Sion's minimax theorem and the proximal point algorithm in Hadamard spaces

TL;DR

This paper extends Sion's minimax theorem to Hadamard spaces, explores properties of saddle function resolvents, and applies these findings to the proximal point algorithm for minimax problems.

Contribution

It introduces a version of Sion's minimax theorem in Hadamard spaces and analyzes resolvent properties, advancing optimization methods in non-Euclidean geometries.

Findings

Established Sion's minimax theorem in Hadamard spaces.

Analyzed properties of resolvents of saddle functions in Hadamard spaces.

Applied results to the proximal point algorithm for minimax problems.

Abstract

We obtain Sion's minimax theorem in Hadamard spaces and discuss its applications. Among other things, we study several fundamental properties of resolvents of saddle functions in Hadamard spaces. An application to the proximal point algorithm for minimax problems in Hadamard spaces are also included.