Set-Valued and Variational Analysis

, Volume 22, Issue 2, pp 375–389 | Cite as

Calmness of the Feasible Set Mapping for Linear Inequality Systems

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


In this paper we deal with parameterized linear inequality systems in the n-dimensional Euclidean space, whose coefficients depend continuosly on an index ranging in a compact Hausdorff space. The paper is developed in two different parametric settings: the one of only right-hand-side perturbations of the linear system, and that in which both sides of the system can be perturbed. Appealing to the backgrounds on the calmness property, and exploiting the specifics of the current linear structure, we derive different characterizations of the calmness of the feasible set mapping, and provide an operative expresion for the calmness modulus when confined to finite systems. In the paper, the role played by the Abadie constraint qualification in relation to calmness is clarified, and illustrated by different examples. We point out that this approach has the virtue of tackling the calmness property exclusively in terms of the system’s data.


Calmness Local error bounds Variational analysis Semi-infinite programming Linear programming Feasible set mapping 

Mathematics Subject Classifications (2010)

90C34 90C31 49J53 90C05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    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
  2. 2.
    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
  3. 3.
    Cánovas, M.J., Gómez-Senent, F.J., Parra, J.: Regularity modulus of arbitrarily perturbed linear inequality systems. J. Math. Anal. Appl. 343, 315–327 (2008)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings: A View from Variational Analysis. Springer, New York (2009)CrossRefGoogle Scholar
  5. 5.
    Fabian, M., Henrion, R., Kruger, A.Y., Outrata, J.: About error bounds in metric spaces. In: Operations Research Proceeding 2011. Selected Paper of the International Conference Operations Research (OR 2011), August 30, September 2, 2011, Zurich, Switzerland, pp. 33–38. Springer, Berlin (2012)Google Scholar
  6. 6.
    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
  7. 7.
    Goberna, M.A., López, M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)MATHGoogle Scholar
  8. 8.
    Henrion, R., Jourani, A., Outrata, J.: On the calmness of a class of multifunctions. SIAM J. Optim. 13, 603–618 (2002)CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Henrion, R., Klatte, D.: Regularity and stability in nonlinear semi-infinite optimization. In: Reemtsen, R., Rückmann, J.-J. (eds.) Semi-Infinite Programming. Kluwer (1998)Google Scholar
  10. 10.
    Henrion, R., Outrata, J.: Calmness of constraint systems with applications. Math. Program. 104B, 437–464 (2005)CrossRefMathSciNetGoogle Scholar
  11. 11.
    Ioffe, A.D.: Necessary and sufficient conditions for a localminimum, part I: a reduction theorem and first order conditions. SIAM J. Control Optim. 17, 245–250 (1979)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspehi Mat. Nauk 55(3), 103–162 (2000). (in Russian), English translation: Russian Math. Surveys 55(3), 501–558 (2000)Google Scholar
  13. 13.
    Jourani, A.: Hoffman’s error bound, local controllability, and sensitivity analysis. SIAM J. Control Optim. 38, 947–970 (2000)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Klatte, D., Kummer, B.: Nonsmooth equations in optimization: regularity, calculus, methods and applications. Nonconvex Optimization and Applications 60. Kluwer, Dordrecht (2002)Google Scholar
  15. 15.
    Klatte, D., Kummer, B.: Optimization methods and stability of inclusions in Banach spaces. Math. Program. B 117, 305–330 (2009)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Klatte, D., Thiere, G.: Error bounds for solutions of linear equations and inequalities. Math. Methods Oper. Res. 41, 191–214 (1995)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.
    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
  19. 19.
    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
  20. 20.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory. Springer, Berlin (2006)Google Scholar
  21. 21.
    Robinson, S.M.: Some continuity properties of polyhedral multifunctions. In: Mathematical programming at Oberwolfach (Proc. Conf. Math. Forschungsinstitut, Oberwolfach, 1979). Math. Program. Stud. No. 14, pp. 206–214 (1981)Google Scholar
  22. 22.
    Rockafellar, R.T., Wets, R.J.-B.: Variational Analysis. Springer, Berlin (1998)CrossRefMATHGoogle Scholar
  23. 23.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)CrossRefMATHGoogle Scholar
  24. 24.
    Zheng, X.Y., Ng, K.F.: Metric regularity and constraint qualifications for convex inequalities on Banach spaces. SIAM J. Optim. 14, 757–772 (2003)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2014

Authors and Affiliations

  • M. J. Cánovas
    • 1
  • M. A. López
    • 2
    • 3
  • J. Parra
    • 1
  • F. J. Toledo
    • 1
  1. 1.Center of Operations ResearchMiguel Hernández University of ElcheElcheSpain
  2. 2.Department of Statistics and Operations ResearchUniversity of AlicanteAlicanteSpain
  3. 3.Federation University of AustraliaBallaratAustralia

Personalised recommendations