Optimality conditions of E-convex programming for an E-differentiable function

Abstract

In this paper we introduce a new definition of an E-differentiable convex function, which transforms a non-differentiable function to a differentiable function under an operator E : R n → R n . By this definition, we can apply Kuhn-Tucker and Fritz-John conditions for obtaining the optimal solution of mathematical programming with a non-differentiable function.

On E-convex and semi-E-convex functions for a linear map E

“…The Theorem (3.1) of Youness [3] concerning the characterization of an Econvex function f in terms of its E-epigraph is modified to the following Theorem.…”

“…Youness in [2] introduced a class of sets and functions which is called Econvex sets and E-convex functions by relaxing the definition of convex sets and convex functions, and applied these functions to non linear programming problems (see for instance [3,4]). In this paper, we give one weak condition for a lower semi-continuous function on R n to be an E-convex function.…”

E-invexity and generalized E-invexity in E-differentiable multiobjective programming

“…Definition 6 [19] Let E : R n → R n and f : M → R be a (not necessarily) differentiable function at a given point u ∈ M. It is said that f is an E-differentiable function at u if and only if f • E is a differentiable function at u (in the usual sense) and, moreover,…”

“…Moreover, the results established by Youness [24] were improved by Yang [25]. Further, Megahed et al [19] presented the concept of an E-differentiable convex function which transforms a (not necessarily) differentiable convex function to a differentiable function based on the effect of an operator E : R n → R n .…”

On strongly E-convex sets and strongly E-convex cone sets

“…Youness [8] studied some properties of -convex programming and established the necessary and sufficient conditions of optimality for nonlinear -convex programming. Recently, Megahed et al [9,10] introduced duality in -convex programming and studied optimality conditions for -convex programming which has -differentiable objective function (see also [11], for more recent results onconvex functions and -convex programming). The initial results of Youness inspired a great deal of subsequent work which has expanded the role ofconvexity for which an extension class of the class of -convex sets and -convex functions, called strongly -convex sets and strongly -convex functions, is established by Youness [12].…”