Extensions of Korpelevich's extragradient method for the variational inequality problem in Euclidean space

We present two extensions of Korpelevich's extragradient method for solving the variational inequality problem (VIP) in Euclidean space. In the first extension, we replace the second orthogonal projection onto the feasible set of the VIP in Korpelevich's extragradient method with a specific subgradient projection. The second extension allows projections onto the members of an infinite sequence of subsets which epi-converges to the feasible set of the VIP. We show that in both extensions the convergence of the method is preserved and present directions for further research.     

New subgradient extragradient methods for solving monotone bilevel equilibrium problems

ABSTRACT

In this paper, we propose new subgradient extragradient methods for finding a solution of a strongly monotone equilibrium problem over the solution set of another monotone equilibrium problem which usually is called monotone bilevel equilibrium problem in Hilbert spaces. The first proposed algorithm is based on the subgradient extragradient method presented by Censor et al. [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]. The strong convergence of the algorithm is established under monotone assumptions of the cost bifunctions with Lipschitz-type continuous conditions recently presented by Mastroeni in the auxiliary problem principle. We also present a modification of the algorithm for solving an equilibrium problem, where the constraint domain is the common solution set of another equilibrium problem and a fixed point problem. Several fundamental experiments are provided to illustrate the numerical behaviour of the algorithms and to compare with others.     

Accelerated hybrid and shrinking projection methods for variational inequality problems

ABSTRACT

In this paper, we introduce several new extragradient-like approximation methods for solving variational inequalities in Hilbert spaces. Our algorithms are based on Tseng's extragradient method, subgradient extragradient method, inertial method, hybrid projection method and shrinking projection method. Strong convergence theorems are established under appropriate conditions. Our results extend and improve some related results in the literature. In addition, the efficiency of our algorithms is shown through numerical examples which are defined by the hybrid projection methods.     

On extragradient-viscosity methods for solving equilibrium and fixed point problems in a Hilbert space

Abstract

In this paper, new numerical algorithms are introduced for finding the solution of a variational inequality problem whose constraint set is the common elements of the set of fixed points of a demicontractive mapping and the set of solutions of an equilibrium problem for a monotone mapping in a real Hilbert space. The strong convergence of the iterates generated by these algorithms is obtained by combining a viscosity approximation method with an extragradient method. First, this is done when the basic iteration comes directly from the extragradient method, under a Lipschitz-type condition on the equilibrium function. Then, it is shown that this rather strong condition can be omitted when an Armijo-backtracking linesearch is incorporated into the extragradient iteration. The particular case of variational inequality problems is also examined.     

关于求解变分不等式问题的2-次梯度外梯度算法收敛性的一个补注

Yair Censor,Aviv Gibali和Simeon Reich为求解变分不等式问题提出了2-次梯度外梯度算法。关于此算法的收敛性,作者给出了部分证明,有一个问题：由算法产生的迭代点列能否收敛到变分不等式问题的一个解上,没有得到解决。此问题作为一个公开问题在文章“Extensions of Korpelevich's extragradient method for the variational inequalityproblem in Euclidean space”(Optimization,61(9)：1119-1132,2012)中被提出。在这篇简短的补注性文章中,对所提出的问题给出了答案：由算法产生的迭代点列能收敛到变分不等式问题的一个解上。给出2-次梯度外梯度算法的全局收敛性的一个完整证明,证明了从任意起始点开始,由算法产生的迭代点列都能收敛到变分不等式问题的一个解上。     

A hybrid extragradient method for general variational inequalities

In this paper, we introduce and study a hybrid extragradient method for finding solutions of a general variational inequality problem with inverse-strongly monotone mapping in a real Hilbert space. An iterative algorithm is proposed by virtue of the hybrid extragradient method. Under two sets of quite mild conditions, we prove the strong convergence of this iterative algorithm to the unique common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the general variational inequality problem, respectively. L. C. Zeng’s research was partially supported by the National Science Foundation of China (10771141), Ph.D. Program Foundation of Ministry of Education of China (20070270004), and Science and Technology Commission of Shanghai Municipality grant (075105118). J. C. Yao’s research was partially supported by a grant from the National Science Council of Taiwan.     

Extragradient methods for differential variational inequality problems and linear complementarity systems

In this paper, 2 extragradient methods for solving differential variational inequality (DVI) problems are presented, and the convergence conditions are derived. It is shown that the presented extragradient methods have weaker convergence conditions in comparison with the basic fixed‐point algorithm for solving DVIs. Then the linear complementarity systems, as an important and practical special case of DVIs, are considered, and the convergence conditions of the presented extragradient methods are adapted for them. In addition, an upper bound for the Lipschitz constant of linear complementarity systems is introduced. This upper bound can be used for adjusting the parameters of the extragradient methods, to accelerate the convergence speed. Finally, 4 illustrative examples are considered to support the theoretical results.     

Modified subgradient extragradient method for variational inequality problems

In this paper, we introduce an algorithm as combination between the subgradient extragradient method and inertial method for solving variational inequality problems in Hilbert spaces. The weak convergence of the algorithm is established under standard assumptions imposed on cost operators. The proposed algorithm can be considered as an improvement of the previously known inertial extragradient method over each computational step. The performance of the proposed algorithm is also illustrated by several preliminary numerical experiments.     

An extragradient iterative scheme by viscosity approximation methods for fixed point problems and variational inequality problems

In this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping and the set of solutions of the variational inequality problem for an α-inverse-strongly-monotone, by combining an modified extragradient scheme with the viscosity approximation method. We prove a strong convergence theorem for the sequences generated by this new iterative process.      

Simultaneous Subgradient Algorithms for the Generalized Split Variational Inclusion Problem in Hilbert Spaces

In this article, we study the generalized split variational inclusion problem. For this purpose, motivated by the projected Landweber algorithm for the split equality problem, we first present a simultaneous subgradient extragradient algorithm and give related convergence theorems for the proposed algorithm. Next, motivated by the alternating CQ-algorithm for the split equality problem, we propose another simultaneous subgradient extragradient algorithm to study the general split variational inclusion problem. As applications, we consider the split equality problem, split feasibility problem, split variational inclusion problem, and variational inclusion problem in Hilbert spaces.     

Some subgradient extragradient type algorithms for solving split feasibility and fixed point problems

In this paper, we consider the split feasibility problem (SFP) in infinite‐dimensional Hilbert spaces and propose some subgradient extragradient‐type algorithms for finding a common element of the fixed‐point set of a strict pseudocontraction mapping and the solution set of a split feasibility problem by adopting Armijo‐like stepsize rule. We derive convergence results under mild assumptions. Our results improve some known results from the literature. Copyright © 2016 John Wiley & Sons, Ltd.     

A modified subgradient extragradient method for solving the variational inequality problem

The subgradient extragradient method for solving the variational inequality (VI) problem, which is introduced by Censor et al. (J. Optim. Theory Appl. 148, 318–335, 2011), replaces the second projection onto the feasible set of the VI, in the extragradient method, with a subgradient projection onto some constructible half-space. Since the method has been introduced, many authors proposed extensions and modifications with applications to various problems. In this paper, we introduce a modified subgradient extragradient method by improving the stepsize of its second step. Convergence of the proposed method is proved under standard and mild conditions and primary numerical experiments illustrate the performance and advantage of this new subgradient extragradient variant.     

Hybrid iterative scheme by a relaxed extragradient method for solutions of equilibrium problems and a general system of variational inequalities with application to optimization

In this paper, we introduce a new iterative process for finding the common element of the set of fixed points of a nonexpansive mapping, the set of solutions of an equilibrium problem and the solutions of the variational inequality problem for two inverse-strongly monotone mappings. We introduce a new viscosity relaxed extragradient approximation method which is based on the so-called relaxed extragradient method and the viscosity approximation method. We show that the sequence converges strongly to a common element of the above three sets under some parametric controlling conditions. Moreover, using the above theorem, we can apply to finding solutions of a general system of variational inequality and a zero of a maximal monotone operator in a real Hilbert space. The results of this paper extended, improved and connected with the results of Ceng et al., [L.-C. Ceng, C.-Y. Wang, J.-C. Yao, Strong convergence theorems by a relaxed extragradient method for a general system of variational inequalities, Math. Meth. Oper. Res. 67 (2008), 375–390], Plubtieng and Punpaeng, [S. Plubtieng, R. Punpaeng, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Appl. Math. Comput. 197 (2) (2008) 548–558] Su et al., [Y. Su, et al., An iterative method of solution for equilibrium and optimization problems, Nonlinear Anal. 69 (8) (2008) 2709–2719], Li and Song [Liwei Li, W. Song, A hybrid of the extragradient method and proximal point algorithm for inverse strongly monotone operators and maximal monotone operators in Banach spaces, Nonlinear Anal.: Hybrid Syst. 1 (3) (2007), 398-413] and many others.     

A New Extragradient Method for Strongly Pseudomonotone Variational Inequalities

This article proposes a new extragradient solution method for strongly pseudomonotone variational inequalities. A detailed analysis of the iterative sequences’ convergence and of the range of applicability of the method is provided. Moreover, an interesting class of strongly pseudomonotone infinite dimensional variational inequality problems is considered.     

Strong Convergence of the Halpern Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Spaces

Building upon the subgradient extragradient method proposed by Censor et al., we prove the strong convergence of the iterative sequence generated by a modification of this method by means of the Halpern method. We also consider the problem of finding a common element of the solution set of a variational inequality and the fixed-point set of a quasi-nonexpansive mapping with a demiclosedness property.     

AN EXTRAGRADIENT METHOD FOR RELAXED COCOERCIVE VARIATIONAL INEQUALITY AND EQUILIBRIUM PROBLEMS

The purpose of this paper is to investigate the problem of finding the common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of an equilibrium problem and the set of solutions of the variational inequality prob- lem for a relaxed cocoercive and Lipschitz continuous mapping in Hilbert spaces. Then, we show that the sequence converges strongly to a common element of the above three sets under some parameter controlling conditions, which are connected with Yao, Liou, Yao[17], Takahashi[12] and many others.     

Relaxed extragradient iterative methods for variational inequalities

In this paper, we suggest and analyze some new relaxed extragradient iterative methods for finding a common element of the solution set of a variational inequality, the solution set of a general system of variational inequalities and the set of fixed points of a strictly pseudo-contractive mapping defined on a real Hilbert space. Strong convergence of the proposed methods under some mild conditions is established.     

The Subgradient Extragradient Method for Solving Variational Inequalities in Hilbert Space

We present a subgradient extragradient method for solving variational inequalities in Hilbert space. In addition, we propose a modified version of our algorithm that finds a solution of a variational inequality which is also a fixed point of a given nonexpansive mapping. We establish weak convergence theorems for both algorithms.     

Hybrid extragradient method for general equilibrium problems and fixed point problems in Hilbert space

In this paper, we introduce an iterative scheme by the hybrid methods for finding a common element of the set of fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of a variational inequality problem in a Hilbert space. Then, we prove the strongly convergent theorem by a hybrid extragradient method to the common element of the set of fixed points of nonexpansive mappings, the set of solutions of an equilibrium problem and the set of solutions of a variational inequality problem. Our results extend and improve the results of Bnouhachem et al. [A. Bnouhachem, M. Aslam Noor, Z. Hao, Some new extragradient iterative methods for variational inequalities, Nonlinear Analysis (2008) doi:10.1016/j.na.2008.02.014] and many others.     

Modified subgradient extragradient algorithms for solving monotone variational inequalities

ABSTRACT

In this paper, we introduce two new algorithms for solving classical variational inequalities problem with Lipschitz continuous and monotone mapping in real Hilbert space. We modify the subgradient extragradient methods with a new step size, the convergence of algorithms are established without the knowledge of the Lipschitz constant of the mapping. Finally, some numerical experiments are presented to show the efficiency and advantage of the proposed algorithms.