Skip to main content

Calmness of the Argmin Mapping in Linear Semi-Infinite Optimization

Abstract

This paper characterizes the calmness property of the argmin mapping in the framework of linear semi-infinite optimization problems under canonical perturbations; i.e., continuous perturbations of the right-hand side of the constraints (inequalities) together with perturbations of the objective function coefficient vector. This characterization is new for semi-infinite problems without requiring uniqueness of minimizers. For ordinary (finitely constrained) linear programs, the calmness of the argmin mapping always holds, since its graph is piecewise polyhedral (as a consequence of a classical result by Robinson). Moreover, the so-called isolated calmness (corresponding to the case of unique optimal solution for the nominal problem) has been previously characterized. As a key tool in this paper, we appeal to a certain supremum function associated with our nominal problem, not involving problems in a neighborhood, which is related to (sub)level sets. The main result establishes that, under Slater constraint qualification, perturbations of the objective function are negligible when characterizing the calmness of the argmin mapping. This result also states that the calmness of the argmin mapping is equivalent to the calmness of the level set mapping.

This is a preview of subscription content, access via your institution.

References

  1. 1.

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

    Google Scholar 

  2. 2.

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

    Book  Google Scholar 

  3. 3.

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

    Google Scholar 

  4. 4.

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

    Google Scholar 

  5. 5.

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

    Book  MATH  Google Scholar 

  6. 6.

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

    Article  MATH  MathSciNet  Google Scholar 

  7. 7.

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

    MATH  Google Scholar 

  8. 8.

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

    Article  MATH  MathSciNet  Google Scholar 

  9. 9.

    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 

  10. 10.

    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 

  11. 11.

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

    Article  MATH  MathSciNet  Google Scholar 

  12. 12.

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

    Article  MATH  MathSciNet  Google Scholar 

  13. 13.

    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  MATH  MathSciNet  Google Scholar 

  14. 14.

    Cánovas, M.J., Dontchev, A.L., López, M.A., Parra, J.: Isolated calmness of solution mappings in convex semi-infinite optimization. J. Math. Anal. Appl. 350, 829–837 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  15. 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  MATH  MathSciNet  Google Scholar 

  16. 16.

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

    Article  MATH  MathSciNet  Google Scholar 

  17. 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  MATH  MathSciNet  Google Scholar 

  18. 18.

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

    Article  MATH  MathSciNet  Google Scholar 

  19. 19.

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

    Chapter  Google Scholar 

  20. 20.

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

    MATH  Google Scholar 

  21. 21.

    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)

    Book  MATH  Google Scholar 

  22. 22.

    Cánovas, M.J., Hantoute, A., López, M.A., Parra, J.: A note on the compactness of the index set in convex optimization. Application to metric regularity. Optimization 59, 447–483 (2010)

    Article  MathSciNet  Google Scholar 

  23. 23.

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

    Article  MATH  Google Scholar 

  24. 24.

    Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer, Dordrecht (1993)

    MATH  Google Scholar 

Download references

Acknowledgements

This research has been partially supported by Grants MTM2008-06695-C03-02 and MTM2011-29064-C03-03 from MICINN/MINECO, Spain. The research of the second author is also partially supported by Fondecyt Project No 1110019 and ECOS-Conicyt project No C10E08.

The authors are indebted to Professor Marco A. López for his valuable hints and also for his comments about the backgrounds on the subject. The authors are also indebted to the anonymous referees and the associate editor for their constructive critical comments.

Author information

Affiliations

Authors

Corresponding author

Correspondence to J. Parra.

Additional information

Communicated by Réne Henrion.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Cánovas, M.J., Hantoute, A., Parra, J. et al. Calmness of the Argmin Mapping in Linear Semi-Infinite Optimization. J Optim Theory Appl 160, 111–126 (2014). https://doi.org/10.1007/s10957-013-0371-z

Download citation

Keywords

  • Calmness
  • Local error bounds
  • Variational analysis
  • Semi-infinite programming
  • Linear programming