An elementary proof of the Fritz-John and Karush-Kuhn-Tucker conditions in nonlinear programming

In this note we give an elementary proof of the Fritz-John and Karush-Kuhn-Tucker conditions for nonlinear finite dimensional programming problems with equality and/or inequality constraints.The proof avoids the implicit function theorem usually applied when dealing with equality constraints and uses a generalization ofFarkas lemma and the Bolzano-Weierstrass property for compact sets.Fritz-John conditions;Karush-Kuhn-Tucker conditions;nonlinear programming