Triple Hierarchical Variational Inequalities with Constraints of Mixed Equilibria, Variational Inequalities, Convex Minimization, and Hierarchical Fixed Point Problems

Lu-Chuan Ceng,Cheng-Wen Liao,Ching-Feng Wen,Chin-Tzong Pang

doi:10.1155/2014/129379

Abstract

We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich's extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element of the fixed point set of infinitely many nonexpansive mappings, the solution set of finitely many generalized mixed equilibrium problems (GMEPs), the solution set of finitely many variational inequality problems (VIPs), the solution set of general system of variational inequalities (GSVI), and the set of minimizers of convex minimization problem (CMP), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space. In addition, we also consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI, and CMP. The results obtained in this paper improve and extend the corresponding results announced by many others.

Highlights

Let C be a nonempty closed convex subset of a real Hilbert space H and let PC be the metric projection of H onto C
We introduce and analyze a hybrid iterative algorithm by virtue of Korpelevich’s extragradient method, viscosity approximation method, hybrid steepest-descent method, and averaged mapping approach to the gradient-projection algorithm
We introduce and study the following triple hierarchical variational inequality (THVI) with constraints of mixed equilibria, variational inequalities, and convex minimization problem

Summary

Introduction

Let C be a nonempty closed convex subset of a real Hilbert space H and let PC be the metric projection of H onto C. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a common element x∗ ∈ Ω := (∩∞ n=1 Fix(Sn)) ∩ (∩M k=1 GMEP(Θk, φk, Bk)) ∩ (∩Ni=1 VI(C, Ai)) ∩ GSVI(G) ∩ Γ of the fixed point set of infinitely many nonexpansive mappings {Sn}∞ n=1, the solution set of finitely many GMEPs, the solution set of finitely many VIPs, the solution set of GSVI (11), and the set of minimizers of CMP (14), which is just a unique solution of the THVI (24). We consider the application of the proposed algorithm to solve a hierarchical fixed point problem with constraints of finitely many GMEPs, finitely many VIPs, GSVI (11), and CMP (14). The results obtained in this paper improve and extend the corresponding results announced by many others

Preliminaries

Strong Convergence Criteria for the THVI and HFPP