Nonlinear Programming via König’s Maximum Theorem

Abstract

Starting from one extension of the Hahn---Banach theorem, the Mazur---Orlicz theorem, and a not very restrictive concept of convexity, that arises naturally in minimax theory, infsup-convexity, we derive an equivalent version of that fundamental result for finite dimensional spaces, which is a sharp generalization of Konig's Maximum theorem. It implies several optimal statements of the Lagrange multipliers, Karush/Kuhn---Tucker, and Fritz John type for nonlinear programs with an objective function subject to both equality and inequality constraints.