Skip to main content
SpringerLink
Log in

On the Lipschitz Modulus of the Argmin Mapping in Linear Semi-Infinite Optimization

  • Published:
Set-Valued Analysis Aims and scope Submit manuscript

Abstract

This paper is devoted to quantify the Lipschitzian behavior of the optimal solutions set in linear optimization under perturbations of the objective function and the right hand side of the constraints (inequalities). In our model, the set indexing the constraints is assumed to be a compact metric space and all coefficients depend continuously on the index. The paper provides a lower bound on the Lipschitz modulus of the optimal set mapping (also called argmin mapping), which, under our assumptions, is single-valued and Lipschitz continuous near the nominal parameter. This lower bound turns out to be the exact modulus in ordinary linear programming, as well as in the semi-infinite case under some additional hypothesis which always holds for dimensions n ⩽ 3. The expression for the lower bound (or exact modulus) only depends on the nominal problem’s coefficients, providing an operative formula from the practical side, specially in the particular framework of ordinary linear programming, where it constitutes the sharp Lipschitz constant. In the semi-infinite case, the problem of whether or not the lower bound equals the exact modulus for n > 3 under weaker hypotheses (or none) remains as an open problem.

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. Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-Linear Parametric Optimization. Akademie, Berlin (1982)

  2. Bonnans, J.F., Shapiro, A.: Nondegenerancy and quantitative stability of parametrized optimization problems with multiple solutions. SIAM J. Optim. 8, 940–946 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  3. Brosowski, B.: Parametric semi-infinite linear programming I. Continuity of the feasible set and of the optimal value. Math. Programming Stud. 21, 18–42 (1984)

    MATH  MathSciNet  Google Scholar 

  4. Cánovas, M.J., Dontchev, A.L., López, M.A., Parra, J.: Metric regularity of semi-infinite constraint systems. Math. Programming 104B, 329–346 (2005)

    Article  Google Scholar 

  5. Cánovas, M.J., Klatte, D., López, M.A., Parra, J.: Metric regularity in convex semi-infinite optimization under canonical perturbations. SIAM J. Optim. (2007) (in press)

  6. Cánovas, M.J., López, M.A., Parra, J., Todorov, M.I.: Stability and well-posedness in linear semi-infinite programming. SIAM J. Optim. 10, 82–98 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  7. Cánovas, M.J., López, M.A., Parra, J., Toledo, F.J.: Lipschitz continuity of the optimal value via bounds on the optimal set in linear semi-infinite optimization. Math. Oper. Res. 31, 478–489 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Cheung, D., Cucker, F., Pena, J.: Unifying condition numbers for linear programming. Math. Oper. Res. 28, 609–624 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dontchev, A.L., Lewis, A.S., Rockafellar, R.T.: The radius of metric regularity. Trans. Amer. Math. Soc. 355, 493–517 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dontchev, A.L., Rockafellar, R.T.: Characterizations of strong regularity for variational inequalities over polyhedral convex sets. SIAM J. Optim. 6, 1087–1105 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  11. Fischer, T.: Contributions to semi-infinite linear optimization. In: Brosowski, B., Martensen, E. (eds.) Approximation and Optimization in Mathematical Physics, pp. 175–199, Peter Lang, Frankfurt-Am-Main (1983)

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

    MATH  Google Scholar 

  13. Goberna, M.A., López, M.A., Todorov, M.I.: Stability theory for linear inequality systems. SIAM J. Matrix Anal. Appl. 17, 730–743 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  14. Henrion, R., Klatte, D.: Metric regularity of the feasible set mapping in semi-infinite optimization. Appl. Math. Optim. 30, 103–109 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  15. Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55(3) (333), 103–162; English translation. Math. Surveys 55, 501–558 (2000)

  16. Klatte, D.: Stability of stationary solutions in semi-infinite optimization via the reduction approach. In: Oettli, W., Pallaschke, D. (eds.) Lecture Notes in Economica and Mathematical Systems 382, pp. 155–170 Springer (1992)

  17. Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications. Kluwer, Dordrecht (2002)

    MATH  Google Scholar 

  18. Klatte, D., Kummer, B.: Strong Lipschitz stability of stationary solutions for nonlinear programs and variational inequalities. SIAM J. Optim. 16, 96–119 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  19. Kojima, M.: Strongly stable stationary solutions in nonlinear programs. In: Robinson, S.M. (ed.) Analysis and Computation of Fixed Points. Academic, New York, pp. 93–138 (1980)

    Google Scholar 

  20. Levy, A.B., Poliquin, R.A.: Characterizing the single-valuedness of multifunctions. Set-Valued Anal. 5, 351–364 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  21. Li, W.: The sharp Lipschitz constants for feasible and optimal solutions of a perturbed linear program. Linear Algebra Appl. 187, 15–40 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  22. Li, W.: Sharp Lipschitz constants for basic optimal solutions and basic feasible solutions of linear programs. SIAM J. Control Optim. 32, 140–153 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  23. Li, W.: Abadie’s constraint qualification, metric regularity, and error bounds for differentiable convex inequalities. SIAM J. Optim. 7, 966–978 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  24. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation (I,II). Springer, Berlin (2006)

    Google Scholar 

  25. Nožička, F., Guddat, J., Hollatz, H., Bank, B.: Theorie der Linearen Parametrischen Optimierung. Akademie-Verlag, Berlin (1974)

    MATH  Google Scholar 

  26. Renegar, J.: Linear programming, complexity theory and elementary functional analysis. Math. Programming 70 Ser. A, 279–351 (1995)

  27. Rockafellar, R.T., Wets, J.-B.R.: Variational Analysis, Springer, Berlin (1997)

    Google Scholar 

  28. Rückmann, J.-J.: On existence and uniqueness of stationary points in semi-infinite optimization. Math. Programming 86, 387–415 (1999)

    Article  MATH  Google Scholar 

  29. Zheng, X.Y., Ng, K.F.: Metric regularity and constraint qualifications for convex inequalities on Banach spaces. SIAM J. Optim. 14, 757–772 (2004)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Operations Research Center, Miguel Hernández University of Elche, 03202, Elche, Alicante, Spain

    M. J. Cánovas, F. J. Gómez-Senent & J. Parra

Authors
  1. M. J. Cánovas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. J. Gómez-Senent
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Parra
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to J. Parra.

Additional information

This research has been partially supported by grants MTM2005-08572-C03-02, from MEC (Spain) and FEDER (E.U.), and ACOMP06/203, from Generalitat Valenciana (Spain).

F.J. Gómez-Senent acknowledges a special permission for studies (‘Licencia por Estudios’) from Consejería de Educación y Cultura de la Región de Murcia (Spain).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cánovas, M.J., Gómez-Senent, F.J. & Parra, J. On the Lipschitz Modulus of the Argmin Mapping in Linear Semi-Infinite Optimization. Set-Valued Anal 16, 511–538 (2008). https://doi.org/10.1007/s11228-007-0052-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-007-0052-x

Keywords

Mathematics Subject Classifications (2000)

Search

Navigation