Modified hybrid projection methods for finding common solutions to variational inequality problems

Abstract

In this paper we propose several modified hybrid projection methods for solving common solutions to variational inequality problems involving monotone and Lipschitz continuous operators. Based on differently constructed half-spaces, the proposed methods reduce the number of projections onto feasible sets as well as the number of values of operators needed to be computed. Strong convergence theorems are established under standard assumptions imposed on the operators. An extension of the proposed algorithm to a system of generalized equilibrium problems is considered and numerical experiments are also presented.

Cyclic subgradient extragradient methods for equilibrium problems

“…. , N }, there exists a subsequence x n j of {x n } converging weakly to p, i.e., x n j p as j → ∞ such that [n j ] = i for all j. Lemma 2.1 and relation (36) ensure that p ∈ F(S). Repeating the proof of Theorem 3.5, we conclude that p ∈ F, hence p ∈ F ∩ F(S) and x n → p as n → ∞.…”

“…which was introduced and studied in [11,21,36]. In 2005, Combettes and Hirstoaga [14] introduced a general procedure for solving CSEPs.…”

Strong Convergence of Modified Inertial Mann Algorithms for Nonexpansive Mappings

“…This method is now referred as the CQ algorithm. For further research, see [6,8,9,10,11]. Recently, Kim and Xu [2] proposed the following modified Mann iteration algorithm based on the Halpern iterative algorithm [12] and the Mann iteration algorithm:…”

A Projection Algorithm for Non-Monotone Variational Inequalities

“…Other projection algorithms for solving variational inequalities are found in [13,15,16,41]. The main difference between our algorithm and [13,15,16,41] lies in the structure of the problem and the techniques used proving the convergence. For instance, [13,41] considers a Lipschitz continuous point-to-point operator, making the analysis of these algorithms substantially different from ours.…”