Constraint Qualifications for Nonlinear Programming

This paper deals with optimality conditions to solve nonlinear programming problems. The classical Karush-Kuhn-Tucker (KKT) optimality conditions are demonstrated through a cone approach, using the well known Farkas’ Lemma. These conditions are valid at a minimizer of a nonlinear programming problem if a constraint qualification is satisfied. First we prove the KKT theorem supposing the equality between the polar of the tangent cone and the polar of the first order feasible variations cone. Although this condition is the weakest assumption, it is extremely difficult to be verified. Therefore, other constraints qualifications, which are easier to be verified, are studied, as: Slater’s, linear independence of gradients, Mangasarian-Fromovitz’s and quasiregularity. The relations among them are discussed.