Hybrid Gradient-Projection Algorithm for Solving Constrained Convex Minimization Problems with Generalized Mixed Equilibrium Problems

Lu-Chuan Ceng,Ching-Feng Wen

doi:10.1155/2012/678353

Abstract

It is well known that the gradient-projection algorithm (GPA) for solving constrained convex minimization problems has been proven to have only weak convergence unless the underlying Hilbert space is finite dimensional. In this paper, we introduce a new hybrid gradient-projection algorithm for solving constrained convex minimization problems with generalized mixed equilibrium problems in a real Hilbert space. It is proven that three sequences generated by this algorithm converge strongly to the unique solution of some variational inequality, which is also a common element of the set of solutions of a constrained convex minimization problem, the set of solutions of a generalized mixed equilibrium problem, and the set of fixed points of a strict pseudocontraction in a real Hilbert space.

Highlights

Let H be a real Hilbert space with inner product ·, · and norm ·
We introduce a new hybrid gradient-projection algorithm for solving constrained convex minimization problems with generalized mixed equilibrium problems in a real Hilbert space
Recall that a ρ-Lipschitz continuous mapping T : C → H is a mapping on C such that

Summary

Introduction

Let H be a real Hilbert space with inner product ·, · and norm ·. It is known that the gradient-projection algorithm has weak convergence, in general, unless the underlying Hilbert space is finite dimensional. The sequence {xn} converges in norm to a minimizer of 1.5 which is the unique solution of the variational inequality of finding x∗ ∈ Ω such that. Peng 10 introduced a variant of Korpelevic’s extragradient method 17 for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of a strict pseudocontraction, and the set of solutions of a variational inequality for a monotone, Lipschitz continuous mapping. It is proven that under very mild conditions, the sequences {xn}, {yn} and {zn} converge strongly to the unique solution of the variational inequality of finding x∗ ∈ Fix S ∩ Ω ∩ GMEP such that.

Preliminaries

Strong Convergence Theorem