Clarke subgradients for directionally Lipschitzian stratifiable functions

TL;DR

This paper provides a unified geometric characterization of Clarke subgradients for stratifiable, directionally Lipschitzian functions, simplifying existing results and offering new insights into their subdifferential structure.

Contribution

It introduces a simple form for Clarke subdifferentials of stratifiable functions using normal cones and gradient limits, unifying various prior results.

Findings

Clarke subdifferential characterized by normal cones and gradient limits

Unified framework for stratifiable, directionally Lipschitzian functions

Simplified proof of uniform local dimension of subdifferential graphs

Abstract

Using a geometric argument, we show that under a reasonable continuity condition, the Clarke subdifferential of a semi-algebraic (or more generally stratifiable) directionally Lipschitzian function admits a simple form: the normal cone to the domain and limits of gradients generate the entire Clarke subdifferential. The characterization formula we obtain unifies various apparently disparate results that have appeared in the literature. Our techniques also yield a simplified proof that closed semialgebraic functions on $\R^n$ have a limiting subdifferential graph of uniform local dimension $n$.