On the theory of semi‐infinite programming and a generalization of the kuhn‐tucker saddle point theorem for arbitrary convex functions

Abstract

AbstractWe first present a survey on the theory of semi‐infinite programming as a generalization of linear programming and convex duality theory. By the pairing of a finite dimensional vector space over an arbitrarily ordered field with a generalized finite sequence space, the major theorems of linear programming are generalized. When applied to Euclidean spaces, semi‐infinite programming theory yields a dual theorem associating as dual problems minimization of an arbitrary convex function over an arbitrary convex set in n‐space with maximization of a linear function in non‐negative variables of a generalized finite sequence space subject to a finite system of linear equations.We then present a new generalization of the Kuhn‐Tucker saddle‐point equivalence theorem for arbitrary convex functions in n‐space where differentiability is no longer assumed.