随机R0张量互补问题的投影Levenberg-Marquardt方法

doi:10.15960/j.cnki.issn.1007-6093.2021.04.006

Abstract

Abstract:

In this paper, we consider a class of stochastic R₀ tensor complementarity problems with finitely many elements. Firstly, we use Fischer-Burmeister function to transform the problem into a constrained optimization problem. Then a projected Levenberg-Marquardt method is used to solve the constrained optimization problem. Under general conditions, the global convergence of this method is proved, and the related numerical results show the efficiency of the method.

Key words: stochastic R₀ tensor complementarity problem, projected Levenberg-Marquardt method, global convergence, line search

CLC Number:

O221

Liyuan CUI, Shouqiang DU. Projected Levenberg-Marquardt method for stochastic R₀ tensor complementarity problems[J]. Operations Research Transactions, 2021, 25(4): 69-79.

Figures/Tables 3

References 23

1	Che M L , Qi L Q , Wei Y M . Stochastic $R_0$ tensors to stochastic tensor complementarity problems[J]. Optimization Letters, 2019, 13, 261- 279. doi: 10.1007/s11590-018-1362-7
2	Song Y S , Qi L Q . Properties of tensor complementarity problem and some classes of structured tensors[J]. Annals of Applied Mathematics, 2017, 33, 308- 323.
3	Song Y S , Yu G H . Properties of solution set of tensor complementarity problem[J]. Journal of Optimization Theory and Applications, 2016, 170, 85- 96. doi: 10.1007/s10957-016-0907-0
4	Song Y S , Qi L Q . Tensor complementarity problem and semi-positive tensors[J]. Journal of Optimization Theory and Applications, 2016, 169 (3): 1069- 1078. doi: 10.1007/s10957-015-0800-2
5	Ding W Y , Luo Z Y , Qi L Q . $P$-tensors, $P_0$-tensors, and their applications[J]. Linear Algebra Applications, 2018, 555, 336- 354. doi: 10.1016/j.laa.2018.06.028
6	Huang Z H , Qi L Q . Formulating an $n$-person noncooperative game as a tensor complementarity problem[J]. Computational Optimization and Applications, 2017, 66 (3): 557- 576. doi: 10.1007/s10589-016-9872-7
7	Bai X L , Hang Z H , Wang Y . Global uniqueness and solvability for tensor complementarity problems[J]. Journal of Optimization Theory and Applications, 2016, 170 (1): 72- 84. doi: 10.1007/s10957-016-0903-4
8	Du S Q , Zhang L P . A mixed integer programming approach to the tensor complementarity problem[J]. Journal of Global Optimization, 2019, 73, 789- 800. doi: 10.1007/s10898-018-00731-4
9	李浙宁, 凌晨, 王宜举, 等. 张量分析和多项式优化的若干进展[J]. 运筹学学报, 2014, 18 (1): 134- 148.
10	徐凤, 凌晨. 高阶张量Pareto-特征值的若干性质[J]. 运筹学学报, 2015, 19 (3): 34- 41. doi: 10.15960/j.cnki.issn.1007-6093.2015.03.005
11	Zhou G L , Caccetta L . Feasible semismooth Newton method for a class of stochastic linear complementarity problems[J]. Journal of Optimization Theory and Applications, 2008, 139, 379- 392. doi: 10.1007/s10957-008-9406-2
12	Liu H W , Huang Y K , Li X L . New reformulation and feasible semismooth Newton method for a class of stochastic linear complementarity problems[J]. Applied Mathematics and Computation, 2011, 217 (23): 9723- 9740. doi: 10.1016/j.amc.2011.04.060
13	Chen X J , Fukushima M . Expected residual minimization method for stochastic linear complementary problems[J]. Mathematics of Operations Research, 2005, 30 (4): 1022- 1038. doi: 10.1287/moor.1050.0160
14	Gurkan G , Yoncaozgea Y , Robinson S M . Sample-path solution of stochastic variational inequalities[J]. Mathematical Programming, 1994, 84 (2): 313- 333.
15	Fischer A . A special Newton-type optimization method[J]. Optimization, 1992, 24, 269- 284. doi: 10.1080/02331939208843795
16	Facchinei F , Pang J H . Finite-dimensional variational inequalities and complementarity problem[M]. New York: Springer, 2003.
17	Qi L Q , Sun J . A nonsmooth version of Newton's method[J]. Mathematical Programming, 1993, 58, 353- 367. doi: 10.1007/BF01581275
18	Sun D F , Qi L Q . On NCP-Functions[J]. Computational Optimization and Applications, 1999, 13, 201- 220.
19	Kanzow C . An unconstrained optimization technique for large-scale linearly constrained convex minimization problems[J]. Computing, 1994, 53 (2): 101- 117.
20	Kanzow C . Global convergence properties of some iterative methods for linear complementarity problems[J]. SIAM Journal on Optimization, 1996, 6 (2): 326- 341.
21	周莎. 离散型随机线性互补问题算法的研究[D]. 桂林: 桂林电子科技大学, 2014.
22	Chen B T , Chen X J , Kanzow C . A penalized Fischer-Burmeister NCP-function[J]. Mathematical Programming, 2000, 88 (1): 211- 216.
23	Birgin E G , Martinez J M , Raydan M . Nonmonotone spectral projected gradient methods on convex sets[J]. SIAM Journal on Optimization, 2000, 10, 196- 211.

$ x^* $	$ \mathrm{val} $	$ K $
$ (0, 0.258\, 8, 0.965\, 9)^{\rm T} $	$ 4.648\, 4\times10^{-17} $	$ 7 $
$ (0, 0.258\, 8, 0.965\, 9)^{\rm T} $	$ 1.941\, 6\times10^{-17} $	$ 6 $
$ (0, 0.258\, 8, 0.965\, 9)^{\rm T} $	$ 1.320\, 9\times10^{-17} $	$ 8 $
$ (0, 0.258\, 8, 0.965\, 9)^{\rm T} $	$ 3.957\, 6\times10^{-17} $	$ 8 $
$ (0, 0.258\, 8, 0.965\, 9)^{\rm T} $	$ 5.090\, 7\times10^{-17} $	$ 7 $

$ x^* $	$ \mathrm{val} $	$ K $
$ (0.001\, 2, 1.000\, 0, 0)^{\rm T} $	$ 2.511\, 0\times10^{-13} $	$ 10 $
$ (0.000\, 8, 1.000\, 0, 0)^{\rm T} $	$ 5.670\, 8\times10^{-14} $	$ 10 $
$ (0.000\, 7, 1.000\, 0, 0)^{\rm T} $	$ 3.154\, 5\times10^{-14} $	$ 10 $
$ (0.001\, 4, 1.000\, 0, 0)^{\rm T} $	$ 4.707\, 1\times10^{-13} $	$ 11 $
$ (0.001\, 3, 1.000\, 0, 0)^{\rm T} $	$ 3.835\, 6\times10^{-13} $	$ 9 $

$ x^* $	$ \mathrm{val} $	$ K $
$ (0.710\, 8, 0.170\, 9, 0.696\, 8, 0.014\, 9)^{\rm T} $	$ 9.832\, 1\times10^{-4} $	$ 340 $
$ (0.710\, 8, 0.170\, 9, 0.696\, 8, 0.014\, 9)^{\rm T} $	$ 9.832\, 1\times10^{-4} $	$ 166 $
$ (0.710\, 8, 0.170\, 9, 0.696\, 9, 0.014\, 9)^{\rm T} $	$ 9.832\, 1\times10^{-4} $	$ 397 $
$ (0.710\, 8, 0.170\, 9, 0.696\, 9, 0.014\, 9)^{\rm T} $	$ 9.832\, 1\times10^{-4} $	$ 200 $
$ (0.710\, 8, 0.170\, 9, 0.696\, 9, 0.014\, 9)^{\rm T} $	$ 9.832\, 1\times10^{-4} $	$ 353 $

[1]	LIU Cong, JIAN Ailun, YUAN Gonglin. A modified conjugate gradient algorithm with its applications in image recovery problems [J]. Operations Research Transactions, 2026, 30(1): 207-216.
[2]	Zilin TAN, Honglin LUO. A second-order splitting method with its application [J]. Operations Research Transactions, 2025, 29(4): 121-140.
[3]	Maoran WANG, Xingju CAI, Zhongming WU, Deren HAN. First-order splitting algorithm for multi-model traffic equilibrium problems [J]. Operations Research Transactions, 2023, 27(2): 63-78.
[4]	LIU Pengjie, JIANG Xianzhen, SONG Dan. A class of spectral conjugate gradient method with sufficient descent property [J]. Operations Research Transactions, 2022, 26(4): 87-97.
[5]	Huiling ZHANG, Naoerzai SAI, Xiaoyun WU. Modified PRP conjugate gradient method for unconstrained optimization [J]. Operations Research Transactions, 2022, 26(2): 64-72.
[6]	Xiquan SHAN, Meixia LI, Jinyu LIU. Smoothing Newton method for the tensor stochastic complementarity problem [J]. Operations Research Transactions, 2022, 26(2): 128-136.
[7]	SUN Cong, ZHANG Ya. A brief review on gradient method [J]. Operations Research Transactions, 2021, 25(3): 119-132.
[8]	LI Jianling, ZHANG Hui, YANG Zhenping, JIAN Jinbao. A globally convergent SSDP algorithm without a penalty function or a filter for nonlinear semidefinite programming [J]. Operations Research Transactions, 2018, 22(4): 1-16.
[9]	SHAO Shuting, DU Shouqiang. Smoothing cautious DPRP conjugate gradient method for solving a kind of special nonsmooth equations with max-type function [J]. Operations Research Transactions, 2018, 22(3): 69-78.
[10]	. A sufficient descent conjugate gradient method for nonlinear unconstrained optimization problems [J]. Operations Research Transactions, 2018, 22(3): 59-68.
[11]	CHEN Yuanyuan, GAO Yan, LIU Zhimin, DU Shouqiang. The smoothing gradient method for a kind of special optimization problem [J]. Operations Research Transactions, 2017, 21(2): 119-125.
[12]	CHEN Zhongwen, ZHAO Qi, BIAN Kai. A successive linearization method with flexible penalty for nonlinear semidefinite programming [J]. Operations Research Transactions, 2017, 21(2): 84-100.
[13]	TIAN Zhaowei, ZHANG Liwei. A look at the convergence of the augmented Lagrange method for nondifferentiable convex programming from the view of a gradient method with constant stepsize [J]. Operations Research Transactions, 2017, 21(1): 111-117.
[14]	HANG Dan, YAN Shijian. Convergence of nonmonotonic Perry-Shanno's memoryless quasi-Newton method with parameters [J]. Operations Research Transactions, 2016, 20(4): 85-92.
[15]	MA Guodong, JIAN Jinbao. A feasible sequential systems of linear equations algorithm for inequality constrained optimization [J]. Operations Research Transactions, 2015, 19(4): 48-58.

Projected Levenberg-Marquardt method for stochastic R₀ tensor complementarity problems

RichHTML

PDF

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 3

References 23

Related Articles 15

Recommended Articles

Metrics

Comments

Projected Levenberg-Marquardt method for stochastic R0 tensor complementarity problems

RichHTML

PDF

Knowledge

Abstract

Cite this article

share this article

Figures/Tables 3

References 23

Related Articles 15

Recommended Articles

Metrics

Comments

Projected Levenberg-Marquardt method for stochastic R₀ tensor complementarity problems