A Direct Proof of the Kuhn–Tucker Necessary Optimality Theorem for Convex and Affine Inequalities

A new proof of the Kuhn–Tucker necessary optimality condition is given for convex and affine inequalities. This proof differs from other existing proofs by relying only on a classical separation theorem for convex sets. Since this statement only assumes the weaker form of the Slater constraint qualification, it contains—without unnecessary restriction—the duality theorem for linear programming as well as the optimality conditions for quadratic and convex-linear programming.