Minimax Theorems for Possibly Nonconvex Functions

TL;DR

This paper develops minimax theorems applicable to possibly nonconvex functions in Euclidean and Hilbert spaces, ensuring saddle point existence and solving an open problem related to differentiable functions.

Contribution

It introduces new minimax theorems for nonconvex functions and provides a complete solution to a longstanding open problem in the field.

Findings

Established three minimax theorems for nonconvex functions

Proved the existence of saddle points under broad conditions

Solved an open problem related to differentiable functions

Abstract

This paper establishes three minimax theorems for possibly nonconvex functions on Euclidean spaces or on infinite-dimensional Hilbert spaces. The theorems also guarantee the existence of saddle points. As a by-product, a complete solution to an interesting open problem related to continuously differentiable functions is obtained. The obtained results are analyzed via a concrete example.