Some properties of E-convex functions

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 andE-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.