applicatae Sinica vol 30 no 3 pp 707ndash720 2014

[2] X Jin Y Liang D Tian and F Zhuang ldquoParticle swarmoptimization using dimension selection methodsrdquo AppliedMathematics and Computation vol 219 no 10 pp 5185ndash51972013

[3] G Tutz and G Schauberger ldquoA penalty approach to differentialitem functioning in Rasch modelsrdquo Psychometrika vol 80 no1 pp 21ndash43 2015

[4] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013

[5] G Ma and B Huang ldquoOptimization of process parameters ofstamping forming of the automotive lower floor boardrdquo Journalof Applied Mathematics vol 2014 Article ID 470320 9 pages2014

[6] F Tramontana A A Elsadany B Xin and H N Agiza ldquoLocalstability of the Cournot solution with increasing heterogeneouscompetitorsrdquo Nonlinear Analysis Real World Applications vol26 pp 150ndash160 2015

[7] M Li X Kao and H Che ldquoRelaxed inertial acceleratedalgorithms for solving split equality feasibility problemrdquo Journalof Nonlinear Sciences and Applications vol 10 no 8 pp 4109ndash4121 2017

[8] M Huang and D Pu ldquoA line search SQP method without apenalty or a filterrdquo Computational amp Applied Mathematics vol34 no 2 pp 741ndash753 2015

[9] A F Izmailov M V Solodov and E I Uskov ldquoCombiningstabilized SQP with the augmented Lagrangian algorithmrdquoComputational Optimization and Applications vol 62 no 2 pp405ndash429 2015

[10] J Gao B Shen E Feng and Z Xiu ldquoModelling and optimalcontrol for an impulsive dynamical system in microbial fed-batch culturerdquoComputational and AppliedMathematics vol 32no 2 pp 275ndash290 2013

[11] J Zhu and B Hao ldquoA new class of smoothing functions anda smoothing Newton method for complementarity problemsrdquoOptimization Letters vol 7 no 3 pp 481ndash497 2013

[12] M Huang and D Pu ldquoA trust-region SQP method withouta penalty or a filter for nonlinear programmingrdquo Journal ofComputational and Applied Mathematics vol 281 pp 107ndash1192015

[13] B Sachsenberg and K Schittkowski ldquoA combined SQPndashIPMalgorithm for solving large-scale nonlinear optimization prob-lemsrdquo Optimization Letters vol 9 no 7 pp 1271ndash1282 2015

[14] J Tang G He and L Fang ldquoA new non-interior continua-tion method for second-order cone programmingrdquo Journal ofNumerical Mathematics vol 21 no 4 pp 301ndash324 2013

[15] C Shen W Shao and W Xue ldquoOn the local convergence ofa penalty-function-free SQP methodrdquo Numerical FunctionalAnalysis and Optimization vol 35 no 5 pp 623ndash647 2014

[16] S Pang T Li F Dai and M Yu ldquoParticle swarm optimizationalgorithm for multi-salesman problem with time and capacityconstraintsrdquo Applied Mathematics amp Information Sciences vol7 no 6 pp 2439ndash2444 2013

[17] Z Z Feng L Fang and G He ldquoAn O Nl iteration primal- dualpath- following method based on wide neighbourhood andlarge update for second- order cone programmingrdquo Optimiza-tion A Journal of Mathematical Programming and OperationsResearch vol 63 no 5 pp 679ndash691 2014

[18] C Han T Feng G He and T Guo ldquoParallel variable distribu-tion algorithm for constrained optimizationwith nonmonotonetechniquerdquo Journal of AppliedMathematics vol 2013 Article ID295147 7 pages 2013

[19] JWang K Liang XHuang ZWang andH Shen ldquoDissipativefault-tolerant control for nonlinear singular perturbed systemswithMarkov jumping parameters based on slow state feedbackrdquoApplied Mathematics and Computation vol 328 pp 247ndash2622018

[20] J B Jian Y L Lai and K C Zhang ldquoA feasible method forsuperlinearly and quadratically convergent sequential systemsof equationsrdquo Acta Mathematica Sinica vol 45 no 6 pp 1137ndash1146 2002

[21] J Che and K Su ldquoA modified SQP method and its globalconvergencerdquo Applied Mathematics and Computation vol 186no 2 pp 945ndash951 2007

[22] P-y Nie ldquoSequential penalty quadratic programming filtermethods for nonlinear programmingrdquo Nonlinear Analysis RealWorld Applications vol 8 no 1 pp 118ndash129 2007

[23] J Yu M Li Y Wang and G He ldquoA decomposition method forlarge-scale box constrained optimizationrdquoAppliedMathematicsand Computation vol 231 no 12 pp 9ndash15 2014

[24] M Chen and G Li ldquoForming mechanism and correction of CTimage artifacts caused by the errors of three system parametersrdquoJournal of Applied Mathematics vol 2013 Article ID 545147 7pages 2013

[25] T Jiang Z Jiang and S Ling ldquoAn algebraic method forquaternion and complex least squares coneigen-problem inquantum mechanicsrdquo Applied Mathematics and Computationvol 249 pp 222ndash228 2014

[26] Z Pang G Liu D Zhou and D Sun ldquoData-based predictivecontrol for networked nonlinear systems with packet dropoutandmeasurement noiserdquo Journal of Systems Science ampComplex-ity vol 30 no 5 pp 1072ndash1083 2017

[27] F Liu Z Fu Y Zheng and Q Yuan ldquoA rough Marcinkiewiczintegral along smooth curvesrdquo Journal of Nonlinear Sciences andApplications vol 9 no 6 pp 4450ndash4464 2016

[28] S Ding H Huang X Xu and J Wang ldquoPolynomial smoothtwin support vector machinesrdquo Applied Mathematics amp Infor-mation Sciences vol 8 no 4 pp 2063ndash2071 2014

[29] C H Liu Y L Shang and P Han ldquoA new infeasible-interior-point algorithm for linear programming over symmetric conesrdquoActaMathematicae Applicatae Sinica vol 33 no 3 pp 771ndash7882017

[30] N T Hang ldquoThe penalty functionsmethod andmultiplier rulesbased on the Mordukhovich subdifferentialrdquo Set-Valued andVariational Analysis vol 22 no 2 pp 299ndash312 2014

8 Mathematical Problems in Engineering

[31] Y Zhang and D Shen ldquoEstimation of semi-parametric varying-coefficient spatial panel data models with random-effectsrdquoJournal of Statistical Planning and Inference vol 159 pp 64ndash802015

[32] S Salman and H S Dawood ldquoApproximate solution of thelinear programming problems by ant system optimizationrdquoEngineering amp Technology Journal no 16 pp 2978ndash2995 2009

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Mathematical Problems in Engineering 3

Table 1 Estimated values by sequential quadratic programming least squares and the ridge estimate

True value Least squares method Ridge method Sequential quadratic programming method

119909 1 28552 08654 122051 -13035 13872 125981 12584 08283 118961 23954 05180 090641 -19306 11375 11034

RMSE 19672 02996 01852The standard deviation 43989 06699 04141

where 119861(119909119896) = 120597119891(119909)120597119909|119909=119909119896 119862(119909119896) = 119891(119909119896) and thepositive definite matrix 119875 is the weight matrix Equation (12)shows convex programming

The penalty function of (12) is (119909 120587) = 119881119879119875119881 + 120587 1003817100381710038171003817119891 (119909) minus 1198711003817100381710038171003817 (14)The steps of the algorithm using the penalty function and

sequential quadratic programming are as follows(1)The initial values are defined as 119909(0) and 1205870 the controlerror is 119890 gt 0 120576 gt 0 and the number of iterations is 119896(119896 = 0)(2)The initial value119909(0) is substituted into (14) to calculatethe iterative value 119891(119909(119896)) minus 119871 If 119891(119909(119896)) minus 119871 lt 119890 theiteration terminates(3) If 119909(119896+1) minus 119909(119896) le 120576 the result is 119909lowast = 119909(119896+1)otherwise set 119909 = 119909(119896) and move to the next step(4) By taking 119909(0) = 119909 as the new initial value(12) is solved using sequential quadratic programming Thecorrection 119909(119896) value can then be obtained(5) If 119909(119896+1) minus 119909(119896) le 120576 the result is 119909lowast = 119909(119896+1)otherwise return to step (4) until the condition is met

In the course of running the algorithm the penaltycoefficient of (6) is as follows [31]

120587 = max(1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816 [119881119879 (Δ119909119896)119879] ( 119875 00 0 ) ( 119881Δ119909119896 )Δ119909119896 1003816100381610038161003816100381610038161003816100381610038161003816100381610038161003816 2120587minus + 120573) (15)

where 120573 ge 0 is the constant and 120587minus is the penalty coefficientof the previous iteration

4 Experimental Results and Analysis

41 Simulated Data In this case 119860119883 = 119871 where119860

=(((((((((((((((((

20000 minus50000 10000 10000 minus95000minus20000 40000 10000 minus10500 85000minus20000 10000 10000 minus10000 24000minus10000 25000 40000 minus05000 70000minus10000 32000 40000 minus05000 8400010000 10000 minus30000 04000 0490030000 70000 minus30000 15000 12700050000 minus10000 minus20000 25000 minus3000040000 20000 minus20000 20100 3000040000 30000 minus20000 20000 50000

)))))))))))))))))

(16)

solu

tion

norm

|| Ｒ

||2

residual norm || A Ｒ - ＜ ||2

L-curve Tikh corner at 145683

25

2

15

1

10minus1

100

101

102

043778

06873

107911694126597

41758

65559

102928

161596

Figure 1 Ridge parameter from the L-curve

The observations are = [minus105 1045 14 12 141 minus011 212 15 901 12]119879119883 = [1199091 1199092 1199093 1199094 1199095]119879are unknown parameters Thetrue values are 119883 = [1 1 1 1 1]119879

The white noise is Δ sim 119873(0 1205902119868) where 119868 denotes theidentity matrix and 120590 = 01 119890 = 10minus2 120576 = 10minus2

The condition number of 119860 is 41397 so this is an ill-conditioned problem Using sequential quadratic program-ming least squares and the ridge estimate the estimatedresults can be compared as shown in Table 1

In Table 1 the ridge parameter of the ridge estimation is14568 Figure 1 shows the ridge parameter from the L-curve[32]

In this ill-conditioned problem all methods obtain esti-mated values The distance between the initial and truevalues is large Sequential quadratic programming obtains theclosest solution (Table 1) The RMSE and standard deviationshow that the sequential quadratic programming model isthe best and the ridge estimate method is better than theleast squares method In sequential quadratic programmingthe iterative point enters the feasible region via the penaltyfunction and the optimal solution can then be obtainedvia sequential quadratic programming The constraints onthe initial values of sequential quadratic programming arereduced however the computational cost is increased Ridge

4 Mathematical Problems in Engineering

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

3

2

1

0

-1

-2

-3

(a) Noise-free wrapped phase

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

3

2

1

0

-1

-2

-3

(b) Wrapped phase after noise

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

25

20

15

10

5

0

-5

-10

-15

-20

(c) Least squares method

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

15

10

5

0

-5

-10

-15

(d) Sequential quadratic programming method

Figure 2 Unwrapping results obtained using the above-mentioned two methods

estimation performs much better than least squares butdetermining the ridge parameters is difficult

42 Phase Unwrapping Experiments

421 Simulated Data To verify the effectiveness of the algo-rithm simulated data were used to conduct experiments In asimulated interferogram with a size of 100 pixels times 100 pixelsa coherence coefficient of 085 and Gaussian noise with astandard deviation of 08160 radians were added to the image119890 = 120576 = 10minus3 The least squares and sequential quadraticprogrammingmethods were used to unwrap images Figure 2shows the unwrapping result of each method

Both methods obtain good unwrapping results withoutlines or islands The image obtained by sequential quadraticprogramming is smoother than that obtained by least squaresand is more similar to the noise-free image

Figure 3 shows a cross-sectional view of the unwrappingphase

As shown in Figure 3 the abscissa represents the hori-zontal position (x-coordinate) of the pixel in the 50th line ofthe image The ordinate denotes the phase The maximumvalue of the cross-sectional view of sequential quadraticprogramming is close to the maximum value of the simulatedsurface The maximum value of the cross-sectional view ofthe least squares method is far from the maximum value of

Mathematical Problems in Engineering 5

100 20 30 40 50 60 70 80 90 100

15

10

5

0

minus5

minus10

minus15

(a) Noise-free100 20 30 40 50 60 70 80 90 100

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

(b) Least squares

100 20 30 40 50 60 70 80 90 100

15

10

5

0

minus5

minus10

minus15

20

(c) Sequential quadratic programming

Figure 3 Fiftieth line cross-sectional view of the unwrapping phase using least squares and sequential quadratic programming

Table 2 Unwrapping time RMSE and 120576 valueMethod Computation time (s) RMSE (rad) 120576 (rad)Least squares 12142 07594 06214Sequential quadratic programming 11866 05393 04228

Table 3 Selected parameters of ASAR image data

Number Sensor Date Type Latitude and longitude of center Polarization Ascending Wavelength022403 ENVISAT 20050306 SLC 116314 6 35305 1 VV Ascending C022404 ENVISAT 20050130 SLC 116314 4 35305 1 VV Ascending C

the simulated surface The curve obtained for of the leastsquares method is also much rougher than that of sequentialquadratic programming

Table 2 shows the running time RMSE and 120576 valueof sequential quadratic programming and least squaresobtained by phase unwrapping Both methods result inrelatively reliable unwrapping The running time is similarindicating that the efficiency of the two algorithms is almostthe same The RMSE of sequential quadratic programming issmaller than that of least squares indicating that sequentialquadratic programming is more precise 120576 indicates the qual-ity of the wrap the smaller the 120576 the better the unwrappingquality Thus the sequential quadratic programming methodprovides better unwrapping than the least squares method

The experimentwith simulated data shows that sequentialquadratic programming is a feasible method the model iseffective and has high precision

422 Observational Data Satellite data collected by ENVI-SAT ASAR covering the Jining area of Shandong provinceChina were used to test the modelrsquos performance

Table 3 shows the basic parameters of the dataThe images are 1024 times 1024 pixels with a resolution of

30m Figure 4 shows the unwrapping result of each method119890 = 120576 = 10minus3Both unwrapping methods were run successfully In the

interferogram the stripe noise is large and the fringes areunclear The phase ripple of sequential quadratic program-ming is smaller than that of least squares demonstrating thatsequential quadratic programming experiences less interfer-ence

Table 4 shows the computation time RMSE and 120576 valueThe RMSE and 120576 values of sequential quadratic program-

ming are smaller than those of the least squares methodThis

6 Mathematical Problems in Engineering

3

2

1

0

-1

-2

-3

100

200

300

400

500

600

700

800

900

1000

200 400 600 800 1000

(a) Wrapped phase

100

200

300

400

500

600

700

800

900

1000

200 400 600 800 1000

15

10

5

0

-5

-10

-15

-20

(b) Least squares method

100

200

300

400

500

600

700

800

900

1000

100 200 300 400 500 600 700 800 900 1000

10

5

0

-5

-10

-15

-20

(c) Sequential quadratic programming method

Figure 4 Unwrapping of ASAR images using least squares and sequential quadratic programming

Table 4 Unwrapping time RMSE and 120576 valuesMethod computation time (s) RMSE (rad) 120576 (rad)Least squares 725940 17594 06214Sequential quadratic programming 1336948 05393 04228

illustrates that the difference between the rewrapped resultsand the original wrapped phase is small The unwrappingphases of the sequential quadratic programming method aresmoother and more reliable However the computation timeof sequential quadratic programming is greater than that ofthe least squares method This is due to high levels of noiseand the penalty function increasing the computation time ofsequential quadratic programming

5 Conclusions

A direction-controlled nonlinear least squares estima-tion algorithm using the sequential quadratic programmingmethod is proposed herein The least squares model is

transformed into the nonlinear programming model andthe iteration direction can then be controlled by sequentialquadratic programming The ill-conditioned matrix andphase unwrapping are used to verify the method Theproposed algorithm achieves better results than alternativeapproaches in terms of RMSE and 120576 values and its effec-tiveness is demonstrated using observational satellite data Infuture research the role change if the inequality constrainsare taken into account will be considered

Data Availability

The 20050130slc and 20050306slc data used to support thefindings of this study are available from the correspondingauthor upon request

Mathematical Problems in Engineering 7

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grants nos 41774002 and 41274007)and the Scientific Research Fund Project for the Introductionof Talents of Shandong University of Science and Technology(Grants nos 2014RCJJ047 and 2015RCJJ022)

References

[1] C Han F Zheng T Guo and G He ldquoParallel algorithms forlarge-scale linearly constrained minimization problemrdquo ActaMathematicae Applicatae Sinica vol 30 no 3 pp 707ndash720 2014

[2] X Jin Y Liang D Tian and F Zhuang ldquoParticle swarmoptimization using dimension selection methodsrdquo AppliedMathematics and Computation vol 219 no 10 pp 5185ndash51972013

[3] G Tutz and G Schauberger ldquoA penalty approach to differentialitem functioning in Rasch modelsrdquo Psychometrika vol 80 no1 pp 21ndash43 2015

[4] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013

[5] G Ma and B Huang ldquoOptimization of process parameters ofstamping forming of the automotive lower floor boardrdquo Journalof Applied Mathematics vol 2014 Article ID 470320 9 pages2014

[6] F Tramontana A A Elsadany B Xin and H N Agiza ldquoLocalstability of the Cournot solution with increasing heterogeneouscompetitorsrdquo Nonlinear Analysis Real World Applications vol26 pp 150ndash160 2015

[7] M Li X Kao and H Che ldquoRelaxed inertial acceleratedalgorithms for solving split equality feasibility problemrdquo Journalof Nonlinear Sciences and Applications vol 10 no 8 pp 4109ndash4121 2017

[8] M Huang and D Pu ldquoA line search SQP method without apenalty or a filterrdquo Computational amp Applied Mathematics vol34 no 2 pp 741ndash753 2015

[9] A F Izmailov M V Solodov and E I Uskov ldquoCombiningstabilized SQP with the augmented Lagrangian algorithmrdquoComputational Optimization and Applications vol 62 no 2 pp405ndash429 2015

[10] J Gao B Shen E Feng and Z Xiu ldquoModelling and optimalcontrol for an impulsive dynamical system in microbial fed-batch culturerdquoComputational and AppliedMathematics vol 32no 2 pp 275ndash290 2013

[11] J Zhu and B Hao ldquoA new class of smoothing functions anda smoothing Newton method for complementarity problemsrdquoOptimization Letters vol 7 no 3 pp 481ndash497 2013

[12] M Huang and D Pu ldquoA trust-region SQP method withouta penalty or a filter for nonlinear programmingrdquo Journal ofComputational and Applied Mathematics vol 281 pp 107ndash1192015

[13] B Sachsenberg and K Schittkowski ldquoA combined SQPndashIPMalgorithm for solving large-scale nonlinear optimization prob-lemsrdquo Optimization Letters vol 9 no 7 pp 1271ndash1282 2015

[14] J Tang G He and L Fang ldquoA new non-interior continua-tion method for second-order cone programmingrdquo Journal ofNumerical Mathematics vol 21 no 4 pp 301ndash324 2013

[15] C Shen W Shao and W Xue ldquoOn the local convergence ofa penalty-function-free SQP methodrdquo Numerical FunctionalAnalysis and Optimization vol 35 no 5 pp 623ndash647 2014

[16] S Pang T Li F Dai and M Yu ldquoParticle swarm optimizationalgorithm for multi-salesman problem with time and capacityconstraintsrdquo Applied Mathematics amp Information Sciences vol7 no 6 pp 2439ndash2444 2013

[17] Z Z Feng L Fang and G He ldquoAn O Nl iteration primal- dualpath- following method based on wide neighbourhood andlarge update for second- order cone programmingrdquo Optimiza-tion A Journal of Mathematical Programming and OperationsResearch vol 63 no 5 pp 679ndash691 2014

[18] C Han T Feng G He and T Guo ldquoParallel variable distribu-tion algorithm for constrained optimizationwith nonmonotonetechniquerdquo Journal of AppliedMathematics vol 2013 Article ID295147 7 pages 2013

[19] JWang K Liang XHuang ZWang andH Shen ldquoDissipativefault-tolerant control for nonlinear singular perturbed systemswithMarkov jumping parameters based on slow state feedbackrdquoApplied Mathematics and Computation vol 328 pp 247ndash2622018

[20] J B Jian Y L Lai and K C Zhang ldquoA feasible method forsuperlinearly and quadratically convergent sequential systemsof equationsrdquo Acta Mathematica Sinica vol 45 no 6 pp 1137ndash1146 2002

[21] J Che and K Su ldquoA modified SQP method and its globalconvergencerdquo Applied Mathematics and Computation vol 186no 2 pp 945ndash951 2007

[22] P-y Nie ldquoSequential penalty quadratic programming filtermethods for nonlinear programmingrdquo Nonlinear Analysis RealWorld Applications vol 8 no 1 pp 118ndash129 2007

[23] J Yu M Li Y Wang and G He ldquoA decomposition method forlarge-scale box constrained optimizationrdquoAppliedMathematicsand Computation vol 231 no 12 pp 9ndash15 2014

[24] M Chen and G Li ldquoForming mechanism and correction of CTimage artifacts caused by the errors of three system parametersrdquoJournal of Applied Mathematics vol 2013 Article ID 545147 7pages 2013

[25] T Jiang Z Jiang and S Ling ldquoAn algebraic method forquaternion and complex least squares coneigen-problem inquantum mechanicsrdquo Applied Mathematics and Computationvol 249 pp 222ndash228 2014

[26] Z Pang G Liu D Zhou and D Sun ldquoData-based predictivecontrol for networked nonlinear systems with packet dropoutandmeasurement noiserdquo Journal of Systems Science ampComplex-ity vol 30 no 5 pp 1072ndash1083 2017

[27] F Liu Z Fu Y Zheng and Q Yuan ldquoA rough Marcinkiewiczintegral along smooth curvesrdquo Journal of Nonlinear Sciences andApplications vol 9 no 6 pp 4450ndash4464 2016

[28] S Ding H Huang X Xu and J Wang ldquoPolynomial smoothtwin support vector machinesrdquo Applied Mathematics amp Infor-mation Sciences vol 8 no 4 pp 2063ndash2071 2014

[29] C H Liu Y L Shang and P Han ldquoA new infeasible-interior-point algorithm for linear programming over symmetric conesrdquoActaMathematicae Applicatae Sinica vol 33 no 3 pp 771ndash7882017

[30] N T Hang ldquoThe penalty functionsmethod andmultiplier rulesbased on the Mordukhovich subdifferentialrdquo Set-Valued andVariational Analysis vol 22 no 2 pp 299ndash312 2014

8 Mathematical Problems in Engineering

[31] Y Zhang and D Shen ldquoEstimation of semi-parametric varying-coefficient spatial panel data models with random-effectsrdquoJournal of Statistical Planning and Inference vol 159 pp 64ndash802015

[32] S Salman and H S Dawood ldquoApproximate solution of thelinear programming problems by ant system optimizationrdquoEngineering amp Technology Journal no 16 pp 2978ndash2995 2009

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

4 Mathematical Problems in Engineering

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

3

2

1

0

-1

-2

-3

(a) Noise-free wrapped phase

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

3

2

1

0

-1

-2

-3

(b) Wrapped phase after noise

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

25

20

15

10

5

0

-5

-10

-15

-20

(c) Least squares method

10

20

30

40

50

60

70

80

90

100

10 20 30 40 50 60 70 80 90 100

15

10

5

0

-5

-10

-15

(d) Sequential quadratic programming method

Figure 2 Unwrapping results obtained using the above-mentioned two methods

estimation performs much better than least squares butdetermining the ridge parameters is difficult

42 Phase Unwrapping Experiments

421 Simulated Data To verify the effectiveness of the algo-rithm simulated data were used to conduct experiments In asimulated interferogram with a size of 100 pixels times 100 pixelsa coherence coefficient of 085 and Gaussian noise with astandard deviation of 08160 radians were added to the image119890 = 120576 = 10minus3 The least squares and sequential quadraticprogrammingmethods were used to unwrap images Figure 2shows the unwrapping result of each method

Both methods obtain good unwrapping results withoutlines or islands The image obtained by sequential quadraticprogramming is smoother than that obtained by least squaresand is more similar to the noise-free image

Figure 3 shows a cross-sectional view of the unwrappingphase

As shown in Figure 3 the abscissa represents the hori-zontal position (x-coordinate) of the pixel in the 50th line ofthe image The ordinate denotes the phase The maximumvalue of the cross-sectional view of sequential quadraticprogramming is close to the maximum value of the simulatedsurface The maximum value of the cross-sectional view ofthe least squares method is far from the maximum value of

Mathematical Problems in Engineering 5

100 20 30 40 50 60 70 80 90 100

15

10

5

0

minus5

minus10

minus15

(a) Noise-free100 20 30 40 50 60 70 80 90 100

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

(b) Least squares

100 20 30 40 50 60 70 80 90 100

15

10

5

0

minus5

minus10

minus15

20

(c) Sequential quadratic programming

Figure 3 Fiftieth line cross-sectional view of the unwrapping phase using least squares and sequential quadratic programming

Table 2 Unwrapping time RMSE and 120576 valueMethod Computation time (s) RMSE (rad) 120576 (rad)Least squares 12142 07594 06214Sequential quadratic programming 11866 05393 04228

Table 3 Selected parameters of ASAR image data

Number Sensor Date Type Latitude and longitude of center Polarization Ascending Wavelength022403 ENVISAT 20050306 SLC 116314 6 35305 1 VV Ascending C022404 ENVISAT 20050130 SLC 116314 4 35305 1 VV Ascending C

the simulated surface The curve obtained for of the leastsquares method is also much rougher than that of sequentialquadratic programming

Table 2 shows the running time RMSE and 120576 valueof sequential quadratic programming and least squaresobtained by phase unwrapping Both methods result inrelatively reliable unwrapping The running time is similarindicating that the efficiency of the two algorithms is almostthe same The RMSE of sequential quadratic programming issmaller than that of least squares indicating that sequentialquadratic programming is more precise 120576 indicates the qual-ity of the wrap the smaller the 120576 the better the unwrappingquality Thus the sequential quadratic programming methodprovides better unwrapping than the least squares method

The experimentwith simulated data shows that sequentialquadratic programming is a feasible method the model iseffective and has high precision

422 Observational Data Satellite data collected by ENVI-SAT ASAR covering the Jining area of Shandong provinceChina were used to test the modelrsquos performance

Table 3 shows the basic parameters of the dataThe images are 1024 times 1024 pixels with a resolution of

30m Figure 4 shows the unwrapping result of each method119890 = 120576 = 10minus3Both unwrapping methods were run successfully In the

interferogram the stripe noise is large and the fringes areunclear The phase ripple of sequential quadratic program-ming is smaller than that of least squares demonstrating thatsequential quadratic programming experiences less interfer-ence

Table 4 shows the computation time RMSE and 120576 valueThe RMSE and 120576 values of sequential quadratic program-

ming are smaller than those of the least squares methodThis

6 Mathematical Problems in Engineering

3

2

1

0

-1

-2

-3

100

200

300

400

500

600

700

800

900

1000

200 400 600 800 1000

(a) Wrapped phase

100

200

300

400

500

600

700

800

900

1000

200 400 600 800 1000

15

10

5

0

-5

-10

-15

-20

(b) Least squares method

100

200

300

400

500

600

700

800

900

1000

100 200 300 400 500 600 700 800 900 1000

10

5

0

-5

-10

-15

-20

(c) Sequential quadratic programming method

Figure 4 Unwrapping of ASAR images using least squares and sequential quadratic programming

Table 4 Unwrapping time RMSE and 120576 valuesMethod computation time (s) RMSE (rad) 120576 (rad)Least squares 725940 17594 06214Sequential quadratic programming 1336948 05393 04228

illustrates that the difference between the rewrapped resultsand the original wrapped phase is small The unwrappingphases of the sequential quadratic programming method aresmoother and more reliable However the computation timeof sequential quadratic programming is greater than that ofthe least squares method This is due to high levels of noiseand the penalty function increasing the computation time ofsequential quadratic programming

5 Conclusions

A direction-controlled nonlinear least squares estima-tion algorithm using the sequential quadratic programmingmethod is proposed herein The least squares model is

transformed into the nonlinear programming model andthe iteration direction can then be controlled by sequentialquadratic programming The ill-conditioned matrix andphase unwrapping are used to verify the method Theproposed algorithm achieves better results than alternativeapproaches in terms of RMSE and 120576 values and its effec-tiveness is demonstrated using observational satellite data Infuture research the role change if the inequality constrainsare taken into account will be considered

Data Availability

The 20050130slc and 20050306slc data used to support thefindings of this study are available from the correspondingauthor upon request

Mathematical Problems in Engineering 7

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grants nos 41774002 and 41274007)and the Scientific Research Fund Project for the Introductionof Talents of Shandong University of Science and Technology(Grants nos 2014RCJJ047 and 2015RCJJ022)

References

[1] C Han F Zheng T Guo and G He ldquoParallel algorithms forlarge-scale linearly constrained minimization problemrdquo ActaMathematicae Applicatae Sinica vol 30 no 3 pp 707ndash720 2014

[2] X Jin Y Liang D Tian and F Zhuang ldquoParticle swarmoptimization using dimension selection methodsrdquo AppliedMathematics and Computation vol 219 no 10 pp 5185ndash51972013

[3] G Tutz and G Schauberger ldquoA penalty approach to differentialitem functioning in Rasch modelsrdquo Psychometrika vol 80 no1 pp 21ndash43 2015

[4] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013

[5] G Ma and B Huang ldquoOptimization of process parameters ofstamping forming of the automotive lower floor boardrdquo Journalof Applied Mathematics vol 2014 Article ID 470320 9 pages2014

[6] F Tramontana A A Elsadany B Xin and H N Agiza ldquoLocalstability of the Cournot solution with increasing heterogeneouscompetitorsrdquo Nonlinear Analysis Real World Applications vol26 pp 150ndash160 2015

[7] M Li X Kao and H Che ldquoRelaxed inertial acceleratedalgorithms for solving split equality feasibility problemrdquo Journalof Nonlinear Sciences and Applications vol 10 no 8 pp 4109ndash4121 2017

[8] M Huang and D Pu ldquoA line search SQP method without apenalty or a filterrdquo Computational amp Applied Mathematics vol34 no 2 pp 741ndash753 2015

[9] A F Izmailov M V Solodov and E I Uskov ldquoCombiningstabilized SQP with the augmented Lagrangian algorithmrdquoComputational Optimization and Applications vol 62 no 2 pp405ndash429 2015

[10] J Gao B Shen E Feng and Z Xiu ldquoModelling and optimalcontrol for an impulsive dynamical system in microbial fed-batch culturerdquoComputational and AppliedMathematics vol 32no 2 pp 275ndash290 2013

[11] J Zhu and B Hao ldquoA new class of smoothing functions anda smoothing Newton method for complementarity problemsrdquoOptimization Letters vol 7 no 3 pp 481ndash497 2013

[12] M Huang and D Pu ldquoA trust-region SQP method withouta penalty or a filter for nonlinear programmingrdquo Journal ofComputational and Applied Mathematics vol 281 pp 107ndash1192015

[13] B Sachsenberg and K Schittkowski ldquoA combined SQPndashIPMalgorithm for solving large-scale nonlinear optimization prob-lemsrdquo Optimization Letters vol 9 no 7 pp 1271ndash1282 2015

[14] J Tang G He and L Fang ldquoA new non-interior continua-tion method for second-order cone programmingrdquo Journal ofNumerical Mathematics vol 21 no 4 pp 301ndash324 2013

[15] C Shen W Shao and W Xue ldquoOn the local convergence ofa penalty-function-free SQP methodrdquo Numerical FunctionalAnalysis and Optimization vol 35 no 5 pp 623ndash647 2014

[16] S Pang T Li F Dai and M Yu ldquoParticle swarm optimizationalgorithm for multi-salesman problem with time and capacityconstraintsrdquo Applied Mathematics amp Information Sciences vol7 no 6 pp 2439ndash2444 2013

[17] Z Z Feng L Fang and G He ldquoAn O Nl iteration primal- dualpath- following method based on wide neighbourhood andlarge update for second- order cone programmingrdquo Optimiza-tion A Journal of Mathematical Programming and OperationsResearch vol 63 no 5 pp 679ndash691 2014

[18] C Han T Feng G He and T Guo ldquoParallel variable distribu-tion algorithm for constrained optimizationwith nonmonotonetechniquerdquo Journal of AppliedMathematics vol 2013 Article ID295147 7 pages 2013

[19] JWang K Liang XHuang ZWang andH Shen ldquoDissipativefault-tolerant control for nonlinear singular perturbed systemswithMarkov jumping parameters based on slow state feedbackrdquoApplied Mathematics and Computation vol 328 pp 247ndash2622018

[20] J B Jian Y L Lai and K C Zhang ldquoA feasible method forsuperlinearly and quadratically convergent sequential systemsof equationsrdquo Acta Mathematica Sinica vol 45 no 6 pp 1137ndash1146 2002

[21] J Che and K Su ldquoA modified SQP method and its globalconvergencerdquo Applied Mathematics and Computation vol 186no 2 pp 945ndash951 2007

[22] P-y Nie ldquoSequential penalty quadratic programming filtermethods for nonlinear programmingrdquo Nonlinear Analysis RealWorld Applications vol 8 no 1 pp 118ndash129 2007

[23] J Yu M Li Y Wang and G He ldquoA decomposition method forlarge-scale box constrained optimizationrdquoAppliedMathematicsand Computation vol 231 no 12 pp 9ndash15 2014

[24] M Chen and G Li ldquoForming mechanism and correction of CTimage artifacts caused by the errors of three system parametersrdquoJournal of Applied Mathematics vol 2013 Article ID 545147 7pages 2013

[25] T Jiang Z Jiang and S Ling ldquoAn algebraic method forquaternion and complex least squares coneigen-problem inquantum mechanicsrdquo Applied Mathematics and Computationvol 249 pp 222ndash228 2014

[26] Z Pang G Liu D Zhou and D Sun ldquoData-based predictivecontrol for networked nonlinear systems with packet dropoutandmeasurement noiserdquo Journal of Systems Science ampComplex-ity vol 30 no 5 pp 1072ndash1083 2017

[27] F Liu Z Fu Y Zheng and Q Yuan ldquoA rough Marcinkiewiczintegral along smooth curvesrdquo Journal of Nonlinear Sciences andApplications vol 9 no 6 pp 4450ndash4464 2016

[28] S Ding H Huang X Xu and J Wang ldquoPolynomial smoothtwin support vector machinesrdquo Applied Mathematics amp Infor-mation Sciences vol 8 no 4 pp 2063ndash2071 2014

[29] C H Liu Y L Shang and P Han ldquoA new infeasible-interior-point algorithm for linear programming over symmetric conesrdquoActaMathematicae Applicatae Sinica vol 33 no 3 pp 771ndash7882017

[30] N T Hang ldquoThe penalty functionsmethod andmultiplier rulesbased on the Mordukhovich subdifferentialrdquo Set-Valued andVariational Analysis vol 22 no 2 pp 299ndash312 2014

8 Mathematical Problems in Engineering

[31] Y Zhang and D Shen ldquoEstimation of semi-parametric varying-coefficient spatial panel data models with random-effectsrdquoJournal of Statistical Planning and Inference vol 159 pp 64ndash802015

[32] S Salman and H S Dawood ldquoApproximate solution of thelinear programming problems by ant system optimizationrdquoEngineering amp Technology Journal no 16 pp 2978ndash2995 2009

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 5

100 20 30 40 50 60 70 80 90 100

15

10

5

0

minus5

minus10

minus15

(a) Noise-free100 20 30 40 50 60 70 80 90 100

10

8

6

4

2

0

minus2

minus4

minus6

minus8

minus10

(b) Least squares

100 20 30 40 50 60 70 80 90 100

15

10

5

0

minus5

minus10

minus15

20

(c) Sequential quadratic programming

Figure 3 Fiftieth line cross-sectional view of the unwrapping phase using least squares and sequential quadratic programming

Table 2 Unwrapping time RMSE and 120576 valueMethod Computation time (s) RMSE (rad) 120576 (rad)Least squares 12142 07594 06214Sequential quadratic programming 11866 05393 04228

Table 3 Selected parameters of ASAR image data

Number Sensor Date Type Latitude and longitude of center Polarization Ascending Wavelength022403 ENVISAT 20050306 SLC 116314 6 35305 1 VV Ascending C022404 ENVISAT 20050130 SLC 116314 4 35305 1 VV Ascending C

the simulated surface The curve obtained for of the leastsquares method is also much rougher than that of sequentialquadratic programming

Table 2 shows the running time RMSE and 120576 valueof sequential quadratic programming and least squaresobtained by phase unwrapping Both methods result inrelatively reliable unwrapping The running time is similarindicating that the efficiency of the two algorithms is almostthe same The RMSE of sequential quadratic programming issmaller than that of least squares indicating that sequentialquadratic programming is more precise 120576 indicates the qual-ity of the wrap the smaller the 120576 the better the unwrappingquality Thus the sequential quadratic programming methodprovides better unwrapping than the least squares method

The experimentwith simulated data shows that sequentialquadratic programming is a feasible method the model iseffective and has high precision

422 Observational Data Satellite data collected by ENVI-SAT ASAR covering the Jining area of Shandong provinceChina were used to test the modelrsquos performance

Table 3 shows the basic parameters of the dataThe images are 1024 times 1024 pixels with a resolution of

30m Figure 4 shows the unwrapping result of each method119890 = 120576 = 10minus3Both unwrapping methods were run successfully In the

interferogram the stripe noise is large and the fringes areunclear The phase ripple of sequential quadratic program-ming is smaller than that of least squares demonstrating thatsequential quadratic programming experiences less interfer-ence

Table 4 shows the computation time RMSE and 120576 valueThe RMSE and 120576 values of sequential quadratic program-

ming are smaller than those of the least squares methodThis

6 Mathematical Problems in Engineering

3

2

1

0

-1

-2

-3

100

200

300

400

500

600

700

800

900

1000

200 400 600 800 1000

(a) Wrapped phase

100

200

300

400

500

600

700

800

900

1000

200 400 600 800 1000

15

10

5

0

-5

-10

-15

-20

(b) Least squares method

100

200

300

400

500

600

700

800

900

1000

100 200 300 400 500 600 700 800 900 1000

10

5

0

-5

-10

-15

-20

(c) Sequential quadratic programming method

Figure 4 Unwrapping of ASAR images using least squares and sequential quadratic programming

Table 4 Unwrapping time RMSE and 120576 valuesMethod computation time (s) RMSE (rad) 120576 (rad)Least squares 725940 17594 06214Sequential quadratic programming 1336948 05393 04228

illustrates that the difference between the rewrapped resultsand the original wrapped phase is small The unwrappingphases of the sequential quadratic programming method aresmoother and more reliable However the computation timeof sequential quadratic programming is greater than that ofthe least squares method This is due to high levels of noiseand the penalty function increasing the computation time ofsequential quadratic programming

5 Conclusions

A direction-controlled nonlinear least squares estima-tion algorithm using the sequential quadratic programmingmethod is proposed herein The least squares model is

transformed into the nonlinear programming model andthe iteration direction can then be controlled by sequentialquadratic programming The ill-conditioned matrix andphase unwrapping are used to verify the method Theproposed algorithm achieves better results than alternativeapproaches in terms of RMSE and 120576 values and its effec-tiveness is demonstrated using observational satellite data Infuture research the role change if the inequality constrainsare taken into account will be considered

Data Availability

The 20050130slc and 20050306slc data used to support thefindings of this study are available from the correspondingauthor upon request

Mathematical Problems in Engineering 7

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grants nos 41774002 and 41274007)and the Scientific Research Fund Project for the Introductionof Talents of Shandong University of Science and Technology(Grants nos 2014RCJJ047 and 2015RCJJ022)

References

[1] C Han F Zheng T Guo and G He ldquoParallel algorithms forlarge-scale linearly constrained minimization problemrdquo ActaMathematicae Applicatae Sinica vol 30 no 3 pp 707ndash720 2014

[2] X Jin Y Liang D Tian and F Zhuang ldquoParticle swarmoptimization using dimension selection methodsrdquo AppliedMathematics and Computation vol 219 no 10 pp 5185ndash51972013

[3] G Tutz and G Schauberger ldquoA penalty approach to differentialitem functioning in Rasch modelsrdquo Psychometrika vol 80 no1 pp 21ndash43 2015

[4] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013

[5] G Ma and B Huang ldquoOptimization of process parameters ofstamping forming of the automotive lower floor boardrdquo Journalof Applied Mathematics vol 2014 Article ID 470320 9 pages2014

[6] F Tramontana A A Elsadany B Xin and H N Agiza ldquoLocalstability of the Cournot solution with increasing heterogeneouscompetitorsrdquo Nonlinear Analysis Real World Applications vol26 pp 150ndash160 2015

[7] M Li X Kao and H Che ldquoRelaxed inertial acceleratedalgorithms for solving split equality feasibility problemrdquo Journalof Nonlinear Sciences and Applications vol 10 no 8 pp 4109ndash4121 2017

[8] M Huang and D Pu ldquoA line search SQP method without apenalty or a filterrdquo Computational amp Applied Mathematics vol34 no 2 pp 741ndash753 2015

[9] A F Izmailov M V Solodov and E I Uskov ldquoCombiningstabilized SQP with the augmented Lagrangian algorithmrdquoComputational Optimization and Applications vol 62 no 2 pp405ndash429 2015

[10] J Gao B Shen E Feng and Z Xiu ldquoModelling and optimalcontrol for an impulsive dynamical system in microbial fed-batch culturerdquoComputational and AppliedMathematics vol 32no 2 pp 275ndash290 2013

[11] J Zhu and B Hao ldquoA new class of smoothing functions anda smoothing Newton method for complementarity problemsrdquoOptimization Letters vol 7 no 3 pp 481ndash497 2013

[12] M Huang and D Pu ldquoA trust-region SQP method withouta penalty or a filter for nonlinear programmingrdquo Journal ofComputational and Applied Mathematics vol 281 pp 107ndash1192015

[13] B Sachsenberg and K Schittkowski ldquoA combined SQPndashIPMalgorithm for solving large-scale nonlinear optimization prob-lemsrdquo Optimization Letters vol 9 no 7 pp 1271ndash1282 2015

[14] J Tang G He and L Fang ldquoA new non-interior continua-tion method for second-order cone programmingrdquo Journal ofNumerical Mathematics vol 21 no 4 pp 301ndash324 2013

[15] C Shen W Shao and W Xue ldquoOn the local convergence ofa penalty-function-free SQP methodrdquo Numerical FunctionalAnalysis and Optimization vol 35 no 5 pp 623ndash647 2014

[16] S Pang T Li F Dai and M Yu ldquoParticle swarm optimizationalgorithm for multi-salesman problem with time and capacityconstraintsrdquo Applied Mathematics amp Information Sciences vol7 no 6 pp 2439ndash2444 2013

[17] Z Z Feng L Fang and G He ldquoAn O Nl iteration primal- dualpath- following method based on wide neighbourhood andlarge update for second- order cone programmingrdquo Optimiza-tion A Journal of Mathematical Programming and OperationsResearch vol 63 no 5 pp 679ndash691 2014

[18] C Han T Feng G He and T Guo ldquoParallel variable distribu-tion algorithm for constrained optimizationwith nonmonotonetechniquerdquo Journal of AppliedMathematics vol 2013 Article ID295147 7 pages 2013

[19] JWang K Liang XHuang ZWang andH Shen ldquoDissipativefault-tolerant control for nonlinear singular perturbed systemswithMarkov jumping parameters based on slow state feedbackrdquoApplied Mathematics and Computation vol 328 pp 247ndash2622018

[20] J B Jian Y L Lai and K C Zhang ldquoA feasible method forsuperlinearly and quadratically convergent sequential systemsof equationsrdquo Acta Mathematica Sinica vol 45 no 6 pp 1137ndash1146 2002

[21] J Che and K Su ldquoA modified SQP method and its globalconvergencerdquo Applied Mathematics and Computation vol 186no 2 pp 945ndash951 2007

[22] P-y Nie ldquoSequential penalty quadratic programming filtermethods for nonlinear programmingrdquo Nonlinear Analysis RealWorld Applications vol 8 no 1 pp 118ndash129 2007

[23] J Yu M Li Y Wang and G He ldquoA decomposition method forlarge-scale box constrained optimizationrdquoAppliedMathematicsand Computation vol 231 no 12 pp 9ndash15 2014

[24] M Chen and G Li ldquoForming mechanism and correction of CTimage artifacts caused by the errors of three system parametersrdquoJournal of Applied Mathematics vol 2013 Article ID 545147 7pages 2013

[25] T Jiang Z Jiang and S Ling ldquoAn algebraic method forquaternion and complex least squares coneigen-problem inquantum mechanicsrdquo Applied Mathematics and Computationvol 249 pp 222ndash228 2014

[26] Z Pang G Liu D Zhou and D Sun ldquoData-based predictivecontrol for networked nonlinear systems with packet dropoutandmeasurement noiserdquo Journal of Systems Science ampComplex-ity vol 30 no 5 pp 1072ndash1083 2017

[27] F Liu Z Fu Y Zheng and Q Yuan ldquoA rough Marcinkiewiczintegral along smooth curvesrdquo Journal of Nonlinear Sciences andApplications vol 9 no 6 pp 4450ndash4464 2016

[28] S Ding H Huang X Xu and J Wang ldquoPolynomial smoothtwin support vector machinesrdquo Applied Mathematics amp Infor-mation Sciences vol 8 no 4 pp 2063ndash2071 2014

[29] C H Liu Y L Shang and P Han ldquoA new infeasible-interior-point algorithm for linear programming over symmetric conesrdquoActaMathematicae Applicatae Sinica vol 33 no 3 pp 771ndash7882017

[30] N T Hang ldquoThe penalty functionsmethod andmultiplier rulesbased on the Mordukhovich subdifferentialrdquo Set-Valued andVariational Analysis vol 22 no 2 pp 299ndash312 2014

8 Mathematical Problems in Engineering

[31] Y Zhang and D Shen ldquoEstimation of semi-parametric varying-coefficient spatial panel data models with random-effectsrdquoJournal of Statistical Planning and Inference vol 159 pp 64ndash802015

[32] S Salman and H S Dawood ldquoApproximate solution of thelinear programming problems by ant system optimizationrdquoEngineering amp Technology Journal no 16 pp 2978ndash2995 2009

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

6 Mathematical Problems in Engineering

3

2

1

0

-1

-2

-3

100

200

300

400

500

600

700

800

900

1000

200 400 600 800 1000

(a) Wrapped phase

100

200

300

400

500

600

700

800

900

1000

200 400 600 800 1000

15

10

5

0

-5

-10

-15

-20

(b) Least squares method

100

200

300

400

500

600

700

800

900

1000

100 200 300 400 500 600 700 800 900 1000

10

5

0

-5

-10

-15

-20

(c) Sequential quadratic programming method

Figure 4 Unwrapping of ASAR images using least squares and sequential quadratic programming

Table 4 Unwrapping time RMSE and 120576 valuesMethod computation time (s) RMSE (rad) 120576 (rad)Least squares 725940 17594 06214Sequential quadratic programming 1336948 05393 04228

illustrates that the difference between the rewrapped resultsand the original wrapped phase is small The unwrappingphases of the sequential quadratic programming method aresmoother and more reliable However the computation timeof sequential quadratic programming is greater than that ofthe least squares method This is due to high levels of noiseand the penalty function increasing the computation time ofsequential quadratic programming

5 Conclusions

A direction-controlled nonlinear least squares estima-tion algorithm using the sequential quadratic programmingmethod is proposed herein The least squares model is

transformed into the nonlinear programming model andthe iteration direction can then be controlled by sequentialquadratic programming The ill-conditioned matrix andphase unwrapping are used to verify the method Theproposed algorithm achieves better results than alternativeapproaches in terms of RMSE and 120576 values and its effec-tiveness is demonstrated using observational satellite data Infuture research the role change if the inequality constrainsare taken into account will be considered

Data Availability

The 20050130slc and 20050306slc data used to support thefindings of this study are available from the correspondingauthor upon request

Mathematical Problems in Engineering 7

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grants nos 41774002 and 41274007)and the Scientific Research Fund Project for the Introductionof Talents of Shandong University of Science and Technology(Grants nos 2014RCJJ047 and 2015RCJJ022)

References

[1] C Han F Zheng T Guo and G He ldquoParallel algorithms forlarge-scale linearly constrained minimization problemrdquo ActaMathematicae Applicatae Sinica vol 30 no 3 pp 707ndash720 2014

[2] X Jin Y Liang D Tian and F Zhuang ldquoParticle swarmoptimization using dimension selection methodsrdquo AppliedMathematics and Computation vol 219 no 10 pp 5185ndash51972013

[3] G Tutz and G Schauberger ldquoA penalty approach to differentialitem functioning in Rasch modelsrdquo Psychometrika vol 80 no1 pp 21ndash43 2015

[4] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013

[5] G Ma and B Huang ldquoOptimization of process parameters ofstamping forming of the automotive lower floor boardrdquo Journalof Applied Mathematics vol 2014 Article ID 470320 9 pages2014

[6] F Tramontana A A Elsadany B Xin and H N Agiza ldquoLocalstability of the Cournot solution with increasing heterogeneouscompetitorsrdquo Nonlinear Analysis Real World Applications vol26 pp 150ndash160 2015

[7] M Li X Kao and H Che ldquoRelaxed inertial acceleratedalgorithms for solving split equality feasibility problemrdquo Journalof Nonlinear Sciences and Applications vol 10 no 8 pp 4109ndash4121 2017

[8] M Huang and D Pu ldquoA line search SQP method without apenalty or a filterrdquo Computational amp Applied Mathematics vol34 no 2 pp 741ndash753 2015

[9] A F Izmailov M V Solodov and E I Uskov ldquoCombiningstabilized SQP with the augmented Lagrangian algorithmrdquoComputational Optimization and Applications vol 62 no 2 pp405ndash429 2015

[10] J Gao B Shen E Feng and Z Xiu ldquoModelling and optimalcontrol for an impulsive dynamical system in microbial fed-batch culturerdquoComputational and AppliedMathematics vol 32no 2 pp 275ndash290 2013

[11] J Zhu and B Hao ldquoA new class of smoothing functions anda smoothing Newton method for complementarity problemsrdquoOptimization Letters vol 7 no 3 pp 481ndash497 2013

[12] M Huang and D Pu ldquoA trust-region SQP method withouta penalty or a filter for nonlinear programmingrdquo Journal ofComputational and Applied Mathematics vol 281 pp 107ndash1192015

[13] B Sachsenberg and K Schittkowski ldquoA combined SQPndashIPMalgorithm for solving large-scale nonlinear optimization prob-lemsrdquo Optimization Letters vol 9 no 7 pp 1271ndash1282 2015

[14] J Tang G He and L Fang ldquoA new non-interior continua-tion method for second-order cone programmingrdquo Journal ofNumerical Mathematics vol 21 no 4 pp 301ndash324 2013

[15] C Shen W Shao and W Xue ldquoOn the local convergence ofa penalty-function-free SQP methodrdquo Numerical FunctionalAnalysis and Optimization vol 35 no 5 pp 623ndash647 2014

[16] S Pang T Li F Dai and M Yu ldquoParticle swarm optimizationalgorithm for multi-salesman problem with time and capacityconstraintsrdquo Applied Mathematics amp Information Sciences vol7 no 6 pp 2439ndash2444 2013

[17] Z Z Feng L Fang and G He ldquoAn O Nl iteration primal- dualpath- following method based on wide neighbourhood andlarge update for second- order cone programmingrdquo Optimiza-tion A Journal of Mathematical Programming and OperationsResearch vol 63 no 5 pp 679ndash691 2014

[18] C Han T Feng G He and T Guo ldquoParallel variable distribu-tion algorithm for constrained optimizationwith nonmonotonetechniquerdquo Journal of AppliedMathematics vol 2013 Article ID295147 7 pages 2013

[19] JWang K Liang XHuang ZWang andH Shen ldquoDissipativefault-tolerant control for nonlinear singular perturbed systemswithMarkov jumping parameters based on slow state feedbackrdquoApplied Mathematics and Computation vol 328 pp 247ndash2622018

[20] J B Jian Y L Lai and K C Zhang ldquoA feasible method forsuperlinearly and quadratically convergent sequential systemsof equationsrdquo Acta Mathematica Sinica vol 45 no 6 pp 1137ndash1146 2002

[21] J Che and K Su ldquoA modified SQP method and its globalconvergencerdquo Applied Mathematics and Computation vol 186no 2 pp 945ndash951 2007

[22] P-y Nie ldquoSequential penalty quadratic programming filtermethods for nonlinear programmingrdquo Nonlinear Analysis RealWorld Applications vol 8 no 1 pp 118ndash129 2007

[23] J Yu M Li Y Wang and G He ldquoA decomposition method forlarge-scale box constrained optimizationrdquoAppliedMathematicsand Computation vol 231 no 12 pp 9ndash15 2014

[24] M Chen and G Li ldquoForming mechanism and correction of CTimage artifacts caused by the errors of three system parametersrdquoJournal of Applied Mathematics vol 2013 Article ID 545147 7pages 2013

[25] T Jiang Z Jiang and S Ling ldquoAn algebraic method forquaternion and complex least squares coneigen-problem inquantum mechanicsrdquo Applied Mathematics and Computationvol 249 pp 222ndash228 2014

[26] Z Pang G Liu D Zhou and D Sun ldquoData-based predictivecontrol for networked nonlinear systems with packet dropoutandmeasurement noiserdquo Journal of Systems Science ampComplex-ity vol 30 no 5 pp 1072ndash1083 2017

[27] F Liu Z Fu Y Zheng and Q Yuan ldquoA rough Marcinkiewiczintegral along smooth curvesrdquo Journal of Nonlinear Sciences andApplications vol 9 no 6 pp 4450ndash4464 2016

[28] S Ding H Huang X Xu and J Wang ldquoPolynomial smoothtwin support vector machinesrdquo Applied Mathematics amp Infor-mation Sciences vol 8 no 4 pp 2063ndash2071 2014

[29] C H Liu Y L Shang and P Han ldquoA new infeasible-interior-point algorithm for linear programming over symmetric conesrdquoActaMathematicae Applicatae Sinica vol 33 no 3 pp 771ndash7882017

[30] N T Hang ldquoThe penalty functionsmethod andmultiplier rulesbased on the Mordukhovich subdifferentialrdquo Set-Valued andVariational Analysis vol 22 no 2 pp 299ndash312 2014

8 Mathematical Problems in Engineering

[31] Y Zhang and D Shen ldquoEstimation of semi-parametric varying-coefficient spatial panel data models with random-effectsrdquoJournal of Statistical Planning and Inference vol 159 pp 64ndash802015

[32] S Salman and H S Dawood ldquoApproximate solution of thelinear programming problems by ant system optimizationrdquoEngineering amp Technology Journal no 16 pp 2978ndash2995 2009

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 7

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was supported by the National Natural ScienceFoundation of China (Grants nos 41774002 and 41274007)and the Scientific Research Fund Project for the Introductionof Talents of Shandong University of Science and Technology(Grants nos 2014RCJJ047 and 2015RCJJ022)

References

[1] C Han F Zheng T Guo and G He ldquoParallel algorithms forlarge-scale linearly constrained minimization problemrdquo ActaMathematicae Applicatae Sinica vol 30 no 3 pp 707ndash720 2014

[2] X Jin Y Liang D Tian and F Zhuang ldquoParticle swarmoptimization using dimension selection methodsrdquo AppliedMathematics and Computation vol 219 no 10 pp 5185ndash51972013

[3] G Tutz and G Schauberger ldquoA penalty approach to differentialitem functioning in Rasch modelsrdquo Psychometrika vol 80 no1 pp 21ndash43 2015

[4] H-jMa andTHou ldquoA separation theorem for stochastic singu-lar linear quadratic control problem with partial informationrdquoActaMathematicae Applicatae Sinica vol 29 no 2 pp 303ndash3142013

[5] G Ma and B Huang ldquoOptimization of process parameters ofstamping forming of the automotive lower floor boardrdquo Journalof Applied Mathematics vol 2014 Article ID 470320 9 pages2014

[6] F Tramontana A A Elsadany B Xin and H N Agiza ldquoLocalstability of the Cournot solution with increasing heterogeneouscompetitorsrdquo Nonlinear Analysis Real World Applications vol26 pp 150ndash160 2015

[7] M Li X Kao and H Che ldquoRelaxed inertial acceleratedalgorithms for solving split equality feasibility problemrdquo Journalof Nonlinear Sciences and Applications vol 10 no 8 pp 4109ndash4121 2017

[8] M Huang and D Pu ldquoA line search SQP method without apenalty or a filterrdquo Computational amp Applied Mathematics vol34 no 2 pp 741ndash753 2015

[9] A F Izmailov M V Solodov and E I Uskov ldquoCombiningstabilized SQP with the augmented Lagrangian algorithmrdquoComputational Optimization and Applications vol 62 no 2 pp405ndash429 2015

[10] J Gao B Shen E Feng and Z Xiu ldquoModelling and optimalcontrol for an impulsive dynamical system in microbial fed-batch culturerdquoComputational and AppliedMathematics vol 32no 2 pp 275ndash290 2013

[11] J Zhu and B Hao ldquoA new class of smoothing functions anda smoothing Newton method for complementarity problemsrdquoOptimization Letters vol 7 no 3 pp 481ndash497 2013

[12] M Huang and D Pu ldquoA trust-region SQP method withouta penalty or a filter for nonlinear programmingrdquo Journal ofComputational and Applied Mathematics vol 281 pp 107ndash1192015

[13] B Sachsenberg and K Schittkowski ldquoA combined SQPndashIPMalgorithm for solving large-scale nonlinear optimization prob-lemsrdquo Optimization Letters vol 9 no 7 pp 1271ndash1282 2015

[14] J Tang G He and L Fang ldquoA new non-interior continua-tion method for second-order cone programmingrdquo Journal ofNumerical Mathematics vol 21 no 4 pp 301ndash324 2013

[15] C Shen W Shao and W Xue ldquoOn the local convergence ofa penalty-function-free SQP methodrdquo Numerical FunctionalAnalysis and Optimization vol 35 no 5 pp 623ndash647 2014

[16] S Pang T Li F Dai and M Yu ldquoParticle swarm optimizationalgorithm for multi-salesman problem with time and capacityconstraintsrdquo Applied Mathematics amp Information Sciences vol7 no 6 pp 2439ndash2444 2013

[17] Z Z Feng L Fang and G He ldquoAn O Nl iteration primal- dualpath- following method based on wide neighbourhood andlarge update for second- order cone programmingrdquo Optimiza-tion A Journal of Mathematical Programming and OperationsResearch vol 63 no 5 pp 679ndash691 2014

[18] C Han T Feng G He and T Guo ldquoParallel variable distribu-tion algorithm for constrained optimizationwith nonmonotonetechniquerdquo Journal of AppliedMathematics vol 2013 Article ID295147 7 pages 2013

[19] JWang K Liang XHuang ZWang andH Shen ldquoDissipativefault-tolerant control for nonlinear singular perturbed systemswithMarkov jumping parameters based on slow state feedbackrdquoApplied Mathematics and Computation vol 328 pp 247ndash2622018

[20] J B Jian Y L Lai and K C Zhang ldquoA feasible method forsuperlinearly and quadratically convergent sequential systemsof equationsrdquo Acta Mathematica Sinica vol 45 no 6 pp 1137ndash1146 2002

[21] J Che and K Su ldquoA modified SQP method and its globalconvergencerdquo Applied Mathematics and Computation vol 186no 2 pp 945ndash951 2007

[22] P-y Nie ldquoSequential penalty quadratic programming filtermethods for nonlinear programmingrdquo Nonlinear Analysis RealWorld Applications vol 8 no 1 pp 118ndash129 2007

[23] J Yu M Li Y Wang and G He ldquoA decomposition method forlarge-scale box constrained optimizationrdquoAppliedMathematicsand Computation vol 231 no 12 pp 9ndash15 2014

[24] M Chen and G Li ldquoForming mechanism and correction of CTimage artifacts caused by the errors of three system parametersrdquoJournal of Applied Mathematics vol 2013 Article ID 545147 7pages 2013

[25] T Jiang Z Jiang and S Ling ldquoAn algebraic method forquaternion and complex least squares coneigen-problem inquantum mechanicsrdquo Applied Mathematics and Computationvol 249 pp 222ndash228 2014

[26] Z Pang G Liu D Zhou and D Sun ldquoData-based predictivecontrol for networked nonlinear systems with packet dropoutandmeasurement noiserdquo Journal of Systems Science ampComplex-ity vol 30 no 5 pp 1072ndash1083 2017

[27] F Liu Z Fu Y Zheng and Q Yuan ldquoA rough Marcinkiewiczintegral along smooth curvesrdquo Journal of Nonlinear Sciences andApplications vol 9 no 6 pp 4450ndash4464 2016

[28] S Ding H Huang X Xu and J Wang ldquoPolynomial smoothtwin support vector machinesrdquo Applied Mathematics amp Infor-mation Sciences vol 8 no 4 pp 2063ndash2071 2014

[29] C H Liu Y L Shang and P Han ldquoA new infeasible-interior-point algorithm for linear programming over symmetric conesrdquoActaMathematicae Applicatae Sinica vol 33 no 3 pp 771ndash7882017

[30] N T Hang ldquoThe penalty functionsmethod andmultiplier rulesbased on the Mordukhovich subdifferentialrdquo Set-Valued andVariational Analysis vol 22 no 2 pp 299ndash312 2014

8 Mathematical Problems in Engineering

[31] Y Zhang and D Shen ldquoEstimation of semi-parametric varying-coefficient spatial panel data models with random-effectsrdquoJournal of Statistical Planning and Inference vol 159 pp 64ndash802015

[32] S Salman and H S Dawood ldquoApproximate solution of thelinear programming problems by ant system optimizationrdquoEngineering amp Technology Journal no 16 pp 2978ndash2995 2009

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

8 Mathematical Problems in Engineering

[31] Y Zhang and D Shen ldquoEstimation of semi-parametric varying-coefficient spatial panel data models with random-effectsrdquoJournal of Statistical Planning and Inference vol 159 pp 64ndash802015

[32] S Salman and H S Dawood ldquoApproximate solution of thelinear programming problems by ant system optimizationrdquoEngineering amp Technology Journal no 16 pp 2978ndash2995 2009

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