Generalized convex functions and generalized differentials

Abstract

We study some classes of generalized convex functions, using a generalized di¤erential approach. By this we mean a set-valued mapping which stands either for a derivative, a subdi¤erential or a pseudodi¤erential in the sense of Jeyakumar and Luc. We establish some links between the corresponding classes of pseudoconvex, quasiconvex and another class of generalized convex functions we introduced. We devise some optimality conditions for constrained optimization problems. In particular, we get Lagrange-KuhnTucker multipliers for mathematical programming problems.

Generalized Affine Functions and Generalized Differentials

“…Since f is @f -pseudoconvex and f is @( f )-pseudoconvex, f and f are quasiconvex by [15,Proposition 6]. Thus, f is quasia¢ ne on C.…”

“…R. We assume that a set-valued map @f : C X is given which stands for a substitute to the derivative of f ; we call it a generalized di¤ erential of f . As observed in [15], the choice for @f is not limited to the subdi¤erentials of nonsmooth analysis; one can also take the convexi…cators of [7], the pseudo-di¤erentials of Jeyakumar and Luc ( [8]), and much more. We assume that @f (x) 6 = ?…”

“…X and f : C ! R. Our purpose here is limited: since optimality conditions for (C) and mathematical programming problems using the concepts of the previous sections are dealt with in [13], [14], [15], [20], [22], [23], we are just concerned with characterizations of solution sets. Let S be the set of solutions to (C) and let @f : C X be a generalized di¤erential of f:…”

“…It is the purpose of the present paper to introduce and study new concepts of generalized a¢ ne functions, as it has been done for generalized convex functions in [15]. Here, to de…ne these classes, we use a generalized di¤erential, i.e.…”