一种新的求解拟单调变分不等式的压缩投影算法

doi:10.15960/j.cnki.issn.1007-6093.2023.01.009

运筹学学报（中英文） ›› 2023, Vol. 27 ›› Issue (1): 127-137.doi: 10.15960/j.cnki.issn.1007-6093.2023.01.009

一种新的求解拟单调变分不等式的压缩投影算法

叶明露¹^,*(), 邓欢¹

1. 西华师范大学数学与信息学院四川高等院校优化理论与应用重点实验室, 四川南充 637009

收稿日期:2022-09-28 出版日期:2023-03-15 发布日期:2023-03-16
通讯作者: 叶明露 E-mail:yml2002cn@aliyun.com
作者简介:叶明露, E-mail: yml2002cn@aliyun.com
基金资助:
国家自然科学基金面上项目(11871059);西华师范大学培育项目(20A024)

A new projection and contraction algorithm for solving quasimonotone variational inequalities

Minglu YE¹^,*(), Huan DENG¹

1. Sichuan Colleges and Universities Key Laboratory of Optimization Theory and Applications, School of Mathematics and Information, China West Normal University, Nanchong, 637009, Sichuan, China

Received:2022-09-28 Online:2023-03-15 Published:2023-03-16
Contact: Minglu YE E-mail:yml2002cn@aliyun.com

摘要/Abstract

摘要：

2020年Liu和Yang提出了求解Hilbert空间中拟单调且Lipschitz连续的变分不等式问题的投影算法，简称LYA。本文在欧氏空间中提出了一种新的求解拟单调变分不等式的压缩投影算法, 简称NPCA。新算法削弱了LYA中映射的Lipschitz连续性。在映射连续、拟单调且对偶变分不等式解集非空的条件下得到了NPCA所生成点列的聚点是解的结论。当变分不等式的解集还满足一定条件时，得到了NPCA的全局收敛性。数值实验结果表明NPCA所需的迭代步数少于LYA的迭代步数，NPCA在高维拟单调例子中所需的计算机耗时也更少。

关键词: 变分不等式, 拟单调, 压缩投影算法, 连续

Abstract:

In 2020, Liu and Yang proposed a projection algorithm (LYA for short) for solving quasi-monotone and Lipschitz continuous variational inequalities problem (VIP for short) in Hilbert space. In this paper, we present a new projection and contraction algorithm (NPCA for short) for solving quasi-monotone VIP in Euclidean space. The new algorithm weakens the Lipschitz continuity of the underlying mapping in LYA. NPCA clusters to the solution of VIP whenever the underlying mapping is continuous, quasi-monotone and the solution set of dual variational inequality is nonempty. The global convergence of NPCA needs an additional assumption about the solution set of VIP. Numerical experiments show that NPCA is more efficient than LYA from the total number of iterative point of view, and the CPU time point of view in high dimensional quasi-monotone VIP.

Key words: variational inequalities, quasi-monotone, projection and contraction algorithm, continuity

中图分类号:

O221.2

叶明露, 邓欢. 一种新的求解拟单调变分不等式的压缩投影算法[J]. 运筹学学报（中英文）, 2023, 27(1): 127-137.

Minglu YE, Huan DENG. A new projection and contraction algorithm for solving quasimonotone variational inequalities[J]. Operations Research Transactions, 2023, 27(1): 127-137.

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参考文献

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$ x^0 $	iter			CPU
$ x^0 $	NPCA	LYA1	LYA2	NPCA	LYA1	LYA2
$ -3 $	32	171	162	0.001 13	0.001 51	0.002 12
$ -2 $	29	162	1	0.001 19	0.001 65	0.000 501
0.5	25	151	151	0.001 16	0.001 69	0.001 72
0.7	28	156	155	0.001 2	0.001 95	0.001 93
0.9	28	161	159	0.001 13	0.001 45	0.001 72
1	1	162	160	0.000 586	0.001 55	0.001 6
4	29	174	172	0.001 17	0.001 68	0.001 72

$ x^0 $	iter			CPU
$ x^0 $	NPCA	LYA1	LYA2	NPCA	LYA1	LYA2
(0, 1)	10	18	20	0.001 21	0.002 63	0.001 13
(0, 0)	7	18	0	0.001 37	0.002 71	0.000 632
(1, 0)	10	18	19	0.001 43	0.001 97	0.001 08
(0.5, 0.5)	7	16	18	0.001 26	0.002 34	0.001 22
(0.2, 0.7)	7	17	19	0.001 22	0.002 22	0.001 3
(0.1, 0.6)	7	17	19	0.001 27	0.001 97	0.001 18
Random	9	17	19	0.001 32	0.002 16	0.001 25
Random	8	15	18	0.001 35	0.001 87	0.001 07

$ x^0 $	$ a $	iter			CPU
$ x^0 $	$ a $	NPCA	LYA1	LYA2	NPCA	LYA1	LYA2
(0, 0, 0, 0, 5)	5	39	155	154	0.295	0.541	0.484
(0, 0, 5, 0, 0)	5	39	155	154	0.295	0.55	0.487
(0, 2, 0, 2, 1)	5	42	150	159	0.273	0.525	0.497
(1, 1, 1, 1, 6)	10	28	309	295	0.101	1.02	0.912
(5, 0, 0, 0, 5)	10	45	313	293	0.156	1.03	0.916

$ n $	iter			CPU
$ n $	NPCA	LYA1	LYA2	NPCA	LYA1	LYA2
50	93	1 369	1 442	0.392	5.72	5.96
100	170	2 646	2 775	0.93	14.4	14.9
200	324	5 093	5 387	2.71	86	91.1
500	769	11 979	12 620	16.9	535	551
1 000	1 482	$ \backslash $	$ \backslash $	73.749 9	$ \backslash $	$ \backslash $

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一种新的求解拟单调变分不等式的压缩投影算法

A new projection and contraction algorithm for solving quasimonotone variational inequalities

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