Hilbert 空间中变分不等式问题的自适应粘性算法

Abstract

Abstract:

In this paper, we propose a new self adaptive subgradient extragradient viscosity algorithm for solving pseudomonotone variational inequality problems in Hilbert space. Using the new stepsize rule, the strong convergence theorem is obtained without any information about the Lipschitz constant. The effectiveness of the suggested algorithm is illustrated through some numerical examples.

Key words: Variational inequality, Pseudomonotone mapping, Self adaptive stepsize, Strong convergence

CLC Number:

O177

Xia Pingjing, Cai Gang. Self Adaptive Viscosity Algorithm for Solving Variational Inequality Problem in Hilbert Spaces[J].Acta mathematica scientia,Series A, 2023, 43(2): 581-592.

Trendmd

References

[1]	Censor Y, Elfving T. A multiprojection algorithms using Bregman projection in a product space. Numer Algorithms, 1994, 8: 221-239
[2]	Censor Y, Elfving T, Kopf N, Bortfeld T. The multiple-sets split feasibility problem and its applications for inverse problems. Inverse Problems, 2005, 21: 2071-2084
[3]	Hieu D V, Cho Y J, Xiao Y. Modified extragradient algorithms for solving equilibrium problems. Optimization, 2018, 67: 2003-2029
[4]	Kinderlehrer D, Stampacchia G. An Introduction to Variational Inequalities and Their Applications. New York: Academic Press, 1980
[5]	Baiocchi C, Capelo A.Variational and Quasivariational Inequalities; Applications to Free Boundary Problems. New York: Wiley, 1984
[6]	Korpelevich G M. The extragradient method for finding saddle points and other problems. Ekonomikai Matematicheskie Metody, 1976, 12: 747-756
[7]	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
[8]	Censor Y, Gibali A, Reich S. Extensions of Korpelevich extragradient method for variational inequality problem in Euclidean space. Optimization, 2011, 61: 1119-1132
[9]	Malitsky Y. Projected reflected gradient methods for monotone variational inequalities. SIAM J Optim, 2015, 25: 502-520
[10]	Thong D V, Hieu D V. Modified subgradient extragradient method for variational inequality problems. Numer Algorithms, 2018, 79: 597-610
[11]	Thong D V, Vinh N T, Cho Y J. A strong convergence theorem for Tseng's extragradient method for solving variational inequality problems. Optim Lett, 2020, 14: 1157-1175
[12]	Censor Y, Gibali A, Reich S. Strong convergence of subgradient extragradient methods for the variational inequality problem in Hilbert space. Optim Methods Softw, 2011, 26: 827-845
[13]	Kraikaew R, Saejung S. Strong convergence of the Halpern subgradient extragradient method for solving variational inequalities in Hilbert spaces. J Optim Theory Appl, 2014, 163: 399-412
[14]	Chuang C S. Hybrid inertial proximal algorithm for the split variational inclusion problem in Hilbert spaces with applications. Optimization, 2017, 66: 777-792
[15]	Majee P, Nahak C. On inertial proximal algorithm for split variational inclusion problems. Optimization, 2018, 67: 1701-1716
[16]	Thong D V. Viscosity approximation methods for solving fixed-point problems and split common fixed-point problems. J Fixed Point Theory Appl, 2017, 19: 1481-1499
[17]	Thong D V, Hieu D V, Rassias T M. Self adaptive inertial subgradient extragradient algorithms for solving pseudomonotone variational inequality problems. Optim Lett, 2020, 14: 115-144
[18]	Moudafi A, Elisabeth E. An approximate inertial proximal method using enlargement of a maximal monotone operator. Int J Pure Appl Math, 2003, 5: 283-299
[19]	Alvarez F, Attouch H. An inertial proximal method for maximal monotone operators via discretization of a nonlinear oscillator with damping. Set-Valued Anal, 2001, 9: 3-11
[20]	Opial Z. Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bull Am Math Soc, 1967, 73: 591-597
[21]	Cottle R W, Yao J C. Pseudo-monotone complementarity problems in Hilbert space. J Optim Theory Appl, 1992, 75: 281-295
[22]	Saejung S, Yotkaew P. Approximation of zeros of inverse strongly monotone operators in Banach spaces. Nonlinear Anal, 2012, 75: 742-750
[23]	Anh P K, Thong D V, Vinh N T. Improved inertial extragradient methods for solving pseudo-monotone variational inequalities. Optimization, doi: 10.1080/02331934.2020.1808
[24]	Cholamjiak P, Thong D V, Cho Y J. A novel inertial projection and contraction method for solving pseudomonotone variational inequality problems. Acta Appl Math, 2020, 169: 217-245

Self Adaptive Viscosity Algorithm for Solving Variational Inequality Problem in Hilbert Spaces

RichHTML

PDF (PC)

Abstract

Cite this article

share this article

References

Related Articles 15

Metrics

Comments

Recommended 0

[1]	Wang Wujing, Zhu Meiling, Zhang Yongle. A New Projection Algorithm for Solving Quasimonotone Variational Inequality Problems and Fixed Point Problems [J]. Acta mathematica scientia,Series A, 2025, 45(1): 236-255.
[2]	Wang Ruyu, Zhao Wenling, Song Daojin. The Finite Termination of Feasible Solution Sequence for Optimization and Variational Inequality [J]. Acta mathematica scientia,Series A, 2024, 44(4): 1037-1051.
[3]	Ma Xiaojun, Chen Fu, Jia Zhifu. Research on a Strong Convergence Theorem for Proximal Split Feasibility Problems with Non-Lipschitz Stepsizes [J]. Acta mathematica scientia,Series A, 2024, 44(4): 1052-1065.
[4]	Yang Lanxiang, Chen Yi, Ye Minglu. Inertial Projection Algorithms for Quasimonotone Variational Inequalities [J]. Acta mathematica scientia,Series A, 2023, 43(2): 593-603.
[5]	Duan Jie, Xia Fuquan. A New Tseng-like Extragradient Algorithm for Common Solutions of Variational Inequalities and Fixed Point Problems [J]. Acta mathematica scientia,Series A, 2023, 43(1): 274-290.
[6]	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.
[7]	Jing Yang,Xianjun Long. A New Projection Algorithm for Solving Pseudo-Monotone Variational Inequality and Fixed Point Problems [J]. Acta mathematica scientia,Series A, 2022, 42(3): 904-919.
[8]	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.
[9]	Guodong Ma. A Strongly Convergent Generalized Gradient Projection Method for Minimax Optimization with General Constraints [J]. Acta mathematica scientia,Series A, 2020, 40(3): 641-649.
[10]	Gang Cai. Viscosity Implicit Algorithms for a Variational Inequality Problem and Fixed Point Problem in Hilbert Spaces [J]. Acta mathematica scientia,Series A, 2020, 40(2): 395-407.
[11]	Xueping Luo, Mengtian Cui. Solvability of Directional Perturbed Generalized Mixed Variational Inequalities in Reflexive Banach Spaces [J]. Acta mathematica scientia,Series A, 2018, 38(6): 1144-1152.
[12]	Cai Gang, Yekini Shehu. Viscosity Iterative Algorithm for Variational Inequality Problems and Fixed Point Problems of Strict Pseudo-Contractions in q-Uniformly Smooth and Uniformly Convex Banach Spaces [J]. Acta mathematica scientia,Series A, 2016, 36(4): 623-638.
[13]	Liu Yuanxing, Zhou Haiyun, Wang Peiyuan, Zhou Yu. Three Kinds of Iterative Algorithms for Solving the Constrained Split Common Fixed Point Problem in Hilbert Spaces [J]. Acta mathematica scientia,Series A, 2015, 35(6): 1115-1126.
[14]	Wang Yuehu, Zhang Congjun. Strong Convergence Based on Generalized Wiener-Hopf Equation Techniques for Solving Generalized Equilibrium Problems [J]. Acta mathematica scientia,Series A, 2015, 35(4): 695-709.
[15]	Cai Gang. Strong Convergence Theorems by A Viscosity Iterative Method for Fixed Point Problems of Nonexpansive Mappings in Banach Spaces [J]. Acta mathematica scientia,Series A, 2015, 35(3): 487-502.