# About Extended Extremal Principle and Regularity of Collections of Sets

Alexander Kruger^{*}

The \emph{Extremal Principle} is recognized as one of the central results of Variational Analysis. It provides a dual necessary condition of local extremality of a collection of closed sets in an Asplund space and can be considered as a generalization of the classical separation theorem to nonconvex sets. In this capacity it has been successfully used for deducing necessary optimality conditions in nonconvex extremal problems, calculus formulas for generalized derivatives, etc.\newline As a necessary condition, the conclusion of the Extremal Principle actually characterizes not just local extremality of collections of sets but a weaker property which can be interpreted as a kind of stationary behavior of such collections. Moreover, for this stationarity property, which can be given clear explicit definition in terms of primal space elements, the conclusion of the Extremal Principle becomes necessary and sufficient (in the Asplund space setting). This statement is referred to as the \emph{Extended Extremal Principle} and like the conventional Extremal Principle provides an extremal characterization of Asplund spaces.\newline When the stationarity condition is not satisfied, this can be interpreted as regularity of the collection of sets. The last property is in a sense equivalent to metric regularity of multifunctions and has important practical applications.

Mathematics Subject Classification: 49J53 49J52

Keywords: Variational analysis; regularity; extremal principle

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