Multi-step extragradient method with regularization for triple hierarchical variational inequalities with variational inclusion and split feasibility constraints

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

doi:10.1186/1029-242x-2014-492

Abstract

AbstractBy combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method, Mann’s iteration method, and the gradient-projection method with regularization, a hybrid multi-step extragradient algorithm with regularization for finding a solution of triple hierarchical variational inequality problem is introduced and analyzed. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a unique solution of a triple hierarchical variational inequality problem which is defined over the set of solutions of a hierarchical variational inequality problem defined over the set of common solutions of finitely many generalized mixed equilibrium problems (GMEP), finitely many variational inclusions, fixed point problems, and the split feasibility problem (SFP). We also prove the strong convergence of the proposed algorithm to a common solution of the SFP, finitely many GMEPs, finitely many variational inclusions, and the fixed point problem of a strict pseudocontraction. The results presented in this paper improve and extend the corresponding results announced by several others.MSC:49J30, 47H09, 47J20, 49M05.

Highlights

The following problems have their own importance because of their applications in diverse areas of science, engineering, social sciences, and management: Equilibrium problems including variational inequalities. Variational inclusion problems. Split feasibility problems. Fixed point problems
Fixed point problem Let C be a nonempty subset of a H and T : C → C be a mapping
Where Ξ denotes the solution set of the hierarchical variational inequality problem (HVIP) of finding z∗ ∈ Fix(T) ∩ VI(C, A) such thatz∗, z – z∗ ≥, ∀z ∈ Fix(T) ∩ VI(C, A)

Summary

Introduction

The following problems have their own importance because of their applications in diverse areas of science, engineering, social sciences, and management:. Let a set-valued mapping R : D(R) ⊂ H → H be maximal monotone. Fixed point problem Let C be a nonempty subset of a H and T : C → C be a mapping. The objective is to find x∗ ∈ Ξ such that (μF – γ V )x∗, x – x∗ ≥ , ∀x ∈ Ξ , where Ξ denotes the solution set of the hierarchical variational inequality problem (HVIP) of finding z∗ ∈ Fix(T) ∩ VI(C, A) such that (μF – γ S)z∗, z – z∗ ≥ , ∀z ∈ Fix(T) ∩ VI(C, A). They proved that the sequence {xn} generated by the proposed algorithm converges strongly to a point x∗ ∈ Fix(T) ∩ VI(C, A) which is a unique solution of Problem . Bm is ηm-inverse strongly monotone, Bm is a monotone and Lipschitz-continuous mapping

BmΛmn un

Since the combination of the boundedness of