Algorithmic construction of the subdifferential from directional derivatives

TL;DR

This paper introduces algorithms to reconstruct the subdifferential of nonsmooth functions from directional derivatives, providing bounds on the number of derivatives needed in low-dimensional spaces and for functions with limited vertices.

Contribution

It presents a novel algorithmic approach for reconstructing polyhedral subdifferentials from directional derivatives, with bounds on the required data in various dimensions.

Findings

Algorithms for subdifferential reconstruction from directional derivatives.

Upper bounds on the number of derivatives needed in low dimensions.

Reconstruction for functions with at most three vertices.

Abstract

The subdifferential of a function is a generalization for nonsmooth functions of the concept of gradient. It is frequently used in variational analysis, particularly in the context of nonsmooth optimization. The present work proposes algorithms to reconstruct a polyhedral subdifferential of a function from the computation of finitely many directional derivatives. We provide upper bounds on the required number of directional derivatives when the space is $\R^1$ and $\R^2$, as well as in $\R^n$ where subdifferential is known to possess at most three vertices.