Some properties of [formula omitted]-convex functions

Abstract

Recently, it was shown by Youness [E.A. Youness, On E -convex sets, E -convex functions and E -convex programming, Journal of Optimization Theory and Applications, 102 (1999) 439–450] that many results for convex sets and convex functions actually hold for a wider class of sets and functions, called E -convex sets and E -convex functions. We introduce the concept of E -quasiconvex functions and strictly E -quasiconvex functions, and develop some basic properties of E -convex and E -quasiconvex functions. For a real-valued function f defined on a nonempty E -convex set M , we show under the convexity condition of E ( M ) , that f is E -quasiconvex (resp. strictly E -quasiconvex) if and only if its restriction to E ( M ) is quasiconvex (resp. strictly quasiconvex). Similarly, we show under the convexity condition of E ( M ) , that f is E -convex (resp. strictly E -convex) if and only if its restriction to E ( M ) is convex (resp. strictly convex). In addition, under the convexity condition of E ( M ) , a characterization of an E -quasiconvex function in terms of the lower level sets of its restriction to E ( M ) is also given. Finally, examples in nonlinear programming problem are used to illustrate the applications of our results.