An elementary proof of linear programming optimality conditions without using Farkas' lemma

TL;DR

This paper provides an elementary proof of the necessary optimality conditions in linear programming that avoids Farkas' lemma, making it potentially more accessible for teaching purposes.

Contribution

It introduces a direct, elementary proof of linear programming optimality conditions that does not depend on Farkas' lemma or advanced LP concepts.

Findings

Proof relies solely on linear algebra and perturbation techniques

Avoids complex topics like separating hyperplanes and duality

Useful for educational purposes in teaching linear programming

Abstract

Although it is easy to prove the sufficient conditions for optimality of a linear program, the necessary conditions pose a pedagogical challenge. A widespread practice in deriving the necessary conditions is to invoke Farkas' lemma, but proofs of Farkas' lemma typically involve "nonlinear" topics such as separating hyperplanes between disjoint convex sets, or else more advanced LP-related material such as duality and anti-cycling strategies in the simplex method. An alternative approach taken previously by several authors is to avoid Farkas' lemma through a direct proof of the necessary conditions. In that spirit, this paper presents what we believe to be an "elementary" proof of the necessary conditions that does not rely on Farkas' lemma and is independent of the simplex method, relying only on linear algebra and a perturbation technique published in 1952 by Charnes. No claim is made that the results are new, but we hope that the proofs may be useful for those who teach linear programming.