Equilibrium problems on Riemannian manifolds with applications

TL;DR

This paper investigates equilibrium problems on Riemannian manifolds, establishing solution existence, convexity properties, and convergence of algorithms, with applications to variational inequalities and Nash equilibria.

Contribution

It introduces a novel approach linking equilibrium problems to variational inequalities on Riemannian manifolds, differing from previous methods.

Findings

Proved existence of solutions on Riemannian manifolds.

Analyzed convex structure of the solution set.

Established convergence of the proximal point algorithm.

Abstract

We study the equilibrium problem on general Riemannian manifolds. The results on existence of solutions and on the convex structure of the solution set are established. Our approach consists in relating the equilibrium problem to a suitable variational inequality problem on Riemannian manifolds, and is completely different from previous ones on this topic in the literature. As applications, the corresponding results for the mixed variational inequality and the Nash equilibrium are obtained. Moreover, we formulate and analyze the convergence of the proximal point algorithm for the equilibrium problem. In particular, correct proofs are provided for the results claimed in J. Math. Anal. Appl. 388, 61-77, 2012 (i.e., Theorems 3.5 and 4.9 there) regarding the existence of the mixed variational inequality and the domain of the resolvent for the equilibrium problem on Hadamard manifolds.