LP Duality and KKT Conditions for LP

Duality is one of the most central concepts in connection with Linear Programming (LP), both from a theoretical and an algorithmic point of view. LP problems always appear in pairs with a primal problem and a corresponding dual problem—a fact which is intimately related to the concept of relaxation and to the necessary and sufficient optimality conditions known as Karush-Kuhn-Tucker (KKT)-conditions for nonlinear programs (NLPs). This article first introduces the concept of LP duality and discusses various ways of presenting duality. The fundamental theorems of Weak and Strong Duality are stated and proven. The Complementary Slackness Conditions for optimality of a set of primal and dual solutions to a set of dual LPs are then introduced, and finally it is demonstrated that these conditions are the Karush-Kuhn-Tucker (KKT) conditions known from NLP restated in a linear setting. Keywords: linear programming; duality; complementary slackness; KKT conditions