A new iterative method for generalized equilibrium and constrained convex minimization problems

M Yazdi

doi:10.17951/a.2020.74.2.81-99

Abstract

The gradient-projection algorithm (GPA) plays an important role in solving constrained convex minimization problems. In this paper, we combine the GPA and averaged mapping approach to propose an explicit composite iterative scheme for finding a common solution of a generalized equilibrium problem and a constrained convex minimization problem. Then, we prove a strong convergence theorem which improves and extends some recent results.

Highlights

Let H be a real Hilbert space and C be a nonempty closed convex subset of H
In this paper, motivated by the above results, we propose an explicit composite iterative scheme for finding the common element of the set of solutions of a generalized equilibrium problem and the solution set of a constrained convex minimization problem
Let H be a real Hilbert space, C be a closed convex subset of H and T : C → C be a nonexpansive mapping with F (T ) = ∅

Summary

Introduction

Let H be a real Hilbert space and C be a nonempty closed convex subset of H. In 2012, Tian and Liu [13] studied the following implicit and explicit composite iterative schemes by the viscosity approximation method for finding the common solution of an equilibrium problem and a constrained convex minimization problem:. L-Lipschitzian mapping with L ≥ 0 such that U ∩ EP (φ) = ∅, f is a contraction, x1 ∈ C, {αn} ⊂ (0, 1), {rn} ⊂ (0, ∞), un = Qrnxn, PC (I−λn g) = They proved that the sequences {xn}, generated by (1.8) and (1.9), converge strongly to a point in U ∩ EP (φ) under certain conditions. Let H be a real Hilbert space, C be a closed convex subset of H and T : C → C be a nonexpansive mapping with F (T ) = ∅.

Since the projection

Applying the conditions sn