On Error Bound Moduli for Locally Lipschitz and Regular Functions

TL;DR

This paper investigates local error bound moduli for locally Lipschitz and regular functions, establishing bounds via subdifferential sets and showing exactness under certain conditions, advancing understanding in variational analysis.

Contribution

It introduces a new upper estimate for local error bound moduli using outer limiting subdifferential sets and proves its tightness for convex functions under regularity.

Findings

Upper estimate of error bound modulus via subdifferential set

Exact characterization of error bound modulus for lower C^1 functions

Tightness of bounds under convexity and regularity conditions

Abstract

In this paper we study local error bound moduli for a locally Lipschitz and regular function via its outer limiting subdifferential set. We show that the distance of 0 from the outer limiting subdifferential of the support function of the subdifferential set, which is essentially the distance of 0 from the end set of the subdifferential set, is an upper estimate of the local error bound modulus. This upper estimate becomes tight for a convex function under some regularity conditions. We show that the distance of 0 from the outer limiting subdifferential set of a lower $\mathcal{C}^1$ function is equal to the local error bound modulus.