A Bregman Regularized Proximal Point Method for Solving Equilibrium Problems on Hadamard Manifolds

TL;DR

This paper introduces a Bregman regularized proximal point algorithm tailored for equilibrium problems on Hadamard manifolds, addressing nonconvexity issues and demonstrating convergence under weaker conditions.

Contribution

It develops a novel Bregman regularization scheme for equilibrium problems on Hadamard manifolds with convergence guarantees and practical effectiveness.

Findings

Convergence to solutions under strong convexity assumptions.

Weaker coercivity conditions than existing methods.

Numerical experiments confirm practical effectiveness.

Abstract

In this paper we develop a Bregman regularized proximal point algorithm for solving monotone equilibrium problems on Hadamard manifolds. It has been shown that the regularization term induced by a Bregman function is, in general, nonconvex on Hadamard manifolds unless the curvature is zero. Nevertheless, we prove that the proposed Bregman regularization scheme does converge to a solution of the equilibrium problem on Hadamard manifolds in the presence of a strong assumption on the convexity of the set formed by the regularization term. Moreover, we employ a coercivity condition on the Bregman function which is weaker than those typically assumed in the existing literature on Bregman regularization. Numerical experiments on illustrative examples demonstrate the practical effectiveness of our proposed method.