Parallel extragradient - viscosity methods for equilibrium problems and fixed point problems

TL;DR

This paper introduces parallel extragradient-viscosity algorithms that efficiently find a unique solution in the intersection of equilibrium and fixed point problems, improving convergence performance using combined iterative methods.

Contribution

It develops novel parallel extragradient-viscosity methods that integrate multiple iterative techniques for solving complex equilibrium and fixed point problems.

Findings

Algorithms achieve strong convergence under standard assumptions.

Methods outperform some existing algorithms in convergence speed.

The approach effectively combines extragradient, Mann, and hybrid steepest-descent methods.

Abstract

In this paper, we propose two parallel extragradient - viscosity methods for finding a particular element in the common solution set of a system of equilibrium problems and finitely many fixed point problems. This particular point is the unique solution of a variational inequality problem on the common solution set. The main idea of the paper is to combine three methods including the extragradient method, the Mann iteration method, the hybrid steepest-descent method with the parallel splitting-up technique to design the algorithms which improve the performance over some existing methods. The strongly convergent theorems are established under the widely used assumptions for equilibrium bifunctions.