Gradients on Sets

TL;DR

This paper introduces a set-valued gradient concept for locally Lipschitz functions to facilitate nonsmooth optimization, focusing on existence, uniqueness, and approximation of descent directions on sets.

Contribution

It develops a new set-valued gradient framework based on Clarke's generalized gradient for nonsmooth functions, aiding in the design of numerical descent algorithms.

Findings

Establishes existence of descent directions on sets.

Provides conditions for uniqueness of these directions.

Lays groundwork for nonsmooth numerical optimization methods.

Abstract

For a locally Lipschitz continuous function $f:X\to\mathbb{R}$ the generalized gradient $\partial f(x)$ of Clarke is used to develop some (set-valued) gradient on a set $A\subset X$. Existence, uniqueness and some approximation are considered for optimal descent directions on set $A$. The results serve as basis for nonsmooth numerical descent algorithms that can be found in subsequent papers.