Some properties of image-convex functions Original Research Article

Recently, it was shown by Youness [E.A. Youness, On EE-convex sets, EE-convex functions and EE-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 EE-convex sets andEE-convex functions. We introduce the concept of EE-quasiconvex functions and strictly EE-quasiconvex functions, and develop some basic properties of EE-convex and EE-quasiconvex functions. For a real-valued function ff defined on a nonempty EE-convex set MM, we show under the convexity condition of E(M)E(M), that ff is EE-quasiconvex (resp. strictly EE-quasiconvex) if and only if its restriction to E(M)E(M) is quasiconvex (resp. strictly quasiconvex). Similarly, we show under the convexity condition of E(M)E(M), that ff is EE-convex (resp. strictly EE-convex) if and only if its restriction to E(M)E(M) is convex (resp. strictly convex). In addition, under the convexity condition of E(M)E(M), a characterization of an EE-quasiconvex function in terms of the lower level sets of its restriction to E(M)E(M) is also given. Finally, examples in nonlinear programming problem are used to illustrate the applications of our results.