求解变分不等式和不动点问题的公共元的修正次梯度外梯度算法

数学物理学报 ›› 2022, Vol. 42 ›› Issue (5): 1517-1536.

求解变分不等式和不动点问题的公共元的修正次梯度外梯度算法

刘丽平(),彭建文*()

^{重庆师范大学数学科学学院重庆 401331}

收稿日期:2021-12-06 出版日期:2022-10-26 发布日期:2022-09-30
通讯作者: 彭建文 E-mail:1318286263@qq.com;jwpeng6@aliyun.com
作者简介:刘丽平, E-mail: 1318286263@qq.com
基金资助:
国家自然科学基金面上项目(11171363);重庆英才·创新创业领军人才·创新创业示范团队项目(CQYC20210309536);重庆市高校创新研究群体项目(CXQT20014);重庆市自然科学基金项目(cstc 2021jcyj-msxmX0300)

Modified Subgradient Extragradient Algorithms for Solving Common Elements of Variational Inequality and Fixed Point Problems

Liping Liu(),Jianwen Peng*()

^{School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331}

Received:2021-12-06 Online:2022-10-26 Published:2022-09-30
Contact: Jianwen Peng E-mail:1318286263@qq.com;jwpeng6@aliyun.com
Supported by:
the National Natural Science Foundation of China(11171363);the Team Project of Innovation Leading Talent in Chongqing(CQYC20210309536);the Chongqing University Innovation Research Group Project(CXQT20014);the Basic and Advanced Research Project of Chongqing(cstc 2021jcyj-msxmX0300)

摘要/Abstract

摘要：

该文在实Hilbert空间中引入了一类新的求解变分不等式问题的惯性次梯度外梯度算法. 在适当的参数假设下, 证明了由该算法所产生的序列强收敛于伪单调变分不等式问题的解集与拟非扩张映射不动点集合的公共元素. 最后, 给出了数值实验来说明所提算法的有效性. 该文所得的结果推广和改进了文献中的一些已有结果.

关键词: 变分不等式, 不动点, 伪单调, 次梯度外梯度算法

Abstract:

In this paper, we introduce a new class of inertial subgradient extragradient algorithms for solving variational inequality problems in the real Hilbert space. Under some appropriate conditions imposed on the parameters, we prove that the sequence generated by the algorithm strongly converges to a common element of the solution set for a pseudomonotone variational inequality problem and the set of fixed points for a quasinonexpansive mapping. Finally, we give numerical experiments to illustrate the effectiveness of our proposed algorithm. The results obtained in this paper extend and improve some existing results in the literature.

Key words: Variational inequality, Fixed point, Pseudomonotone, Subgradient extragradient method

中图分类号:

刘丽平,彭建文. 求解变分不等式和不动点问题的公共元的修正次梯度外梯度算法[J]. 数学物理学报, 2022, 42(5): 1517-1536.

Liping Liu,Jianwen Peng. Modified Subgradient Extragradient Algorithms for Solving Common Elements of Variational Inequality and Fixed Point Problems[J]. Acta mathematica scientia,Series A, 2022, 42(5): 1517-1536.

图/表 4

参考文献 37

1	Aubin J P , Ekeland I . Applied Nonlinear Analysis. New York: Wiley, 1984
2	Karamardian S . Complementarity problems over cones with monotone and pseudomonotone maps. J Optim Theory Appl, 1976, 18, 445- 454 doi: 10.1007/BF00932654
3	Kinderlehrer D , Stampacchia G . An Introduction to Variational Inequalities and Their Applications. New York: Academic Press, 1980
4	Baiocchi C , Capelo A . Variational and Quasivariational Inequalities Applications to Free Boundary Problems. New York: Wiley, 1984
5	Konnov I V . Combined Relaxation Methods for Variational Inequalities. Berlin: Springer, 2001
6	Sibony M . Methodes iteratives pour les equations et inequations aux derivees partielles non lineares de type monotone. Calcolo, 1970, 7 (1): 65- 183
7	Korpelevich G M . The extragradient method for finding saddle points and other problems. Matecon, 1976, 12, 747- 756
8	Cai G , Bu S Q . Modified extragradient methods for variational inequality problems and fixed point problems for an infinite family of nonexpansive mappings in Banach spaces. J Global Optim, 2013, 55 (2): 437- 457 doi: 10.1007/s10898-012-9883-6
9	Cai G , Gibali A , Iyiola O S , Shehu Y . A new double-projection method for solving variational inequalities in Banach spaces. J Optim Theory Appl, 2018, 178, 219- 239 doi: 10.1007/s10957-018-1228-2
10	Iusem A N , Nasri M . Korpelevich's method for variational inequality problems in Banach spaces. J Global Optim, 2011, 50, 59- 76 doi: 10.1007/s10898-010-9613-x
11	Yao Y H , Noor M A , Noor K I , Liou Y C , Yaqoob H . Modified extragradient methods for a system of variational inequalities in Banach spaces. Acta Appl Math, 2010, 110, 1211- 1224 doi: 10.1007/s10440-009-9502-9
12	Shehu Y . Single projection algorithm for variational inequalities in Banach spaces with application to contact problem. Acta Math Sci, 2020, 40B (4): 1045- 1063
13	Popov L D . A modification of the Arrow-Hurwicz method for the search of saddle points. Mathematical notes of the Academy of Sciences of the USSR, 1980, 28, 845- 848
14	Censor Y , Gibali A , Reich S . The subgradient extragradient method for solving variational inequalities in Hilbert space. J Optim Theory Appl, 2011, 148, 318- 335 doi: 10.1007/s10957-010-9757-3
15	Ceng L C , Yao J C . Strong convergence theorem by an extragradient method for fixed point problems and variational inequality problems. Taiwan J Math, 2006, 10 (5): 1293- 1303
16	Ceng L C , Hadjisavvas N , Wong N C . Strong convergence theorem by a hybrid extragradient-like approximation method for variational inequalities and fixed point problems. J Global Optim, 2010, 46, 635- 646 doi: 10.1007/s10898-009-9454-7
17	Iiduka H , Takahashi W . Strong convergence theorems for nonexpansive mappings and inverse strongly monotone mappings. Nonlinear Anal, 2005, 61 (3): 341- 350 doi: 10.1016/j.na.2003.07.023
18	Thong D V , Hieu D V . Inertial subgradient extragradient algorithms with line-search process for solving variational inequality problems and fixed point problems. Numer Algor, 2019, 80, 1283- 1307 doi: 10.1007/s11075-018-0527-x
19	Cai G , Dong Q L , Peng Y . Strong convergence theorems for solving variational inequality problems with pseudo-monotone and non-Lipschitz operators. J Optim Theory Appl, 2021, 188, 447- 472 doi: 10.1007/s10957-020-01792-w
20	Yang H , Agarwal R P , Nashine H K , Liang Y . Fixed point theorems in partially ordered Banach spaces with applications to nonlinear fractional evolution equations. J Fixed Point Theory Appl, 2017, 19, 1661- 1678 doi: 10.1007/s11784-016-0316-x
21	Takahashi W , Toyoda M . Weak convergence theorems for nonexpansive mappings and monotone mappings. J Optim Theory Appl, 2003, 118, 417- 428 doi: 10.1023/A:1025407607560
22	Nadezhkina N , Takahashi W . Weak convergence theorem by an extragradient method for nonexpansive mappings and monotone mappings. J Optim Theory Appl, 2006, 128, 191- 201 doi: 10.1007/s10957-005-7564-z
23	Diaz J B , Metcalf F T . Subsequential limit points of successive approximations. Trans Amer Math Soc, 1969, 135, 459- 485
24	Thong D V , Li X H , Dong Q L , Cho Y J , Rassias T M . An inertial Popov's method for solving pseudomonotone variational inequalities. Optim Lett, 2021, 15, 757- 777 doi: 10.1007/s11590-020-01599-8
25	Bot R I , Csetnek E R . An inertial Tseng's type proximal algorithm for nonsmooth and nonconvex optimization problems. J Optim Theory Appl, 2016, 171, 600- 616 doi: 10.1007/s10957-015-0730-z
26	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. 数学物理学报, 2021, 41A (6): 1897- 1911 doi: 10.3969/j.issn.1003-3998.2021.06.026
	He Y H , Long X J . An inertial contraction and projection algorithm for pseudomonotone variational inequality problems. Acta Math Sci, 2021, 41A (6): 1897- 1911 doi: 10.3969/j.issn.1003-3998.2021.06.026
27	Dong Q L , Yuan H B , Cho Y J , Rassias T M . Modified inertial Mann algorithm and inertial CQ-algorithm for nonexpansive mappings. Optim Lett, 2018, 12, 87- 102 doi: 10.1007/s11590-016-1102-9
28	Thong D V , Hieu D V . An inertial method for solving split common fixed point problems. J Fixed Point Theory Appl, 2017, 19, 3029- 3051 doi: 10.1007/s11784-017-0464-7
29	Goebel K , Reich S . Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. New York: Marcel Dekker, 1984
30	Cottle R W , Yao J C . Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl, 1992, 75, 281- 295 doi: 10.1007/BF00941468
31	Xu H K . Iterative algorithm for nonlinear operators. Journal of the London Mathematical Society, 2002, 66 (1): 240- 256 doi: 10.1112/S0024610702003332
32	Mainge P E . The viscosity approximation process for quasi-nonexpansive mapping in Hilbert space. Computers and Mathematics with Applications, 2010, 59, 74- 79 doi: 10.1016/j.camwa.2009.09.003
33	Denisov S V , Semenov V V , Chabak L M . Convergence of the modified extragradient method for variational inequalities with non-Lipschitz operators. Cybern Syst Anal, 2015, 51, 757- 765 doi: 10.1007/s10559-015-9768-z
34	Iusem A N , Otero R G . Inexact versions of proximal point and augmented Lagrangian algorithms in Banach spaces. Numer Funct Anal Optim, 2001, 22, 609- 640 doi: 10.1081/NFA-100105310
35	Yang J . Self-adaptive inertial subgradient extragradient algorithm for solving pseudomonotone variational inequalities. Applicable Analysis, 2021, 100 (5): 1067- 1078 doi: 10.1080/00036811.2019.1634257
36	Harker P T , Pang J S . A damped-Newton method for the linear complementarity problem. Lect Appl Math, 1990, 26, 265- 284
37	Solodov M V , Svaiter B F . A new projection method for variational inequality problems. SIAM J Control Optim, 1999, 37 (3): 765- 776 doi: 10.1137/S0363012997317475

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