Characterizing Approximate Global Minimizers of the Difference of two Abstract Convex Functions with Applications

Abstract

In this paper, we first investigate characterizations of maximal elements
of abstract convex functions under a mild condition. Also, we give various characterizations for
global $\e$-minimum of the difference of two abstract convex functions and, by using the abstract Rockafellar's
antiderivative, we present the abstract $\e$-subdifferential of abstract convex functions
in terms of their abstract subdifferential. Finally, as an application, a necessary and sufficient condition for global $\e$-minimum of the difference of two increasing and positively homogeneous (IPH) functions is presented.