Algebraic Lipschitz and Subdifferential Calculus in General Vector Spaces

TL;DR

This paper extends Lipschitz and subdifferential calculus to convex functions in general vector spaces without topological constraints, broadening the scope of Clarke's calculus.

Contribution

It introduces an algebraic approach to Lipschitz and subdifferentiability for convex functions in vector spaces, generalizing Clarke's subdifferential calculus.

Findings

Convex functions with non-empty relative algebraic interior are Lipschitz and subdifferentiable algebraically.

Clarke's subdifferential calculus is generalized to vector and topological vector spaces.

The approach does not require additional topological constraints.

Abstract

The main contribution of this paper is that every convex function with non-empty relative algebraic interior of its domain is Lipschitz and subdifferentiable in some algebraic sense without any additional topological constraints. The proposed approach uses slightly modified Clarke's subdifferential for functions defined on a convex symmetric set and Lipschitz with respect to a Minkowski functional. Following this, Clarke's subdifferential calculus is generalized to vector spaces and, where continuity properties are claimed, to topological vector spaces.