求解伪单调变分不等式问题的惯性收缩投影算法

数学物理学报 ›› 2021, Vol. 41 ›› Issue (6): 1897-1911.

求解伪单调变分不等式问题的惯性收缩投影算法

贺月红(),龙宪军*()

^{重庆工商大学数学与统计学院重庆 400067}

收稿日期:2021-03-01 出版日期:2021-12-26 发布日期:2021-12-02
通讯作者: 龙宪军 E-mail:heyuehong1111@163.com;xianjunlong@ctbu.edu.cn
作者简介:贺月红, E-mail: heyuehong1111@163.com
基金资助:
国家自然科学基金(11471059);重庆市基础与前沿研究计划项目(cstc2021jcyj-msxmX0721);重庆市教育委员会科学技术研究重点项目(KJZDK201900801);重庆工商大学科研团队项目(ZDPTTD201908)

A Inertial Contraction and Projection Algorithm for Pseudomonotone Variational Inequality Problems

Yuehong He(),Xianjun Long*()

^{College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067}

Received:2021-03-01 Online:2021-12-26 Published:2021-12-02
Contact: Xianjun Long E-mail:heyuehong1111@163.com;xianjunlong@ctbu.edu.cn
Supported by:
the NSFC(11471059);the Basic and Advanced Research Project of Chongqing(cstc2021jcyj-msxmX0721);the Education Committee Project Research Foundation of Chongqing(KJZDK201900801);the Innovative Project of CTBU(ZDPTTD201908)

摘要/Abstract

摘要：

该文提出了一种新的惯性收缩投影算法，在不依赖Lipschitz常数L的条件下，证明了伪单调变分不等式问题解的强收敛性定理.最后给出了数值实验结果.

关键词: 变分不等式, 惯性收缩投影法, 伪单调, 强收敛

Abstract:

In this paper, we introduce a new inertial contraction and projection algorithm for pseudomonotone variational inequality problems. We prove the strong convergence theorem without the knowledge of the Lipschitz constant of the mapping. Finally, we give some numerical experiments to show the efficiency of the algorithm.

Key words: Variational inequality, Inertial contraction and projection method, Pseudomonotone, Strong convergence

中图分类号:

O224

贺月红,龙宪军. 求解伪单调变分不等式问题的惯性收缩投影算法[J]. 数学物理学报, 2021, 41(6): 1897-1911.

Yuehong He,Xianjun Long. A Inertial Contraction and Projection Algorithm for Pseudomonotone Variational Inequality Problems[J]. Acta mathematica scientia,Series A, 2021, 41(6): 1897-1911.

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参考文献 15

1	Facchinei F, Pang J S. Finite-Dimensional Variational Inequalities and Complementarity Problems. New York: Springer, 2003
2	Goebel K, Reich S. Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. New York: Marcel Dekker, 1983
3	He B . A class of projection and contraction methods for monotone variational inequalities. Appl Math Optim, 1997, 35 (1): 69- 76 doi: 10.1007/s002459900037
4	Cai X , Gu G , He B . On the O(1/t) convergence rate of the projection and contraction methods for variational inequalities with Lipschitz continuous monotone operators. Comput Optim Appl, 2014, 57 (2): 339- 363 doi: 10.1007/s10589-013-9599-7
5	Censor Y , Gibali A , Reich S . The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl, 2011, 148 (2): 318- 335 doi: 10.1007/s10957-010-9757-3
6	万升联. 解变分不等式的一种二次投影算法. 数学物理学报, 2021, 41A(1): 237-244
	Wan S L. A double projection algorithm for solving variational inequalities. 2021, 41A(1): 237-244
7	Tseng P . A modified forward-backward splitting method for maximal monotone mappings. SIAM J Control Optim, 2000, 38 (2): 431- 446 doi: 10.1137/S0363012998338806
8	Karamardian S . Complementarity problems over cones with monotone and pseudo-monotone maps. J Optim Theory Appl, 1976, 18 (4): 445- 454 doi: 10.1007/BF00932654
9	Korpelevich G M . The extragradient method for finding saddle points and other problems. Ekonomika i Mat Metody, 1976, 12, 747- 756
10	Dong L Q , Cho Y J , Zhong L L , Rassias M T . Inertial projection and contraction algorithms for variational inequalities. J Glob Optim, 2018, 70 (3): 687- 704 doi: 10.1007/s10898-017-0506-0
11	Thong D V , Vinh N T , Cho Y J . New strong convergence theorem of the inertial projection and contraction method for variational inequality problems. Numer Algor, 2020, 84 (1): 285- 305 doi: 10.1007/s11075-019-00755-1
12	Yang J , Liu H W . Strong convergence result for solving monotone variational inequalities in Hilbert space. Numer Algor, 2019, 80 (3): 741- 752 doi: 10.1007/s11075-018-0504-4
13	Gibali A , Thong D V , Tuan P A . Two simple projection-type methods for solving variational inequalities. Anal Math Phys, 2019, 9, 2203- 2225 doi: 10.1007/s13324-019-00330-w
14	Maingé P E . A hybrid extragradient-viscosity method for monotone operators and fixed point problems. SIAM J Control Optim, 2008, 47 (3): 1499- 1515 doi: 10.1137/060675319
15	Xu H K . Iterative algorithms for nonlinear operators. J Lond Math Soc, 2002, 66 (1): 240- 256 doi: 10.1112/S0024610702003332

	$u_0 = {(t^3+1)e^{-t}}$		$u_0 = \frac{1}{500}[{\rm cost}+\sin(-5t)]$		$u_0 = \frac{1}{600}(t^2-e^{-t})$
	Iter	Time	Iter	Time	Iter	Time
NIPCM	6	0.0429	8	0.0535	7	0.0427
VPTA	33	0.0238	125	0.1098	78	0.0573
Y.L	340	0.0437	340	0.0319	340	0.0525
IPCM	533	0.0567	533	0.0612	533	0.0633

	$\epsilon = 10^{-4}$		$\epsilon = 10^{-10}$		$\epsilon = 10^{-20}$		$\epsilon = 10^{-30}$
	Iter	Time	Iter	Time	Iter	Time	Iter	Time
NIPCM	32	0.7108	61	1.1787	79	1.4880	88	2.1495
VTPA	40	9.0375	40	10.3125	40	10.8281	40	9.8906
Y.L	64	3.2096	66	2.4719	69	3.3800	68	2.9023
IPCM	-	$>50$	-	$>50$	-	$>50$	-	$>50$

[1]	万升联. 解变分不等式的一种二次投影算法[J]. 数学物理学报, 2021, 41(1): 237-244.
[2]	马国栋. 一般约束极大极小优化问题一个强收敛的广义梯度投影算法[J]. 数学物理学报, 2020, 40(3): 641-649.
[3]	蔡钢. Hilbert空间中关于变分不等式问题和不动点问题的粘性隐式中点算法[J]. 数学物理学报, 2020, 40(2): 395-407.
[4]	胡雨贤. 求解变分不等式的一种双投影算法[J]. 数学物理学报, 2019, 39(6): 1492-1498.
[5]	丘小玲,贾文生. 有限理性下变分不等式的逼近定理[J]. 数学物理学报, 2019, 39(4): 730-737.
[6]	罗雪萍, 崔梦天. 自反巴拿赫空间中方向扰动的广义混合变分不等式的可解性[J]. 数学物理学报, 2018, 38(6): 1144-1152.
[7]	李茹, 余国林, 刘伟, 刘三阳. 一类广义不变凸函数和向量变分不等式[J]. 数学物理学报, 2017, 37(6): 1029-1039.
[8]	蔡钢, Yekini Shehu. q-一致光滑、一致凸Banach空间中关于变分不等式问题和严格伪压缩映射的不动点问题的粘性迭代算法[J]. 数学物理学报, 2016, 36(4): 623-638.
[9]	李择均, 孙淑芹. 一个扰动变分不等式的可解性[J]. 数学物理学报, 2016, 36(3): 473-480.
[10]	刘元星, 周海云, 王培元, 周宇. Hilbert空间中求解带约束的分裂公共不动点问题的三种迭代算法[J]. 数学物理学报, 2015, 35(6): 1115-1126.
[11]	魏利, Agarwal Ravi P. 含有广义p-Laplacian算子的双曲型非线性微分方程的单调性方法[J]. 数学物理学报, 2015, 35(5): 867-877.
[12]	王月虎, 张从军. 广义均衡问题的算法及强收敛性——基于Wiener-Hopf方程技巧[J]. 数学物理学报, 2015, 35(4): 695-709.
[13]	蔡钢. 关于非扩张映射的不动点问题的粘性迭代算法的强收敛定理[J]. 数学物理学报, 2015, 35(3): 487-502.
[14]	万成高. 随机环境中马氏链函数加权和的极限定理[J]. 数学物理学报, 2015, 35(1): 163-171.
[15]	闻道君. 有限簇伪压缩映象和单调映象广义迭代法的强收敛性[J]. 数学物理学报, 2014, 34(5): 1123-1132.