Regularized Extragradient Methods for Solving Equilibrium Problems on Hadamard Manifolds

TL;DR

This paper introduces two regularized extragradient algorithms for equilibrium problems on Hadamard manifolds, ensuring convergence without Lipschitz conditions and demonstrating effectiveness through numerical experiments.

Contribution

The paper proposes novel regularized extragradient methods that guarantee convergence on Hadamard manifolds without requiring Lipschitz continuity.

Findings

Algorithms converge to solutions without Lipschitz assumptions

Global error bounds established for the methods

Demonstrated $R$-linear convergence under strong pseudomonotonicity

Abstract

Employing two distinct types of regularization terms, we propose two regularized extragradient methods for solving equilibrium problems on Hadamard manifolds. The sequences generated by these extragradient algorithms converge to a solution of the equilibrium problem without requiring the Lipschitz continuity of the bifunction or imposing additional conditions on the parameters. We establish convergence results for both algorithms and derive global error bounds along with $R$-linear convergence rates in cases where the bifunction is strongly pseudomonotone. Finally, we present numerical experiments to demonstrate the effectiveness of our methods.