On (p, r)-invexity-type nonlinear programming problems

Abstract

Mathematical programming problems involving (p, r)-invex functions with respect to η are considered. We introduce new classes of nonlinear programming problems, called KT-(p, r)-invex, WD-(p, r)-invex, and HC-(p, r)-invex problems (where p, r are some real numbers). It is shown that for these types of problems Kuhn-Tucker conditions are both necessary and sufficient for optimality. Furthermore, these (p, r)-invexity-type problems with r ≠ 0 are not sufficient for Wolfe weak duality. In this way it was shown that the optimization problems possessing "some kind" of invexity need not be equivalent to the class of optimization problems for which Kuhn-Tucker necessary conditions for optimality are also sufficient and Wolfe weak duality holds. © 2001 Elsevier Science.