A Quest for Simple and Unified Proofs in Regularity Theory: Perturbation Stability

TL;DR

This paper advocates for using a direct, unified approach based on Ekeland's variational principle and the distance function to simplify proofs of metric regularity and its variants, avoiding unnecessary detours.

Contribution

It demonstrates that the general criterion derived from Ekeland's principle suffices for metric regularity proofs, streamlining existing methods and covering various regularity properties.

Findings

Simplifies proofs of metric regularity using the general criterion.

Shows no need for slope-based or envelope-based arguments in perturbation stability.

Provides unified criteria applicable to multiple regularity properties.

Abstract

Ioffe's criterion and various reformulations of it have become a~standard tool in proving theorems guaranteeing metric regularity of a (set-valued) mapping. First, we demonstrate that one should always use directly the so-called general criterion which follows, for example, from Ekeland's variational principle, and that there is no need to make a detour through the slope-based consequences of this general statement. Second, we argue that when proving perturbation stability results, in the spirit of Lyusternik-Graves theorem, there is no need to employ the concept of a lower semi-continuous envelope even in the case of an incomplete target space. The gist is to use the "correct" function to which Ekeland's variational principle is applied; namely, the distance function to the graph of the set-valued mapping under consideration. This approach originates in the notion of graphical regularity introduced by L. Thibault, which is equivalent to the property of metric regularity. Our criteria cover also both metric subregularity and metric semiregularity, which are weaker properties obtained by fixing one of the points in the definition of metric regularity.