Hybrid Iterative Scheme for Triple Hierarchical Variational Inequalities with Mixed Equilibrium, Variational Inclusion, and Minimization Constraints

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

doi:10.1155/2014/767109

Abstract

We introduce and analyze a hybrid iterative algorithm by combining Korpelevich's extragradient method, the hybrid steepest-descent method, and the 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 finitely many nonexpansive mappings, the solution set of a generalized mixed equilibrium problem (GMEP), the solution set of finitely many variational inclusions, and the solution set of a convex minimization problem (CMP), which is also 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 solving a hierarchical variational inequality problem with constraints of the GMEP, the CMP, and finitely many variational inclusions.

Highlights

Let H be 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
We introduce and study the following triple hierarchical variational inequality (THVI) with constraints of generalized mixed equilibrium problem (GMEP) (3), convex minimization problem (CMP) (7), and finitely many variational inclusions
We prove the strong convergence of the proposed algorithm to a unique solution of THVI (16) under suitable conditions

Summary

Introduction

Let H be 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. Assume that (i) K : H → R is strongly convex with a constant σ > 0 and its derivative K󸀠 is Lipschitz continuous with a constant ] > 0 such that the function x 󳨃→ ⟨y − x, K󸀠(x)⟩ is weakly upper semicontinuous for each y ∈ H; (ii) for each x ∈ H, there exist a bounded subset Dx ⊂ C and zx ∈ C such that, for any y ∉ Dx, Θ (y, zx)

Results

Conclusion