Skip to main content
SpringerLink
Log in

Outer Limit of Subdifferentials and Calmness Moduli in Linear and Nonlinear Programming

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

With a common background and motivation, the main contributions of this paper are developed in two different directions. Firstly, we are concerned with functions, which are the maximum of a finite amount of continuously differentiable functions of n real variables, paying special attention to the case of polyhedral functions. For these max-functions, we obtain some results about outer limits of subdifferentials, which are applied to derive an upper bound for the calmness modulus of nonlinear systems. When confined to the convex case, in addition, a lower bound on this modulus is also obtained. Secondly, by means of a Karush–Kuhn–Tucker index set approach, we are also able to provide a point-based formula for the calmness modulus of the argmin mapping of linear programming problems, without any uniqueness assumption on the optimal set. This formula still provides a lower bound in linear semi-infinite programming. Illustrative examples are given.

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

Access this article

Log in via an institution

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. Cánovas, M.J., Hantoute, A., Parra, J., Toledo, F.J.: Boundary of subdifferentials and calmness moduli in linear semi-infinite optimization. Optim. Lett. 9, 513–521 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cánovas, M.J., Hantoute, A., Parra, J., Toledo, F.J.: Calmness of fully perturbed linear programs. Math. Program. Ser. A (2015). doi:10.1007/s10107-015-0926-x

  3. Cánovas, M.J., Kruger, A.Y., López, M.A., Parra, J., Thera, M.A.: Calmness modulus of linear semi-infinite programs. SIAM J. Optim. 24, 29–48 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  4. Cánovas, M.J., López, M.A., Parra, J., Toledo, F.J.: Calmness of the feasible set mapping for linear inequality systems. Set Valued Var. Anal. 22, 375–389 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Robinson, S.M.: Some continuity properties of polyhedral multifunctions. Mathematical programming at Oberwolfach (Proc. Conf., Math. Forschungsinstitut, Oberwolfach, 1979). Math. Program. Stud. 14, 206–214 (1981)

  6. Robinson, S.M.: A characterization of stability in linear programming. Oper. Res. 25, 435–447 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  7. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings: A View from Variational Analysis. Springer, New York (2009)

    Book  MATH  Google Scholar 

  8. Klatte, D. Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications. Nonconvex Optim. Appl. 60. Kluwer Academic, Dordrecht (2002)

  9. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)

    Google Scholar 

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

    Book  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  12. Henrion, R., Outrata, J.: Calmness of constraint systems with applications. Math. Program. B 104, 437–464 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Klatte, D., Kummer, B.: Optimization methods and stability of inclusions in Banach spaces. Math. Program. B 117, 305–330 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Klatte, D., Thiere, G.: Error bounds for solutions of linear equations and inequalities. Math. Methods Oper. Res. 41, 191–214 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  15. 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  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Kruger, A., Van Ngai, H., Théra, M.: Stability of error bounds for convex constraint systems in Banach spaces. SIAM J. Optim. 20, 3280–3296 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Jourani, A.: Hoffman’s error bound, local controllability, and sensitivity analysis. SIAM J. Control Optim. 38, 947–970 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  20. Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)

    MATH  Google Scholar 

  21. Goberna, M.A., López, M.A.: Post-Optimal Analysis in Linear Semi-Infinite Optimization. Springer Briefs in Optimization. Springer, New York (2014)

  22. Clarke, F.H.: Optimization and Nonsmooth Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. A Wiley-Interscience Publication. Wiley, New York (1983)

  23. Luenberger, D.G., Ye, Y.: Linear and Nonlinear Programming, 3rd edn. Springer, Berlin (2008)

    MATH  Google Scholar 

  24. Hiriart-Urruty, J.B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I. Springer, Berlin (1991)

    MATH  Google Scholar 

  25. Klatte, D., Kummer, B.: On calmness of the argmin mapping in parametric optimization problems. J. Optim. Theory Appl. 165, 708–719 (2015)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

This research has been partially supported by Grants MTM2011-29064-C03 (02-03) and MTM2014-59179-C2-(1,2)-P from MINECO, Spain.

Author information

Authors and Affiliations

  1. Center of Operations Research, Miguel Hernández University of Elche, Elche (Alicante), Spain

    M. J. Cánovas & J. Parra

  2. Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstr. 39, 10117, Berlin, Germany

    R. Henrion

  3. Department of Statistics and Operations Research, University of Alicante, 03071, Alicante, Spain

    M. A. López

Authors
  1. M. J. Cánovas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. R. Henrion
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. M. A. López
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. Parra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to M. A. López.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cánovas, M.J., Henrion, R., López, M.A. et al. Outer Limit of Subdifferentials and Calmness Moduli in Linear and Nonlinear Programming. J Optim Theory Appl 169, 925–952 (2016). https://doi.org/10.1007/s10957-015-0793-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-015-0793-x

Keywords

Mathematics Subject Classification

Search

Navigation