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Transversality Properties: Primal Sufficient Conditions

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Abstract

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.

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Acknowledgments

The authors wish to thank the referee and the handling editor for their careful reading of the manuscript and valuable comments and suggestions.

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Authors and Affiliations

  1. Centre for Informatics and Applied Optimization, School of Science, Engineering and Information Technology, Federation University Australia, POB 663, Ballarat, Vic, 3350, Australia

    Nguyen Duy Cuong & Alexander Y. Kruger

  2. Department of Mathematics, College of Natural Sciences, Can Tho University, Can Tho, Vietnam

    Nguyen Duy Cuong

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  1. Nguyen Duy Cuong
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  2. Alexander Y. Kruger
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Correspondence to Alexander Y. Kruger.

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Dedicated to the memory of Prof. Rafail Gabasov, a great person and teacher

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The research was supported by the Australian Research Council, project DP160100854. The second author benefited from the support of the FMJH Program PGMO and from the support of EDF.

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Cuong, N.D., Kruger, A.Y. Transversality Properties: Primal Sufficient Conditions. Set-Valued Var. Anal 29, 221–256 (2021). https://doi.org/10.1007/s11228-020-00545-1

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