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

运筹学学报 ›› 2021, Vol. 25 ›› Issue (4): 69-79.doi: 10.15960/j.cnki.issn.1007-6093.2021.04.006

随机R₀张量互补问题的投影Levenberg-Marquardt方法

崔丽媛¹, 杜守强¹^,*()

1. 青岛大学数学与统计学院, 山东青岛 266071

收稿日期:2019-10-29 出版日期:2021-12-15 发布日期:2021-12-11
通讯作者: 杜守强 E-mail:sqdu@qdu.edu.cn
作者简介:杜守强, E-mail: sqdu@qdu.edu.cn
基金资助:
国家自然科学基金(11671220)

Projected Levenberg-Marquardt method for stochastic R₀ tensor complementarity problems

Liyuan CUI¹, Shouqiang DU¹^,*()

1. School of Mathematics and Statistics, Qingdao University, Qingdao 266071, Shandong, China

Received:2019-10-29 Online:2021-12-15 Published:2021-12-11
Contact: Shouqiang DU E-mail:sqdu@qdu.edu.cn

摘要/Abstract

摘要：

本文考虑一类离散型随机$R_0$张量互补问题，利用Fischer-Burmeister函数将问题转化为约束优化问题，并用投影Levenberg-Marquardt方法对其进行了求解。在一般的条件下得到了该方法的全局收敛性，相关的数值实验表明了该方法的有效性。

关键词: 随机R₀张量互补问题, 投影Levenberg-Marquardt方法, 全局收敛, 线搜索

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

中图分类号:

O221

崔丽媛, 杜守强. 随机R₀张量互补问题的投影Levenberg-Marquardt方法[J]. 运筹学学报, 2021, 25(4): 69-79.

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

图/表 3

表1

表2

表3

参考文献 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 $

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随机R₀张量互补问题的投影Levenberg-Marquardt方法

Projected Levenberg-Marquardt method for stochastic R₀ tensor complementarity problems

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