Directed Subdifferentiable Functions and the Directed Subdifferential without Delta-Convex Structure

TL;DR

This paper introduces a new way to construct the directed subdifferential for a broader class of functions using only directional derivatives, removing the need for delta-convex structure, thus expanding its applicability.

Contribution

It generalizes the directed subdifferential to functions beyond delta-convex, including Lipschitz and quasidifferentiable functions, using a novel construction from directional derivatives.

Findings

Directed subdifferential constructed without delta-convex structure

Applicable to Lipschitz functions definable on o-minimal structures

Extends to quasidifferentiable functions

Abstract

We show that the directed subdifferential introduced for differences of convex (delta-convex, DC) functions by Baier and Farkhi can be constructed from the directional derivative without using any information on the DC structure of the function. The new definition extends to a more general class of functions, which includes Lipschitz functions definable on o-minimal structure and quasidifferentiable functions.