Directional metric regularity of multifunctions

TL;DR

This paper investigates directional metric regularity of set-valued mappings, providing new characterizations without completeness assumptions and applying these to criteria for robustness and coderivative conditions.

Contribution

It introduces novel characterizations of relative metric regularity using lower semicontinuous envelopes, extending understanding of directional regularity without completeness constraints.

Findings

Characterizations of relative metric regularity without completeness assumptions

Coderivative criteria for directional metric regularity

Results on robustness of metric regularity

Abstract

In this paper, we study relative metric regularity of set-valued mappings with emphasis on directional metric regularity. We establish characterizations of relative metric regularity without assuming the completeness of the image spaces, by using the relative lower semicontinuous envelopes of the distance functions to set-valued mappings. We then apply these characterizations to establish a coderivative type criterion for directional metric regularity as well as for the robustness of metric regularity.