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Explicit Formula for Preimages of Relaxed One-Sided Lipschitz Mappings with Negative Lipschitz Constants: A Geometric Approach

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Abstract

This paper addresses Lipschitzian stability issues, that play an important role in both theoretical and numerical aspects of variational analysis, optimization, and their applications. We particularly concentrate on the so-called relaxed one-sided Lipschitz property of set-valued mappings with negative Lipschitz constants. This property has been much less investigated than more conventional Lipschitzian behavior, while being well recognized in a variety of applications. Recent work has revealed that set-valued mappings satisfying the relaxed one-sided Lipschitz condition with negative Lipschitz constant possess a localization property, that is stronger than uniform metric regularity, but does not imply strong metric regularity. The present paper complements this fact by providing a characterization, not only of one specific single point of a preimage, but of entire preimages of such mappings. Developing a geometric approach, we derive an explicit formula to calculate preimages of relaxed one-sided Lipschitz mappings between finite-dimensional spaces and obtain a further specification of this formula via extreme points of image sets.

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Acknowledgements

Research of the second author was partly supported by the USA National Science Foundation under Grants DMS-1512846 and DMS-1808978, by the USA Air Force Office of Scientific Research under Grant #15RT0462, and by the Australian Research Council under Discovery Project DP-190100555.

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Authors and Affiliations

  1. RMIT, Melbourne, Australia

    Andrew S. Eberhard

  2. Wayne State University, Detroit, USA

    Boris S. Mordukhovich

  3. Monash University, Melbourne, Australia

    Janosch Rieger

Authors
  1. Andrew S. Eberhard
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  2. Boris S. Mordukhovich
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  3. Janosch Rieger
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Correspondence to Janosch Rieger.

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Communicated by Asen L. Dontchev.

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Eberhard, A.S., Mordukhovich, B.S. & Rieger, J. Explicit Formula for Preimages of Relaxed One-Sided Lipschitz Mappings with Negative Lipschitz Constants: A Geometric Approach. J Optim Theory Appl 185, 34–43 (2020). https://doi.org/10.1007/s10957-020-01644-7

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  • DOI: https://doi.org/10.1007/s10957-020-01644-7

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