Journal of Optimization Theory and Applications

, Volume 160, Issue 1, pp 111–126 | Cite as

Calmness of the Argmin Mapping in Linear Semi-Infinite Optimization

  • M. J. Cánovas
  • A. Hantoute
  • J. Parra
  • F. J. Toledo


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.


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


  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) CrossRefGoogle 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) CrossRefMATHGoogle Scholar
  6. 6.
    Wu, Z., Ye, J.J.: Sufficient conditions for error bound. SIAM J. Optim. 12, 421–435 (2001) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Pang, J.-S.: Error bounds in mathematical programming. Math. Program. 79, 299–332 (1997) MATHGoogle Scholar
  8. 8.
    Klatte, D., Thiere, G.: Error bounds for solutions of linear equations and inequalities. Math. Methods Oper. Res. 41, 191–214 (1995) CrossRefMATHMathSciNetGoogle 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) CrossRefMATHMathSciNetGoogle 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) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Henrion, R., Outrata, J.: Calmness of constraint systems with applications. Math. Program. B 104, 437–464 (2005) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Klatte, D., Kummer, B.: Optimization methods and stability of inclusions in Banach spaces. Math. Program. B 117, 305–330 (2009) CrossRefMATHMathSciNetGoogle 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) CrossRefMATHMathSciNetGoogle 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) CrossRefMATHMathSciNetGoogle 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) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Jourani, A.: Hoffman’s error bound, local controllability, and sensitivity analysis. SIAM J. Control Optim. 38, 947–970 (2000) CrossRefMATHMathSciNetGoogle 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) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    Henrion, R., Jourani, A., Outrata, J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13, 603–618 (2002) CrossRefMATHMathSciNetGoogle 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) CrossRefGoogle Scholar
  20. 20.
    Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998) MATHGoogle Scholar
  21. 21.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002) CrossRefMATHGoogle 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) CrossRefMathSciNetGoogle 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) CrossRefMATHGoogle Scholar
  24. 24.
    Beer, G.: Topologies on Closed and Closed Convex Sets. Kluwer, Dordrecht (1993) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • M. J. Cánovas
    • 1
  • A. Hantoute
    • 2
  • J. Parra
    • 1
  • F. J. Toledo
    • 1
  1. 1.Center of Operations ResearchMiguel Hernández University of ElcheElche (Alicante)Spain
  2. 2.Departamento de Ingeniería Matemática, Centro de Modelamiento Matemático (CMM)Universidad de ChileSantiagoChile

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