γ-Subdifferential and γ-convexity of functions on the real line

Abstract

In this paper the γ-subdifferential and γ-convexity of real-valued functions on the real line are introduced. By means of the γ-subdifferential, a new necessary condition for global minima (or maxima) is formulated which many local minima (or maxima) cannot satisfy. The γ-convexity is used to state sufficient conditions for global minima. The class of γ-convex functions is relatively large. For example, there are γ-convex functions which are not continuous anywhere. Nevertheless, a γ-local minimum of a γ-convex function is always a global minimum. Furthermore, if a γ-convex function attains its global minimum, then it does so near the boundary of its domain.