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

Official URL: http://dx.doi.org/10.1016/j.ejor.2006.04.012

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 of Farkas lemma and the Bolzano-Weierstrass property for compact sets.