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

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