Skip to main content

Stability Analysis for Parameterized Variational Systems with Implicit Constraints


In the paper we provide new conditions ensuring the isolated calmness property and the Aubin property of parameterized variational systems with constraints depending, apart from the parameter, also on the solution itself. Such systems include, e.g., quasi-variational inequalities and implicit complementarity problems. Concerning the Aubin property, possible restrictions imposed on the parameter are also admitted. Throughout the paper, tools from the directional limiting generalized differential calculus are employed enabling us to impose only rather weak (non- restrictive) qualification conditions. Despite the very general problem setting, the resulting conditions are workable as documented by some academic examples.


  1. Benko, M., Gfrerer, H., Outrata, J.V.: Calculus for directional limiting normal cones and subdifferentials, Set-Valued Var. Anal.,

  2. Baiocchi, C., Capelo, A.: Variational and quasivariational inequalities. Applications to free boundary problems. Wiley, New York (1984)

    MATH  Google Scholar 

  3. Bonnans, J.F., Shapiro, A.: Perturbation analysis of optimization problems. Springer, New York (2000)

    Book  Google Scholar 

  4. Dontchev, A.L., Rockafellar, R.T.: Regularity and conditioning of solution mappings in variatonal analysis. Set-Valued Anal. 12, 79–109 (2004)

    MathSciNet  Article  Google Scholar 

  5. Dontchev, A.L., Rockafellar, R.T.: Implicit functions and solution mappings. Springer, Heidelberg (2014)

    MATH  Google Scholar 

  6. Dontchev, A.L., Gfrerer, H., Kruger, A.Y., Outrata, J.V.: The radius of metric subregularity, submitted, arXiv:1807.02198

  7. Fabian, M., Henrion, R., Kruger, A.Y., Outrata, J.V.: Error bounds: necessary and sufficient conditions. Set-Valued Var. Anal. 18, 121–149 (2010)

    MathSciNet  Article  Google Scholar 

  8. Fabian, M., Henrion, R., Kruger, A.Y., Outrata, J.V.: About error bounds in metric spaces. In: Klatte, D., Lüthi, H.-J., Schmedders, K. (eds.) Operation Research Proceedings 2011. Springer, Berlin (2012)

  9. Gfrerer, H.: First order and second order characterizations of metric subregularity and calmness of constraint set mappings. SIAM J. Optim. 21, 1439–1474 (2011)

    MathSciNet  Article  Google Scholar 

  10. Gfrerer, H.: On directional metric regularity, subregularity and optimality conditions for nonsmooth mathematical programs. Set-Valued Var. Anal. 21, 151–176 (2013)

    MathSciNet  Article  Google Scholar 

  11. Gfrerer, H.: On directional metric subregularity and second-order optimality conditions for a class of nonsmooth mathematical programs. SIAM J. Optim. 23, 632–665 (2013)

    MathSciNet  Article  Google Scholar 

  12. Gfrerer, H.: On metric pseudo-(sub)regularity of multifunctions and optimality conditions for degenerated mathematical programs. Set-Valued Var. Anal. 22, 79–115 (2014)

    MathSciNet  Article  Google Scholar 

  13. Gfrerer, H., Klatte, D.: Lipschitz and Hölder stability of optimization problems and generalized equations. Math. Program. Ser. A 158, 35–75 (2016)

    Article  Google Scholar 

  14. Gfrerer, H., Mordukhovich, B.S.: Robinson stability of parametric constraint systems via variational analysis. SIAM J. Optim. 27, 438–465 (2017)

    MathSciNet  Article  Google Scholar 

  15. Gfrerer, H., Mordukhovich, B.S.: Second-order variational analysis of parametric constraint and variational systems, (2017), to appear in SIAM. J. Optim., arXiv:1711.07082

  16. Gfrerer, H., Outrata, J.V.: On Lipschitzian properties of implicit multifunctions. SIAM J. Optim. 26, 2160–2189 (2016)

    MathSciNet  Article  Google Scholar 

  17. Gfrerer, H., Outrata, J.V.: On the Aubin property of a class of parameterized variational systems. Math. Methods Oper. Res. 86, 443–467 (2017)

    MathSciNet  Article  Google Scholar 

  18. Gfrerer, H., Outrata, J.V.: On the Aubin property of solution maps to parameterized variational systems with implicit constraints, (2018), submitted, arXiv:1810.12604

  19. Ginchev, I., Mordukhovich, B.S.: On directionally dependent subdifferentials. C.R. Bulg. Acad. Sci. 64, 497–508 (2011)

    MathSciNet  MATH  Google Scholar 

  20. Henrion, R., Jourani, A., Outrata, J.V.: On the calmness of a class of multifunctions. SIAM J. Optim. 13, 603–618 (2002)

    MathSciNet  Article  Google Scholar 

  21. Henrion, R., Outrata, J.V.: Calmness of constraint systems with applications. Math. Programming, Ser. B 104, 437–464 (2005)

    MathSciNet  Article  Google Scholar 

  22. Ioffe, A.D.: Necessary and sufficient conditions for a local minimum 1: a reduction theorem and first order conditions. SIAM J. Control Optim. 17, 245–250 (1979)

    MathSciNet  Article  Google Scholar 

  23. Ioffe, A.D.: Regular points of Lipschitz functions. Trans. Amer. Math Soc. 251, 61–69 (1979)

    MathSciNet  Article  Google Scholar 

  24. Ioffe, A.D., Outrata, J.V.: On metric and calmness qualification conditions in subdifferential calculus. Set-Valued Anal. 16, 199–227 (2008)

    MathSciNet  Article  Google Scholar 

  25. 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)

    MATH  Google Scholar 

  26. Kruger, A.Y.: Error bounds and metric subregularity. Optimization 64, 49–79 (2015)

    MathSciNet  Article  Google Scholar 

  27. Levy, A.B.: Implicit multifunction theorems for the sensitivity analysis of variational conditons. Math Program. 74, 333–350 (1996)

    MATH  Google Scholar 

  28. Mordukhovich, B.S: Variational analysis and generalized differentiation, vol I: basic theory. Springer, Berlin (2006)

    Book  Google Scholar 

  29. Mordukhovich, B.S., Outrata, J.V.: Coderivative analysis of quasi-variational inequalities with applications to stability and optimization. SIAM J. Optim. 18, 389–412 (2007)

    MathSciNet  Article  Google Scholar 

  30. Mordukhovich, B.S.: Variational analysis and applications. Springer, Cham (2018)

    Book  Google Scholar 

  31. Robinson, S.M.: Stability theory for systems of inequalities, II: differentiable nonlinear systems. SIAM J. Numer. Anal. 13, 497–513 (1976)

    MathSciNet  Article  Google Scholar 

  32. Robinson, S.M.: Constraint nondegeneracy in variational analysis. Math. of Oper. Res. 28, 201–232 (2003)

    MathSciNet  Article  Google Scholar 

  33. Rockafellar, R.T.: Convex analysis. Princeton, New Jersey (1970)

    Book  Google Scholar 

  34. Rockafellar, R.T., Wets, R.J.-B.: Variational analysis. Springer, Berlin (1998)

    Book  Google Scholar 

  35. Zheng, X.Y., Ng, K.F.: Metric subregularity and calmness for nonconvex generalized equations in Banach spaces. SIAM J. Optim. 20, 2119–2136 (2010)

    MathSciNet  Article  Google Scholar 

Download references


The research of the first two authors was supported by the Austrian Science Fund (FWF) under grant P29190-N32. The research of the third author was supported by the Grant Agency of the Czech Republic, Project 17-08182S and the Australian Research Council, Project DP160100854.


Open access funding provided by Austrian Science Fund (FWF).

Author information



Corresponding author

Correspondence to Helmut Gfrerer.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Benko, M., Gfrerer, H. & Outrata, J.V. Stability Analysis for Parameterized Variational Systems with Implicit Constraints. Set-Valued Var. Anal 28, 167–193 (2020).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Parameterized variational system
  • Solution map
  • Aubin property
  • Isolated calmness property

Mathematics Subject Classification (2010)

  • 49J53
  • 90C31
  • 90C46