γ-Subdifferential and γ-convexity of functions on a normed space

Abstract

In this paper, the γ-subdifferential ∂_γ is introduced for investigating the global behavior of real-valued functions on a normed spaceX. Iff: D⊂X → ℝ attains its global minimum onD atx ^*, then 0∈∂_γ f(x ^*). This necessary condition always holds, even iff is not continuous orx ^* is at the boundary of its domain. Nevertheless, it is useful because, by choosing a suitable γ∈ℝ₊, many local minima cannot satisfy this necessary condition. For the sufficient conditions, the so-called γ-convex functions are defined. The class of these functions is rather large. For example, every periodic function on the real line is a γ-convex function. There are γ-convex functions which are not continuous everywhere. Every function of bounded variation can be represented as the difference of two γ-convex functions. For all that, γ-convex functions still have properties similar to those of convex functions. For instance, each γ-local minimizer off is at the same time a global one. Iff attains its global minimum onD, then it does so at least at one point of its γ-boundary.