Solving Generalized Mixed Equilibria, Variational Inequalities, and Constrained Convex Minimization

A E Al-Mazrooei,A Latif,J C Yao

doi:10.1155/2014/587865

A E Al-Mazrooei, A Latif + Show 1 more

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https://doi.org/10.1155/2014/587865

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Abstract

We propose implicit and explicit iterative algorithms for finding a common element of the set of solutions of the minimization problem for a convex and continuously Fréchet differentiable functional, the set of solutions of a finite family of generalized mixed equilibrium problems, and the set of solutions of a finite family of variational inequalities for inverse strong monotone mappings in a real Hilbert space. We prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets under very mild conditions.

Highlights

Introduction and Problems FormulationLet 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
Motivated and inspired by the above facts, in this paper we introduce implicit and explicit iterative algorithms for finding a common element of the set of solutions of the convex minimization problem (CMP) (19) for a convex functional f : C → R with L
Throughout this paper, we assume that H is a real Hilbert space with inner product and norm denoted by ⟨⋅, ⋅⟩ and ‖ ⋅ ‖, respectively

Summary

Introduction and Problems Formulation

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. Motivated and inspired by the above facts, in this paper we introduce implicit and explicit iterative algorithms for finding a common element of the set of solutions of the CMP (19) for a convex functional f : C → R with L-. Lipschitz continuous gradient ∇f, the set of solutions of a finite family of GMEPs, and the set of solutions of a finite family of VIPs for inverse strong monotone mappings in a real. Under very mild control conditions, we prove that the sequences generated by the proposed algorithms converge strongly to a common element of three sets, which is the unique solution of a variational inequality defined over the intersection of three sets. The results obtained in this paper improve and extend the corresponding results announced by many others

Preliminaries

Implicit Iterative Algorithm and Its Convergence Criteria

Explicit Iterative Algorithm and Its Convergence Criteria