Generalized Differentiable -Invex Functions and Their Applications in Optimization

The concept of -convex function and its generalizations is studied with differentiability assumption. Generalized differentiable -convexity and generalized differentiable -invexity are used to derive the existence of optimal solution of a general optimization problem. 1. Introduction convex function was introduced by Youness [1] and revised by Yang [2]. Chen [3] introduced Semi- -convex function and studied some of its properties. Syau and Lee [4] defined -quasi-convex function, strictly -quasi-convex function and studied some basic properties. Fulga and Preda [5] introduced the class of -preinvex and -prequasi-invex functions. All the above -convex and generalized -convex functions are defined without differentiability assumptions. Since last few decades, generalized convex functions like quasiconvex, pseudoconvex, invex, -vex, -invex, and so forth, have been used in nonlinear programming to derive the sufficient optimality condition for the existence of local optimal point. Motivated by earlier works on convexity and convexity, we have introduced the concept of differentiable -convex function and its generalizations to derive sufficient optimality condition for the existence of local optimal solution of a nonlinear programming problem. Some preliminary definitions and results regarding -convex function are discussed below, which will be needed in the sequel. Throughout this paper, we consider functions , , and are nonempty subset of . Definition 1.1 (see [1]). is said to be -convex set if for , . Definition 1.2 (see [1]). is said to be -convex on if is an -convex set and for all and , Definition 1.3 (see [3]). Let be an -convex set. is said to be semi- -convex on if for and , Definition 1.4 (see [5]). is said to be -invex with respect to if for and ,？？ . Definition 1.5 (see [6]). Let be an -invex set with respect to . ？Also ？ is said to be -preinvex with respect to on if for and , Definition 1.6 (see [7]). Let be an -invex set with respect to . Also ？ is said to be semi- -invex with respect to at if for all and . Definition 1.7 (see [7]). Let be a nonempty -invex subset of with respect to , . Let and be an open set in ？Also ？ and are differentiable on . Then, is said to be semi- -quasiinvex at if or Lemma 1.8 (see [1]). If a set is -convex, then . Lemma 1.9 (see [5]). If is -invex, then . Lemma 1.10 (see [5]). If is a collection of -invex sets and , for all , then is -invex. 2. -Convexity and Its Generalizations with Differentiability Assumption -convexity and convexity are different from each other in several contests. From the previous results on