Dual sufficient characterizations of transversality properties
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Positivity
Dual sufficient characterizations of transversality properties Nguyen Duy Cuong1 · Alexander Y. Kruger1 Received: 28 July 2019 / Accepted: 24 December 2019 © Springer Nature Switzerland AG 2020
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 (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 Mathematics Subject Classification 49J52 · 49J53 · 49K40 · 90C30 · 90C46
The research was supported by the Australian Research Council, Project DP160100854, and the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie Grant Agreement No. 823731 CONMECH. The second author benefited from the support of the FMJH Program PGMO and from the support of EDF.
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Alexander Y. Kruger [email protected] Nguyen Duy Cuong [email protected]
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Centre for Informatics and Applied Optimization, School of Science, Engineering and Information Technology, Federation University Australia, POB 663, Ballarat, VIC 3350, Australia
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N. D. Cuong, A. Y. Kruger
1 Introduction In this paper, we continue our study of ‘good arrangements’ of collections of sets in normed vector spaces near a point in their intersection, known as transversality (regularity) properties and playing an important role in optimization and variational analysis, e.g., as constraint qualifications in optimality conditions, and qualification conditions in subdifferential, normal cone and coderivative calculus, and convergence analysis of computational algorithms [1–5,7,11,16,20,23–25,27–37,39– 41,43,44,50,51]. Following Ioffe [20], such arrangements are now commonly referred to as transversality properties. Here we refer to transversality broadly as a group of ‘good arrangement’ properties, which includes semitransversality, subtransversality, transversality (a specific property) and some others. The term regularity was extensively used for the same purpose in the earlier publications by the second author, and is still preferred b
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