Hybrid Extragradient Method with Regularization for Convex Minimization, Generalized Mixed Equilibrium, Variational Inequality and Fixed Point Problems

Lu-Chuan Ceng,Juei-Ling Ho

doi:10.1155/2014/436069

Abstract

We introduce two iterative algorithms by the hybrid extragradient method with regularization for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings and the set of fixed points of an asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions.

Highlights

Throughout this paper, we assume that H is a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖; let C be a nonempty closed convex subset of H and let PC be the metric projection of H onto C
Let C be a nonempty subset of a Hilbert space H
In this paper, inspired by the above facts, we introduce two iterative algorithms by hybrid extragradient method with regularization for finding a common element of the set of solutions of the CMP (2) for a convex functional f : C → R with L-Lipschitz continuous gradient ∇f, the set of solutions of finite generalized mixed equilibrium problem (GMEP), the set of solutions of finite variational inequality problem (VIP) for inverse strong monotone mappings, and the set of fixed points of an asymptotically κ-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space

Summary

Introduction

Throughout this paper, we assume that H is a real Hilbert space with inner product ⟨⋅, ⋅⟩ and norm ‖ ⋅ ‖; let C be a nonempty closed convex subset of H and let PC be the metric projection of H onto C. Very recently, motivated by Yao et al [26], Cai and Bu [3] introduced two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inequalities for inverse strong monotone mappings, and the set of fixed points of an asymptotically κ-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space They proved some strong and weak convergence theorems for the proposed iterative algorithms under appropriate conditions. In this paper, inspired by the above facts, we introduce two iterative algorithms by hybrid extragradient method with regularization for finding a common element of the set of solutions of the CMP (2) for a convex functional f : C → R with L-Lipschitz continuous gradient ∇f, the set of solutions of finite GMEPs, the set of solutions of finite VIPs for inverse strong monotone mappings, and the set of fixed points of an asymptotically κ-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. See, for example, [7, 24, 27,28,29,30,31] and ther references therein

Preliminaries

Strong Convergence Theorem

Weak Convergence Theorem