Necessary Conditions for Non-Intersection of Collections of Sets

This paper continues studies of non-intersection properties of finite collections of sets initiated 40 years ago by the extremal principle. We study elementary non-intersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish new primal (slope) and dual (generalized separation) necessary conditions for these non-intersection properties. The results are applied to convergence analysis of alternating projections.Comment: 26 page

Necessary conditions for non-intersection of collections of sets

This paper continues studies of non-intersection properties of finite collections of sets initiated 40 years ago by the extremal principle. We study elementary non-intersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish new primal (slope) and dual (generalized separation) necessary conditions for these non-intersection properties. The results are applied to convergence analysis of alternating projections. © 2021 Informa UK Limited, trading as Taylor & Francis Group

Extremality and stationarity of collections of sets : metric, slope and normal cone characterisations

Variational analysis, a relatively new area of research in mathematics, has become one of the most powerful tools in nonsmooth optimisation and neighbouring areas. The extremal principle, a tool to substitute the conventional separation theorem in the general nonconvex environment, is a fundamental result in variational analysis. There have seen many attempts to generalise the conventional extremal principle in order to tackle certain optimisation models. Models involving collections of sets, initiated by the extremal principle, have proved their usefulness in analysis and optimisation, with non-intersection properties (or their absence) being at the core of many applications: recall the ubiquitous convex separation theorem, extremal principle, Dubovitskii Milyutin formalism and various transversality/regularity properties. We study elementary nonintersection properties of collections of sets, making the core of the conventional definitions of extremality and stationarity. In the setting of general Banach/Asplund spaces, we establish nonlinear primal (slope) and linear/nonlinear dual (generalised separation) characterisations of these non-intersection properties. We establish a series of consequences of our main results covering all known formulations of extremality/ stationarity and generalised separability properties. This research develops a universal theory, unifying all the current extensions of the extremal principle, providing new results and better understanding for the exquisite theory of variational analysis. This new study also results in direct solutions for many open questions and new future research directions in the fields of variational analysis and optimisation. Some new nonlinear characterisations of the conventional extremality/stationarity properties are obtained. For the first time, the intrinsic transversality property is characterised in primal space without involving normal cones. This characterisation brings a new perspective on intrinsic transversality. In the process, we thoroughly expose and classify all quantitative geometric and metric characterisations of transversality properties of collections of sets and regularity properties of set-valued mappings.Doctor of Philosoph

Stationarity and regularity of infinite collections of sets

This article investigates extremality, stationarity, and regularity properties of infinite collections of sets in Banach spaces. Our approach strongly relies on the machinery developed for finite collections. When dealing with an infinite collection of sets, we examine the behavior of its finite subcollections. This allows us to establish certain primal-dual relationships between the stationarity/regularity properties some of which can be interpreted as extensions of the Extremal principle. Stationarity criteria developed in the article are applied to proving intersection rules for Fréchet normals to infinite intersections of sets in Asplund spaces. © 2012 Springer Science+Business Media, LLC

About uniform regularity of collections of sets

We further investigate the uniform regularity property of collections of sets via primal and dual characterizing constants. These constants play an important role in determining convergence rates of projection algorithms for solving feasibility problems

Rated Extremal Principles for Finite and Infinite Systems

In this paper we introduce new notions of local extremality for finite and infinite systems of closed sets and establish the corresponding extremal principles for them called here rated extremal principles. These developments are in the core geometric theory of variational analysis. We present their applications to calculus and optimality conditions for problems with infinitely many constraints

About regularity of collections of sets

The paper continues investigations of stationarity and regularity properties of collections of sets in normed spaces. It contains a summary of different characterizations (both primal and dual) of regularity and a list of sufficient conditions for a collection of sets to be regular

Stationarity and regularity of set systems

Extremality, stationarity and regularity notions for a system of closed sets in a normed linear space are investigated. The equivalence of different abstract “extremal” settings in terms of set systems and multifunctions is proved. The dual necessary and sufficient conditions of weak stationarity (the Extended extremal principle) are presented for the case of an Asplund space

About uniform regularity of collection sets

We further investigate the uniform regularity property of collections of sets via primal and dual characterizing constants. These constants play an important role in determining convergence rates of projection algorithms for solving feasibility problems