SOME PROPERTIES OF NONCONVEX FUNCTIONS

Abstract

In this paper, we introduce and study a new class of convex functions with respect to an arbitrary function, which is called the k-convex function. These functions are nonconvex functions and include the convex function and φ-convex convex as special cases. We study some properties of k-convex functions. It is shown that the minimum of k-convex functions on the k-convex sets can be characterized by a class of variational inequalities, which is called the k-directional variational inequalities. Some open problems are also suggested for future research.