The Behavior of Error Bounds via Moreau Envelopes

TL;DR

This paper explores the relationships between different error bounds for prox-regular functions, demonstrating their equivalence and analyzing their behavior via Moreau envelopes, supported by illustrative examples.

Contribution

It establishes the equivalence of three error bounds for prox-regular functions and studies their behavior through Moreau envelopes under certain conditions.

Findings

Proves the equivalence of u-KL, u-LSEB, and u-HEB for prox-regular functions.

Analyzes the behavior of LSEB and LHEB via Moreau envelopes.

Provides examples illustrating the theoretical results.

Abstract

In this paper, we first establish the equivalence of three types of error bounds: uniformized Kurdyka-{\L}ojasiewicz (u-KL) property, uniformized level-set subdifferential error bound (u-LSEB) and uniformized H\"{o}lder error bound (u-HEB) for prox-regular functions. Then we study the behavior of the level-set subdifferential error bound (LSEB) and the local H\"{o}lder error bound (LHEB) which is expressed respectively by Moreau envelopes, under suitable assumptions. Finally, in order to illustrate our main results and to compare them with those of recent references, some examples are also given.