An implicit multifunction theorem for the hemiregularity of mappings with application to constrained optimization

TL;DR

This paper introduces a new stability property called uniform hemiregularity for set-valued mappings, with applications to constrained optimization, including an implicit multifunction theorem and insights on penalty function exactness.

Contribution

It develops a quantitative stability concept for set-valued mappings and establishes an implicit multifunction theorem linking solution stability to problem data.

Findings

Introduces the concept of uniform hemiregularity.

Establishes an implicit multifunction theorem for parameterized problems.

Discusses implications for penalty function exactness.

Abstract

The present paper contains some investigations about a uniform variant of the notion of metric hemiregularity, the latter being a less explored property obtained by weakening metric regularity. The introduction of such a quantitative stability property for set-valued mappings is motivated by applications to the penalization of constrained optimization problems, through the notion of problem calmness. As a main result, an implicit multifunction theorem for parameterized inclusion problems is established, which measures the uniform hemiregularity of the related solution mapping in terms of problem data. A consequence on the exactness of penalty functions is discussed.