Relaxed extragradient methods for systems of variational inequalities

Lu-Chuan Ceng,Ching-Feng Wen

doi:10.1186/s13660-015-0648-x

Abstract

In this paper, we introduce a relaxed extragradient iterative algorithm for finding a common element of the set of solutions of a general mixed equilibrium problem, the set of solutions of general system of variational inequalities, the set of solutions of finitely many variational inclusions, and the set of common fixed points of finitely many nonexpansive mappings and a strict pseudocontraction in a real Hilbert space. The iterative algorithm is based on Korpelevich’s extragradient method, the viscosity approximation method, Mann’s iterative method, and the strongly positive bounded linear operator approach. We derive the strong convergence of the iterative algorithm to a common element of these sets, which also solves some hierarchical minimization.

Highlights

Let H be a real Hilbert space with inner product ·, · and norm ·, C be a nonempty closed convex subset of H, and PC be the metric projection of H onto C
We introduce a relaxed extragradient iterative algorithm for finding a common element of the solution set general mixed equilibrium problem (GMEP)(Θ, h) of GMEP ( . ), the solution set general system of variational inequalities (GSVI)(C, F, F ) (i.e., Ξ ) of GSVI ( . ), the solution set of finitely many variational inclusions for maximal monotone mappings {Rk}Mk= and inverse-strongly monotone mappings {Bk}Mk=, and the common fixed point set of finitely many nonexpansive mappings Si : C → C, i =, . . . , N, and a strictly pseudocontractive mapping
Theorems . and . of this paper show that the proposed algorithm converges strongly to a unique solution of a variational inequality problem (VIP) defined over the set

Summary

Introduction

Let H be a real Hilbert space with inner product ·, · and norm · , C be a nonempty closed convex subset of H, and PC be the metric projection of H onto C. Let a set-valued mapping R : D(R) ⊂ H → H be maximal monotone. The generalized mixed equilibrium problem is defined as follows: Let φ : C → R be a realvalued function, A : C → H be a nonlinear mapping and Θ : C × C → R be a bifunction.

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