G-pre-invex functions in mathematical programming

Abstract

In the present paper, we introduce the concept of G-pre-invex functions with respect to defined on an invex set with respect to . These function unify the concepts of nondifferentiable convexity, pre-invexity and r-pre-invexity. Furthermore, relationships of Gpre-invex functions to various introduced earlier pre-invexity concepts are also discussed. Some (geometric) properties of this class of functions are also derived. Finally, optimality results are established for optimization problems under appropriate G-pre-invexity conditions.

On the relationships between G-preinvex functions and semistrictly G-preinvex functions

“…Yang and Li [8] presented some properties of preinvex functions; they in [9] introduced the semistrictly preinvex function and established relationships between preinvex functions and semistrictly preinvex functions under a certain set of conditions. Recently, Antczak [2,3] introduced the concept of the G-preinvex function, which includes the preinvex function [7] and r -preinvex function [1] as special cases. Relations of this G-preinvex function to preinvex functions and some properties of this class of functions were studied in [2].…”

“…Recently, Antczak [2,3] introduced the concept of the G-preinvex function, which includes the preinvex function [7] and r -preinvex function [1] as special cases. Relations of this G-preinvex function to preinvex functions and some properties of this class of functions were studied in [2].…”

“…Therefore, to consider wider and wider classes of generalized convex functions is important, but it is also important to seek practical criteria for convexity or generalized convexity (see [1][2][3][4][5][6][7][8][9][10] and the references therein). Two significant generalizations of convex functions are the so-called preinvex functions introduced in [7] and prequasi-invex functions given in [6].…”

“…First, we shall prove that f is a G-preinvex function on K . By Theorem 10 in [2], we need to show that for each x, y ∈ K , there exists λ ∈ (0, 1) such that…”

Modi ed objective function approach for multitime variational problems

“…Among these generalizations, the notion of invexity was first introduced by Hanson [10]. The results developed by Hanson inspired a great deal of subsequent works, which have greatly expanded the role of invexity in optimization (see, for example, [2,6,7,12]). …”