Generalized convexity in non-regular programming problems with inequality-type constraints

Abstract

Convexity plays a very important role in optimization for establishing optimality conditions. Different works have shown that the convexity property can be replaced by a weaker notion, the invexity. In particular, for problems with inequality-type constraints, Martin defined a weaker notion of invexity, the Karush-Kuhn-Tucker-invexity (hereafter KKTinvexity), that is both necessary and sufficient to obtain Karush-Kuhn-Tucker-type optimality conditions. It is well known that for this result to hold the problem has to verify a constraint qualification, i.e., it must be regular or non-degenerate. In non-regular problems, the classical optimality conditions are totally inapplicable. Meaningful results were obtained for problems with inequality-type constraints by Izmailov. They are based on the 2-regularity condition of the constraints at a feasible point. In this work, we generalize Martin's result to non-regular problems by defining an analogous concept, the 2-KKTinvexity, and using the characterization of the tangent cone in the 2-regular case and the necessary optimality condition given by Izmailov.

Condições de Otimalidade para Problemas Multiobjetivos Irregulares

“…Neste trabalho, os problemas dessa natureza serão denominados irregulares, seguindo a terminologia utilizada por Brezhneva e Tretyakov em [5]. Observamos ainda que o tratamento das condições de otimalidade para problemas irregulares tem sido um tópico de pesquisa bastante atual [1,2,3,4,7]. A relevância de tais problemas e algumas importantes referências que contêm exemplos significativos e aplicações podem ser encontrados em [1] e suas referências.…”