Karush-Kuhn-Tucker Optimality Conditions and Constraint Qualifications trhough a Cone Approach
Resumo
This paper deals with optimality conditions to solve nonlinear programming
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 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 discussed, as: Slaters, linear
independence of gradients, Mangasarian-Fromovitzs and quasiregularity
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 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 discussed, as: Slaters, linear
independence of gradients, Mangasarian-Fromovitzs and quasiregularity
Palavras-chave
Optimaliy conditions, Karush-Kuhn-Tucker, constraint qualifications