An extension of hybrid method without extrapolation step to equilibrium problems

TL;DR

This paper presents a new hybrid algorithm for equilibrium problems that simplifies previous methods by eliminating extra steps, using only an optimization program per iteration, and demonstrates its convergence through theoretical proofs and numerical experiments.

Contribution

It introduces a hybrid algorithm that avoids extrapolation steps, combining extragradient and hybrid methods with only one optimization per iteration.

Findings

The algorithm converges strongly to the solution.

Numerical experiments confirm the effectiveness and convergence.

The method outperforms some existing algorithms in efficiency.

Abstract

In this paper, we introduce a new hybrid algorithm for solving equilibrium problems. The algorithm combines the extragradient method and the hybrid (outer approximation) method. In this algorithm, only an optimization program is solved at each iteration without the extra-steps like as in the extragradient method and the Armijo linesearch method. A specially constructed half-space in the hybrid method is the reason for the absence of an optimization program in our algorithm. The strong convergence theorem is established and several numerical experiments are implemented to illustrate the convergence of the algorithm and compare it with others.