Karush-Kuhn-Tucker optimality conditions and constraint qualifications through a cone approach Rodrigo G. Eust ´ aquio 1 , Elizabeth W. Karas 2 , Ademir A. Ribeiro 2 1 Department of Mathematics, Federal Technological University of Paran´ a Curitiba – PR – Brazil 2 Department of Mathematics, Federal University of Paran´ a CP 19.081 – 81.531-980 – Curitiba – PR – Brazil eustaquio@utfpr.edu.br,{ewkaras,ademir}@ufpr.br Abstract. This paper deals with optimality conditions to solve nonlinear pro- gramming problems. The classical Karush-Kuhn-Tucker (KKT) 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 sup- posing 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 as- sumption, it is extremely difficult to be verified. Therefore, other constraints qualifications, which are easier to be verified, are discussed, as: Slater’s, linear independence of gradients, Mangasarian-Fromovitz’s and quasiregularity. Key words. Optimality conditions, Karush-Kuhn-Tucker, constraint qualifica- tions. 1. Introduction We shall study the nonlinear programming problem (P ) minimize f (x) subject to h(x)=0 g(x) ≤ 0, where the functions f : IR n → IR, g : IR n → IR p and h : IR n → IR m are continuously differentiable. The feasible set is Ω= {x ∈ IR n | h(x)=0, g(x) ≤ 0}. Given x * ∈ Ω, the classical Karush-Kuhn-Tucker (KKT) conditions say that there exist Lagrangian multipliers λ * ∈ IR m and μ * ∈ IR p such that: -∇f (x * )= m  i=1 λ * i ∇h i (x * )+ p  j =1 μ * j ∇g j (x * ), μ * j ≥ 0, j =1,...,p, μ * j g j (x * )=0, j =1,...,p. In nonlinear programming we would like that KKT are necessary conditions for a given point to be a solution to the problem. When the problem is unconstrained (Ω=IR n ), the KKT conditions reduce to ∇f (x * )=0 which is a necessary optimality condition. However, this not always happens, as shown in the following example.