Variational properties of the abstract subdifferential operator

TL;DR

This paper explores the properties of the abstract subdifferential in abstract convexity, establishing calculus rules, maximality conditions, and highlighting limitations in separation theorems that impact numerical methods.

Contribution

It introduces calculus rules for the abstract subdifferential, proves maximality under certain conditions, and provides a counterexample showing separation theorems do not generally hold.

Findings

Subdifferential calculus rules established

Maximality of subdifferential proved under conditions

Counterexample shows separation theorem generally fails

Abstract

Abstract convexity generalises classical convexity by considering the suprema of functions taken from an arbitrarily defined set of functions. These are called the abstract linear (abstract affine) functions. The purpose of this paper is to study the abstract subdifferential. We obtain a number of results on the calculus of this subdifferential: summation and composition rules, and prove that under some reasonable conditions the subdifferential is a maximal abstract monotone operator. Another contribution of this paper is a counterexample that demonstrates that the separation theorem between two abstract convex sets is generally not true. The lack of the extension of separation results to the case of abstract convexity is one of the obstacles in the development of abstract convexity based numerical methods.