Hybrid Steepest-Descent Methods for Triple Hierarchical Variational Inequalities

L C Ceng,A Latif,A E Al-Mazrooei,C F Wen

doi:10.1155/2015/980352

Abstract

We introduce and analyze a relaxed iterative algorithm by combining Korpelevich’s extragradient method, hybrid steepest-descent method, and Mann’s iteration method. We prove 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 inclusions, and the solution set of general system of variational inequalities (GSVI), 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 for solving a hierarchical variational inequality problem with constraints of finitely many GMEPs, finitely many variational inclusions, and the GSVI. 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
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 inclusions, and the solution set of general system of variational inequalities (GSVI), which is just a unique solution of a triple hierarchical variational inequality (THVI) in a real Hilbert space
We introduce and study the following triple hierarchical variational inequality (THVI) with constraints of finitely many GMEPs, finitely many variational inclusions, and general system of variational inequalities

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. We will introduce and analyze a relaxed iterative algorithm for finding a solution of the THVI (19) with constraints of several problems: finitely many GMEPs, finitely many variational inclusions, and GSVI (10) in a real Hilbert space.

Results

Conclusion