Nonlinear Transversality Properties of Collections of Sets: Dual Space Sufficient Characterizations

Nguyen Duy Cuong(ndcuongctu.edu.vn)
Alexander Y. Kruger(a.krugerfederation.edu.au)

Abstract: This paper continues the study of 'good arrangements' of collections of sets near a point in their intersection. Our aim is to develop a general scheme for quantitative analysis of several transversality properties within the same framework. We consider a general nonlinear setting and establish dual space (subdifferential and normal cone) sufficient characterizations of transversality properties of collections of sets in Banach/Asplund spaces. Besides quantitative estimates for the rates/moduli of the corresponding properties, we establish here also estimates for the other parameters involved in the definitions, particularly the size of the neighbourhood where a property holds. Interpretations of the main general nonlinear characterizations for the case of Hölder transversality are provided. Some characterizations are new even in the linear setting. As an application, we provide dual sufficient conditions for nonlinear extensions of the new transversality properties of a set-valued mapping to a set in the range space due to Ioffe.

Keywords: Transversality · Subtransversality · Semitransversality · Regularity · Subregularity · Semiregularity · Sum rule · Chain rule

Category 1: Convex and Nonsmooth Optimization

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Entry Submitted: 07/28/2019
Entry Accepted: 07/28/2019
Entry Last Modified: 07/28/2019

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