The Radius of Metric Subregularity

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


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.


Well-posedness Metric subregularity Generalized differentiation Radius theorems Constraint system 

Mathematics Subject Classification (2010)

49J52 49J53 49K40 90C31 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.



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


  1. 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
  2. 2.
    Cibulka, R., Dontchev, A.L., Kruger, A.Y.: Strong metric subregularity of mappings in variational analysis and optimization. J. Math. Anal. Appl. 457(2), 1247–1282 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Dontchev, A.L., Lewis, A.S., Rockafellar, R.T.: The radius of metric regularity. Trans. Amer. Math. Soc. 355(2), 493–517 (2003)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Dontchev, A.L., Rockafellar, R.T.: Regularity and conditioning of solution mappings in variational analysis. Set-Valued Anal. 12(1-2), 79–109 (2004)MathSciNetCrossRefGoogle Scholar
  5. 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
  6. 6.
    Durea, M., Strugariu, R.: Metric subregularity of composition set-valued mappings with applications to fixed point theory. Set-Valued Var. Anal. 24(2), 231–251 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Eckart, C., Young, G.: The approximation of one matrix by another of lower rank. Psychometrica 1, 211–218 (1936)CrossRefGoogle Scholar
  8. 8.
    Gfrerer, H.: First order and second order characterizations of metric subregularity and calmness of constraint set mappings. SIAM J. Optim. 21(4), 1439–1474 (2011)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Gfrerer, H.: On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs. Set-Valued Var. Anal. 21(2), 151–176 (2013)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gfrerer, H., Mordukhovich, B.S.: Complete characterizations of tilt stability in nonlinear programming under weakest qualification conditions. SIAM J. Optim. 25(4), 2081–2119 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Gfrerer, H., Outrata, J.V.: On Lipschitzian properties of implicit multifunctions. SIAM J. Optim. 26(4), 2160–2189 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ginchev, I., Mordukhovich, B.S.: On directionally dependent subdifferentials. C. R. Acad. Bulgare Sci. 64(4), 497–508 (2011)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Graves, L.M.: Some mapping theorems. Duke Math. J. 17, 111–114 (1950)MathSciNetCrossRefGoogle Scholar
  14. 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. 15.
    Ioffe, A.D.: Variational Analysis of Regular Mappings. Theory and Applications. Springer Monographs in Mathematics Springer (2017)Google Scholar
  16. 16.
    Ioffe, A.D., Sekiguchi, Y.: Regularity estimates for convex multifunctions. Math. Program., Ser. B 117(1-2), 255–270 (2009). MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications Nonconvex Optimization and its Applications, vol. 60. Kluwer Academic Publishers, Dordrecht (2002)zbMATHGoogle Scholar
  18. 18.
    Kruger, A.Y.: Error bounds and Hölder metric subregularity. Set-Valued Var. Anal. 23(4), 705–736 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Kruger, A.Y.: Error bounds and metric subregularity. Optimization 64(1), 49–79 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Kruger, A.Y., Luke, D.R., Thao, N.H.: About subtransversality of collections of sets. Set-Valued Var. Anal. 25(4), 701–729 (2017). MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Kruger, A.Y., Thao, N.H.: Quantitative characterizations of regularity properties of collections of sets. J. Optim. Theory Appl. 164(1), 41–67 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Maréchal, M.: Metric subregularity in generalized equations. J. Optim. Theory Appl. 176(3), 527–540 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Maréchal, M., Correa, R.: Error bounds, metric subregularity and stability in generalized Nash equilibrium problems with nonsmooth payoff functions. Optimization 65(10), 1829–1854 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15(6), 959–972 (1977)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation. I: Basic Theory Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330. Springer, Berlin (2006)CrossRefGoogle Scholar
  26. 26.
    Ngai, H.V., Phan, N.T.: Metric subregularity of multifunctions: First and second order infinitesimal characterizations. Math. Oper. Res. 40(3), 703–724 (2015). MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Ngai, H.V., Tron, N.H., Tinh, P.N.: Directional Hölder metric subregularity and application to tangent cones. J. Convex Anal. 24(2), 417–457 (2017)MathSciNetzbMATHGoogle Scholar
  28. 28.
    Peña, J.: On the block-structured distance to non-surjectivity of sublinear mappings. Math. Program., Ser. A 103(3), 561–573 (2005). MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Robinson, S.M.: Some continuity properties of polyhedral multifunctions. Math. Programming Stud 14, 206–214 (1981). Mathematical Programming at Oberwolfach (Proc. Conf., Math. Forschungsinstitut, Oberwolfach, 1979)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, Berlin (1998)CrossRefGoogle Scholar
  31. 31.
    Uderzo, A.: A strong metric subregularity analysis of nonsmooth mappings via steepest displacement rate. J. Optim. Theory Appl. 171(2), 573–599 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Ye, J.J., Zhou, J.: Verifiable sufficient conditions for the error bound property of second-order cone complementarity problems. Math. Program., Ser. A 171(1-2), 361–395 (2018). MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Zheng, X.Y.: Metric subregularity for a multifunction. J. Math. Study 49(4), 379–392 (2016). MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zheng, X.Y., Zhu, J.: Generalized metric subregularity and regularity with respect to an admissible function. SIAM J. Optim. 26(1), 535–563 (2016). MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© 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

Personalised recommendations