Hybrid extragradient method for hierarchical variational inequalities

Lu-Chuan Ceng,Jen-Chih Yao,Qamrul Hasan Ansari,Abdul Latif

doi:10.1186/1687-1812-2014-222

Abstract

AbstractIn this paper, we consider a hierarchical variational inequality problem (HVIP) defined over a common set of solutions of finitely many generalized mixed equilibrium problems, finitely many variational inclusions, a general system of variational inequalities, and the fixed point problem of a strictly pseudocontractive mapping. By combining Korpelevich’s extragradient method, the viscosity approximation method, the hybrid steepest-descent method and Mann’s iteration method, we introduce and analyze a multistep hybrid extragradient algorithm for finding a solution of our HVIP. It is proven that under appropriate assumptions, the proposed algorithm converges strongly to a solution of a general system of variational inequalities defined over a common set of solutions of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, and the fixed point problem of a strictly pseudocontractive mapping. In the meantime, we also prove the strong convergence of the proposed algorithm to a unique solution of our HVIP. The results obtained in this paper improve and extend the corresponding results announced by many others.MSC:49J30, 47H09, 47J20.

Highlights

Introduction and formulations1.1 Variational inequalities and equilibrium problems Let C be a nonempty closed convex subset of a real Hilbert space H and A : C → H be a nonlinear mapping
1.5 Problem to be considered In this paper, we introduce and study the following hierarchical variational inequality problem (HVIP) defined over a common set of solutions of finitely many generalized mixed equilibrium problems (GMEPs), finitely many variational inclusions, a general system of variational inequalities, and a fixed point of a strictly pseudocontractive mapping
We prove the strong convergence of the proposed algorithm to a unique solution of HVIP ( . ) under suitable conditions

Summary

Then we know that κ

Lemma . [ ] Let {sn} be a sequence of nonnegative numbers satisfying the conditions sn+ ≤ ( – αn)sn + αnβn, ∀n ≥ , where {αn} and {βn} are sequences of real numbers such that (a) {αn} ⊂ [ , ] and. (This shows that G : C → C is a nonexpansive mapping.) Since (γn + δn)ξ ≤ γn for all n ≥ and T is ξ -strictly pseudocontractive, utilizing Lemma . Since xn – yn → , {βn} ⊂ [a, b] ⊂ ( , ), νj ∈ ( , ζj), j = , , and {xn}, {yn} are bounded sequences, we have lim n→∞. Bm is ηm-inverse-strongly monotone, Bm is a monotone and Lipschitz continuous mapping

Next we prove that w

It is clear that in this case the mapping Γ