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Minimax Theorems for Possibly Nonconvex Functions

Nguyen Nang Thieu, Nguyen Dong Yen

TL;DR

The paper addresses minimax theorems for possibly nonconvex functions on Euclidean or Hilbert spaces and investigates the existence of saddle points. It introduces a Lagrangian-type transform $\mathcal{L}(x,y)= J(x+y)-\dfrac{L}{2}\|y\|^2$ and proves a minimax equality under coercivity of $J$ and a Lipschitz gradient. Extensions to a Hilbert space setting and to a perturbed function with a concave $\gamma$ preserve the equality and saddle-point property through convex-analytic tools such as subdifferentials and the Moreau–Rockafellar theorem. The results provide a complete solution to Ricceri’s open problem and are illustrated by a concrete one-dimensional example, with potential extensions to non-$C^1$ $J$.

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.

Minimax Theorems for Possibly Nonconvex Functions

TL;DR

The paper addresses minimax theorems for possibly nonconvex functions on Euclidean or Hilbert spaces and investigates the existence of saddle points. It introduces a Lagrangian-type transform and proves a minimax equality under coercivity of and a Lipschitz gradient. Extensions to a Hilbert space setting and to a perturbed function with a concave preserve the equality and saddle-point property through convex-analytic tools such as subdifferentials and the Moreau–Rockafellar theorem. The results provide a complete solution to Ricceri’s open problem and are illustrated by a concrete one-dimensional example, with potential extensions to non- .

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.

Paper Structure

This paper contains 4 sections, 3 theorems, 61 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main Results
  3. An Illustrative Example
  4. Concluding Remarks

Key Result

Theorem 2.3

Let $Y \subset \Bbb R^n$, with $n\geq 1$, be a nonempty closed convex set and let $J:\Bbb R^n \to \Bbb R$ be a Fréchet differentiable function satisfying the following conditions: Then, one has or, equivalently, with Furthermore, the function ${\mathcal{L}}(x,y)$ has a saddle point on $\Bbb R^n\times Y$, that is, there exists $(\bar{x},\bar{y})\in \Bbb R^n\times Y$ satisfying for all $(x,y)\i

Figures (1)

  • Figure 1: Graph of $J$ defined in Example \ref{['example1']}

Theorems & Definitions (6)

  • Remark 2.1
  • Remark 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Example 3.1