The Radius of Metric Subregularity
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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.
KeywordsWell-posedness Metric subregularity Generalized differentiation Radius theorems Constraint system
Mathematics Subject Classification (2010)49J52 49J53 49K40 90C31
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The authors wish to thank the referees for their comments and suggestions.
- 1.Bürgisser, P., Cucker, F.: Condition. The Geometry of Numerical Algorithms Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 349. Springer, Heidelberg (2013)Google Scholar
- 5.Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. A View from Variational Analysis, 2 edn. Springer Series in Operations Research and Financial Engineering. Springer, New York (2014)Google Scholar
- 14.Ioffe, A.D.: On stability estimates for the regularity property of maps. In: Topological Methods, Variational Methods and their Applications (Taiyuan, 2002), pp 133–142. World Sci. Publ, River Edge (2003)Google Scholar
- 15.Ioffe, A.D.: Variational Analysis of Regular Mappings. Theory and Applications. Springer Monographs in Mathematics Springer (2017)Google Scholar