Metric regularity of composition set-valued mappings: metric setting and coderivative conditions

TL;DR

This paper introduces a new unified method to prove metric regularity of composite set-valued maps in metric spaces, utilizing coderivative conditions and refining previous techniques for broader applicability.

Contribution

It provides a direct proof approach for metric regularity of compositions, synthesizing and specializing existing methods with new coderivative-based techniques.

Findings

New proofs of metric regularity results using coderivative conditions

Unification and refinement of existing techniques for set-valued map regularity

Clarification of the role of (co)derivatives in establishing openness in general settings

Abstract

The paper concerns a new method to obtain a direct proof of the openness at linear rate/metric regularity of composite set-valued maps on metric spaces by the unification and refinement of several methods developed somehow separately in several works of the authors. In fact, this work is a synthesis and a precise specialization to a general situation of some techniques explored in the last years in the literature. In turn, these techniques are based on several important concepts (like error bounds, lower semicontinuous envelope of a set-valued map, local composition stability of multifunctions) and allow us to obtain two new proofs of a recent result having deep roots in the topic of regularity of mappings. Moreover, we make clear the idea that it is possible to use (co)derivative conditions as tools of proof for openness results in very general situations.