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The Radius of Metric Subregularity

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Abstract

There is a basic paradigm, called here the radius of well-posedness, which quantifies the “distance” from a given well-posed problem to the set of ill-posed problems of the same kind. In variational analysis, well-posedness is often understood as a regularity property, which is usually employed to measure the effect of perturbations and approximations of a problem on its solutions. In this paper we focus on evaluating the radius of the property of metric subregularity which, in contrast to its siblings, metric regularity, strong regularity and strong subregularity, exhibits a more complicated behavior under various perturbations. We consider three kinds of perturbations: by Lipschitz continuous functions, by semismooth functions, and by smooth functions, obtaining different expressions/bounds for the radius of subregularity, which involve generalized derivatives of set-valued mappings. We also obtain different expressions when using either Frobenius or Euclidean norm to measure the radius. As an application, we evaluate the radius of subregularity of a general constraint system. Examples illustrate the theoretical findings.

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Acknowledgements

The authors wish to thank the referees for their comments and suggestions.

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Authors and Affiliations

  1. Department of Aerospace Engineering, The University of Michigan, Ann Arbor, MI, USA

    Asen L. Dontchev

  2. Institute of Computational Mathematics, Johannes Kepler University Linz, A-4040, Linz, Austria

    Helmut Gfrerer

  3. Centre for Informatics and Applied Optimization, Federation University Australia, PO Box 663, Ballarat, VIC, 3353, Australia

    Alexander Y. Kruger

  4. Institute of Information Theory and Automation, Czech Academy of Science, Pod Vodárenskou věží 4, 182 08, Prague, Czech Republic

    Jiří V. Outrata

Authors
  1. Asen L. Dontchev
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  2. Helmut Gfrerer
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  3. Alexander Y. Kruger
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  4. Jiří V. Outrata
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Correspondence to Alexander Y. Kruger.

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Dedicated to Professor Alexander Ioffe on the occasion of his 80th birthday

Supported by the National Science Foundation (NSF) grant 156229; the Austrian Science Fund (FWF) grants P26640-N25, P26132-N25 and P29190-N32; the Australian Research Council (ARC) grant DP160100854 and the Grant Agency of the Czech Republic (GACR) grants 17-04301S and 17-08182S.

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Dontchev, A.L., Gfrerer, H., Kruger, A.Y. et al. The Radius of Metric Subregularity. Set-Valued Var. Anal 28, 451–473 (2020). https://doi.org/10.1007/s11228-019-00523-2

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  • DOI: https://doi.org/10.1007/s11228-019-00523-2

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