Skip to main content

Advertisement

SpringerLink
Log in

Calmness of constraint systems with applications

  • Published:
Mathematical Programming Submit manuscript

Abstract

The paper is devoted to the analysis of the calmness property for constraint set mappings. After some general characterizations, specific results are obtained for various types of constraints, e.g., one single nonsmooth inequality, differentiable constraints modeled by polyhedral sets, finitely and infinitely many differentiable inequalities. The obtained conditions enable the detection of calmness in a number of situations, where the standard criteria (via polyhedrality or the Aubin property) do not work. Their application in the framework of generalized differential calculus is explained and illustrated by examples associated with optimization and stability issues in connection with nonlinear complementarity problems or continuity of the value-at-risk.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubin, J.-P.: Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9, 87–111 (1984)

    Google Scholar 

  2. Azé, D., Corvellec, J.-N.: Characterizations of error bounds for lower semicontinuous functions on metric spaces. ESAIM Control Optim. Calc. Var. 10, 409–425 (2004)

    Article  Google Scholar 

  3. Bazaraa, M.S., Shetty C.M.: Nonlinear Programming. Theory and Algorithms. Wiley, Chichester, 1979

  4. Burke, J. V.: Calmness and exact penalization. SIAM J. Control Optim. 29, 493–497 (1991)

    Article  Google Scholar 

  5. Burke, J. V., Deng, S.: Weak sharp minima revisited. I. Basic theory. Control Cybernet. 31, 439–469 (2002)

    Google Scholar 

  6. Clarke, F.H.: A new approach to Lagrange multipliers. Math. Oper. Res. 2, 165–174 (1976)

    Google Scholar 

  7. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York, 1983

  8. Dontchev, A.L., Hager, W.W.: Implicit functions, Lipschitz maps and stability in optimization. Math. Oper. Res. 19, 753–768 (1994)

    Google Scholar 

  9. Dontchev, A.L., Rockafellar, R.T.: Characterization of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 7, 1087–1105 (1996)

    Article  Google Scholar 

  10. Dontchev, A.L., Rockafellar, R.T.: Ample parameterization of variational inclusions, SIAM J. Optim. 12, 170–187 (2001)

    Google Scholar 

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

    Google Scholar 

  12. Henrion, R., Römisch, W.: Metric regularity and quantitative stability in stochastic programs with probabilistic constraints. Math. Program. 84, 55–88 (1999)

    Google Scholar 

  13. Henrion, R.: Qualitative stability in convex programs with probabilistic constraints. In: V.H. Nguyen et al. (eds.), Optimization, Lecture Notes in Economics and Mathematical Systems, 481, Springer, Berlin, 2000, pp. 164–180

  14. Henrion, R., Outrata, J.V.: A subdifferential criterion for calmness of multifunctions. J. Math. Anal. Appl. 258, 110–130 (2001)

    Article  Google Scholar 

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

    Article  Google Scholar 

  16. Henrion, R.: Perturbation Analysis of chance-constrained programs under variation of all constraint data. In: K. Marti et al. (eds.), Dynamic Stochastic Optimization, Lecture Notes in Economics and Mathematical Systems, 532, Springer, Berlin, 2004, pp. 257–274

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

    Article  Google Scholar 

  18. Ioffe, A.D.: Metric regularity and subdifferential calculus. Russian Math. Surveys 55, 501–558 (2000)

    Article  Google Scholar 

  19. King, A.J., Rockafellar, R.T.: Sensitivity analysis for nonsmooth generalized equations, Math. Programming 55, 193–212 (1992)

    Article  Google Scholar 

  20. Klatte, D., Kummer, B.: Constrained minima and Lipschitzian penalties in metric spaces, SIAM J. Optim. 13, 619–633 (2002)

    Google Scholar 

  21. Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization-Regularity, Calculus, Methods and Applications. Kluwer, Dordrecht, 2002

  22. Levy, A.B.: Implicit multifunction theorems for the sensitivity analysis of variational conditions, Math. Programming 74, 333–350 (1996)

    Article  Google Scholar 

  23. Mifflin, R.: Semismooth and semiconvex functions in constrained optimization. SIAM J. Control Optim. 15, 959–972 (1979)

    Article  Google Scholar 

  24. Mordukhovich, B.S.: Approximation Methods in Problems of Optimization and Control. Nauka, Moscow, 1988 (in Russian)

  25. Mordukhovich, B.S.: Generalized differential calculus for nonsmooth and set-valued mappings. J. Math. Anal. Appl. 183, 250–288 (1994)

    Article  Google Scholar 

  26. Ng, K.F., Zheng, X.Y.: Error bounds for lower semicontinuous functions in normed spaces. SIAM J. Optim. 12, 1–17 (2001)

    Article  Google Scholar 

  27. Outrata, J.V.: A generalized mathematical program with equilibrium constraints. SIAM J. Control Optim. 38, 1623–1638 (2000)

    Article  Google Scholar 

  28. Pang, J.-S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997)

    Article  Google Scholar 

  29. Pflug, G.Ch.: Stochastic Optimization and Statistical Inference. In: A. Ruszczynski and A. Shapiro (eds.) Stochastic Programming. Handbooks in Operations Research and Management Science, Vol. 10, Elsevier, Amsterdam 2003, chapter 7

  30. Prékopa, A.: Stochastic Programming. Kluwer, Dordrecht, 1995

  31. Robinson, S.M.: Generalized equations and their solutions, Part I: Basic theory, Math. Programming Study 10, 128–141 (1979)

    Google Scholar 

  32. Robinson, S.M.: Some continuity properties of polyhedral multifunctions. Math. Programming Study 14, 206–214 (1981)

    Google Scholar 

  33. Robinson, S.M.: An implicit-function theorem for a class of nonsmooth functions. Math. Oper. Res. 16, 292–309 (1991)

    Google Scholar 

  34. Rockafellar, R.T.: Lipschitzian properties of multifunctions, Nonlinear Anal. 9, 867–885 (1985)

    Google Scholar 

  35. Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, New York, 1998

  36. Studniarski, M., Ward, D.E.: Weak sharp minima: characterizations and sufficient conditions. SIAM J. Control and Optim. 38, 219–236 (1999)

    Article  Google Scholar 

  37. Thibault, L.: Propriétés des Sous-Différentiels de Fonctions Localement Lipschitziennes Définies sur un espace de Banach Séparable. PhD Thesis, Department of Mathematics, University of Montpellier, 1975

  38. Wu, Z., Ye, J.J.: Sufficient conditions for error bounds. SIAM J. Optim. 12, 421–435 (2001)

    Article  Google Scholar 

  39. Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977–997 (1997)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Weierstrass Institute Berlin, D-10117, Berlin, Germany

    René Henrion

  2. Institute of Information Theory and Automation, 18208, Prague, Czech Republic

    Jiří V. Outrata

Authors
  1. René Henrion
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiří V. Outrata
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jiří V. Outrata.

Additional information

This research was supported by the DFG Research center Matheon Mathematics for key technologies in Berlin

Support by grant IAA1030405 of the Grant Agency of the Academy of Sciences of the Czech Republic is acknowledged

Rights and permissions

Reprints and permissions

About this article

Cite this article

Henrion, R., Outrata, J. Calmness of constraint systems with applications. Math. Program. 104, 437–464 (2005). https://doi.org/10.1007/s10107-005-0623-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-005-0623-2

Keywords

Mathematics Subject Classification (2000)

Search

Navigation