About Extensions of the Extremal Principle

TL;DR

This paper explores extensions of the extremal principle in set collections, introducing new concepts of relative extremality and stationarity to broaden the theoretical framework and improve applicability in variational analysis.

Contribution

It introduces universal concepts of relative extremality and stationarity, formulates a relative extended extremal principle, and links these properties to regularity and stability of set-valued mappings.

Findings

Stability of the relative approximate stationarity is established.

Relationships between extremality properties and regularity conditions are clarified.

New universal concepts expand the applicability of the extremal principle.

Abstract

In this article, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets and the corresponding (extended) extremal principle, we focus on extensions of these properties and the corresponding dual conditions with the goal to refine the main arguments used in this type of results, clarify the relationships between different extensions and expand the applicability of the generalised separability results. We introduce and study new more universal concepts of relative extremality and stationarity and formulate the relative extended extremal principle. Among other things, certain stability of the relative approximate stationarity is proved. Some links are established between the relative extremality and stationarity properties of collections of sets and (the absence of) certain regularity, lower semicontinuity and Lipschitz-like properties of set-valued mappings.