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 [q]-regularity properties of collections of sets

We examine three primal space local Hoelder type regularity properties of finite collections of sets, namely, [q]-semiregularity, [q]-subregularity, and uniform [q]-regularity as well as their quantitative characterizations. Equivalent metric characterizations of the three mentioned regularity properties as well as a sufficient condition of [q]-subregularity in terms of Frechet normals are established. The relationships between [q]-regularity properties of collections of sets and the corresponding regularity properties of set-valued mappings are discussed.Comment: arXiv admin note: substantial text overlap with arXiv:1309.700

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

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

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

Transversality Properties: Primal Sufficient Conditions

The paper studies 'good arrangements' (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Holder case is given a special attention. Our main objective is not formally extending our earlier results from the Holder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties. The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic. Unlike the well-studied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case. Quantitative relations between nonlinear transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe are also discussed.Comment: 33 page

Transversality properties : primal sufficient conditions

The paper studies ‘good arrangements’ (transversality properties) of collections of sets in a normed vector space near a given point in their intersection. We target primal (metric and slope) characterizations of transversality properties in the nonlinear setting. The Hölder case is given a special attention. Our main objective is not formally extending our earlier results from the Hölder to a more general nonlinear setting, but rather to develop a general framework for quantitative analysis of transversality properties. The nonlinearity is just a simple setting, which allows us to unify the existing results on the topic. Unlike the well-studied subtransversality property, not many characterizations of the other two important properties: semitransversality and transversality have been known even in the linear case. Quantitative relations between nonlinear transversality properties and the corresponding regularity properties of set-valued mappings as well as nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe are also discussed. © 2020, Springer Nature B.V

Immobile indices and CQ-free optimality criteria for linear copositive programming problems

We consider problems of linear copositive programming where feasible sets consist of vectors for which the quadratic forms induced by the corresponding linear matrix combinations are nonnegative over the nonnegative orthant. Given a linear copositive problem, we define immobile indices of its constraints and a normalized immobile index set. We prove that the normalized immobile index set is either empty or can be represented as a union of a finite number of convex closed bounded polyhedra. We show that the study of the structure of this set and the connected properties of the feasible set permits to obtain new optimality criteria for copositive problems. These criteria do not require the fulfillment of any additional conditions (constraint qualifications or other). An illustrative example shows that the optimality conditions formulated in the paper permit to detect the optimality of feasible solutions for which the known sufficient optimality conditions are not able to do this. We apply the approach based on the notion of immobile indices to obtain new formulations of regularized primal and dual problems which are explicit and guarantee strong duality.publishe