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The Radius of Metric Subregularity

  • Asen L. Dontchev
  • Helmut Gfrerer
  • Alexander Y. KrugerEmail author
  • Jiří V. Outrata
Article
  • 13 Downloads

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.

Keywords

Well-posedness Metric subregularity Generalized differentiation Radius theorems Constraint system 

Mathematics Subject Classification (2010)

49J52 49J53 49K40 90C31 

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Notes

Acknowledgements

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

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© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of Aerospace EngineeringThe University of MichiganAnn ArborUSA
  2. 2.Institute of Computational MathematicsJohannes Kepler University LinzLinzAustria
  3. 3.Centre for Informatics and Applied OptimizationFederation University AustraliaBallaratAustralia
  4. 4.Institute of Information Theory and AutomationCzech Academy of SciencePragueCzech Republic

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