Linear regularity, equirregularity, and intersection mappings for convex semi-infinite inequality systems

Full Length Paper Series B

Abstract

In this paper, we introduce the concepts of linear regularity and equirregularity for an arbitary family of set-valued mappings between (extended) metric spaces. The concept of linear regularity is inspired in the same property for a family of sets. Then we analyze the relationship between the (metric) regularity moduli of the mappings in the family and the modulus of the associated intersection mapping. We are particularly concerned with the solution set of a system of infinitely many convex inequalities. Our framework allows for right hand side perturbations as well as for linear perturbations of the left hand side of all the inequalities.

Mathematics Subject Classification (2000)

49J53 90C34 90C25 52A41 

References

  1. 1.
    Bauschke H.H., Borwein J.M., Li W.: Strong conical hull intersection property, bounded linear regularity, Jameson’s property (G), and error bounds in convex optimization. Math. Program. Ser. A 86, 135–160 (1999)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Cánovas M.J., Dontchev A.L., López M.A., Parra J.: Metric regularity of semi-infinite constraint systems. Math. Program. 104B, 329–346 (2005)CrossRefGoogle 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)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    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. Optim. (2009). doi:10.1080/0233193081950985
  5. 5.
    Dinh N., Goberna M.A., López M.A.: From linear to convex system: consistency, Farkas’ lemma and applications. J. Convex Anal. 13, 113–133 (2006)MATHMathSciNetGoogle Scholar
  6. 6.
    Dontchev A.L., Lewis A.S., Rockafellar R.T.: The radius of metric regularity. Trans. Am. Math. Soc. 355, 493–517 (2003)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Goberna M.A., López M.A.: Linear Semi-Infinite Optimization. Wiley, Chichester (1998)MATHGoogle Scholar
  8. 8.
    Hantoute A., López M.A.: A complete characterization of the subdifferential set of the supremum of an arbitrary family of convex functions. J. Convex Anal. 15, 831–858 (2008)MATHMathSciNetGoogle Scholar
  9. 9.
    Hiriart-Urruty J.-B., Lemaréchal C.: Convex Analysis and Minimization Algorithms I. Springer, Berlin (1993)Google Scholar
  10. 10.
    Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55(3) (333), pp. 103–162 (2000) (English translation Math. Surveys, 55 (2000), pp. 501–558)Google Scholar
  11. 11.
    Ioffe A.D., Sekiguchi Y.: Exact formulae for regularity estimates. In: Takahashi, W., Tanaka, T. (eds) Proceedings of the 4th International Conference on Nonlinear Analysis and Convex Analysis, Okinawa, 2005, pp. 185–198. Yokohama Publisher, Yokohama (2007)Google Scholar
  12. 12.
    Ioffe, A.D., Sekiguchi, Y.: Regularity estimates for convex multifunctions. Math. Program. (to appear 2009)Google Scholar
  13. 13.
    Jeyakumar V.: Asymptotic dual conditions characterizing optimality for infinite convex programs. J. Optim. Theory Appl. 93, 153–165 (1997)MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Klatte D., Kummer B.: Nonsmooth Equations in Optimization. Regularity, Calculus, Methods and Applications. Kluwer, Dordrecht (2002)MATHGoogle Scholar
  15. 15.
    Lewis, A.S., Pang, J.-S.: Error bounds for convex inequality systems. In: Crouzeix, J.-P. (ed.), Generalized Convexity, Proceeding of the 5th Symposium on Generalized Convexity, pp. 75–110. Luminy-Marseille (1997)Google Scholar
  16. 16.
    Luenberger D.G.: Optimization by Vector Space Methods. Wiley, New York (1969)MATHGoogle Scholar
  17. 17.
    Martínez-Legaz J.-E., Rubinov A., Singer I.: Downward sets and their separation and approximation properties. J. Glob. Optim. 23, 111–137 (2002)MATHCrossRefGoogle Scholar
  18. 18.
    Mordukhovich B.S.: Variational Analysis and Generalized Differentiation (I, II). Springer, Berlin (2006)Google Scholar
  19. 19.
    Rockafellar R.T.: Convex Analysis. Princeton University Press, Princeton (1970)MATHGoogle Scholar
  20. 20.
    Rockafellar R.T., Wets R.J.-B.: Variational Analysis. Springer, Berlin (1997)Google Scholar
  21. 21.
    Zălinescu C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)MATHGoogle Scholar

Copyright information

© Springer and Mathematical Programming Society 2009

Authors and Affiliations

  • M. J. Cánovas
    • 1
  • F. J. Gómez-Senent
    • 1
  • J. Parra
    • 1
  1. 1.Operations Research CenterMiguel Hernández University of ElcheElcheSpain

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