Sequential Extremal Principle and Necessary Conditions for Minimizing Sequences

TL;DR

This paper extends the extremal principle to sequences of points in collections of sets, enabling analysis of unbounded sets with empty intersections and deriving necessary conditions for optimization problems.

Contribution

It introduces a sequential extremal principle and related concepts, broadening the scope of extremality analysis to sequences and unbounded sets.

Findings

Extended extremal principle for sequences of points

Sequential necessary conditions for constrained optimization

Application to unbounded sets with empty intersection

Abstract

The conventional definition of extremality of a finite collection of sets is extended by replacing a fixed point (extremal point) in the intersection of the sets by a collection of sequences of points in the individual sets with the distances between the corresponding points tending to zero. This allows one to consider collections of unbounded sets with empty intersection. Exploiting the ideas behind the conventional extremal principle, we derive an extended sequential version of the latter result in terms of Fr\'echet and Clarke normals. Sequential versions of the related concepts of stationarity, approximate stationarity and transversality of collections of sets are also studied. As an application, we establish sequential necessary conditions for minimizing (and more general firmly stationary, stationary and approximately stationary) sequences in a constrained optimization problem.