Revisiting the Hahn-Banach Theorem and Nonlinear Infinite Programming

TL;DR

This paper presents a refined version of the K"onig supremum theorem, linking it to the Hahn-Banach theorem, and applies it to derive optimality conditions for nonlinear infinite programming, emphasizing infsup-convexity.

Contribution

It introduces a sharp reformulation of the K"onig supremum theorem and demonstrates its application to nonlinear infinite programming with a focus on infsup-convexity.

Findings

A new sharp version of the K"onig supremum theorem.

Derivation of Lagrange multiplier and KKT conditions for infinite programs.

Identification of infsup-convexity as the key convexity concept.

Abstract

[REVISED VERSION] The aim of this paper is to state a sharp version of the K\"onig supremum theorem, an equivalent reformulation of the Hahn--Banach theorem. We apply it to derive statements of the Lagrange multipliers, Karush-Kuhn-Tucker and Fritz John type, for nonlinear infinite programs. We also show that a weak concept of convexity coming from minimax theory, infsup-convexity, is the adequate one for this kind of results.