Set-Valued and Variational Analysis

, Volume 21, Issue 1, pp 93–126 | Cite as

On Subregularity Properties of Set-Valued Mappings

Applications to Solid Vector Optimization
Article

Abstract

In this work we classify the at-point regularities of set-valued mappings into two categories and then we analyze their relationship through several implications and examples. After this theoretical tour, we use the subregularity properties to deduce implicit theorems for set-valued maps. Finally, we present some applications to the study of multicriteria optimization problems.

Keywords

Set-valued maps At-point regularity Around-point regularity Implicit multifunction theorems Solid vector optimization 

Mathematics Subject Classifications (2010)

90C30 49J52 49J53 

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References

  1. 1.
    Aragon Artacho, F.J., Mordukhovich, B.S.: Metric regularity and Lipscithian stability of parametric variational systems. Nonlinear Anal. 72, 1149–1170 (2010)MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    Aragon Artacho, F.J., Mordukhovich, B.S.: Enhanced metric regularity and Lipscithian stability of variational systems. J. Glob. Optim. 50, 145–167 (2010)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Arutyunov, A.V.: Covering mapping in metric spaces, and fixed points. Dokl. Math. 76, 665–668 (2007)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Arutyunov, A.V.: Stability of coincidence points and properties of covering mappings. Math. Notes 86, 153–158 (2009)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Arutyunov, A., Avakov, E., Gel’man, B., Dmitruk, A., Obukhovskii, V.: Locally covering maps in metric spaces and coincidence points. J. Fixed Point Theory Appl. 5, 105–127 (2009)MathSciNetMATHCrossRefGoogle Scholar
  6. 6.
    Arutyunov, A.V., Zhukovskiy, E.S., Zhukovskiy, S.E.: Covering mappings and well-posedness of nonlinear Volterra equations. Nonlinear Anal. 75, 1026–1044 (2012)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    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)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. Serie A, 122, 301–347 (2010)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Chuong, T.D., Kruger, A.Y., Yao, J.-C.: Calmness of efficient solution maps in parametric vector optimization. J. Glob. Optim. 51, 677–688 (2011)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    De Giorgi, E., Marino, A., Tosques, M.: Problemi di evoluzione in spazi metrici e curve di massima pendenza. Atti. Accad. Naz. Lincei rend. Cl. Sci. Fis. Mat. Natur. 68, 180–187 (1980)MathSciNetMATHGoogle Scholar
  11. 11.
    Dmitruk, A.V.: On a nonlocal metric regularity of nonlinear operators. Control Cybern. 34, 723–746 (2005)MathSciNetMATHGoogle Scholar
  12. 12.
    Dmitruk, A.V., Milyutin, A.A., Osmolovskii, N.P.: Lyusternik’s theorem and the theory of extrema. Usp. Mat. Nauk 35, 11–46 (1980)MathSciNetGoogle Scholar
  13. 13.
    Dontchev, A.L., Frankowska, H.: Lyusternik–Graves theorem and fixed points. Proc. Am. Math. Soc. 139, 521–534 (2011)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Dontchev, A.L., Frankowska, H.: Lyusternik–Graves theorem and fixed points II. J. Convex Anal. (accepted)Google Scholar
  15. 15.
    Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Berlin (2009)MATHCrossRefGoogle Scholar
  16. 16.
    Durea, M., Nguyen, H.T., Strugariu, R.: Metric regularity of epigraphical multivalued mappings and applications to vector optimization. Math. Program. Serie B (accepted)Google Scholar
  17. 17.
    Durea, M., Strugariu, R.: On some Fermat rules for set-valued optimization problems. Optimization 60, 575–591 (2011)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Durea, M., Strugariu, R.: On parametric vector optimization via metric regularity of constraint systems. Math. Methods Oper. Res. 74, 409–425 (2011)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Durea, M., Strugariu, R.: Openness stability and implicit multifunction theorems: applications to variational systems. Nonlinear Anal. 75, 1246–1259 (2012)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Durea, M., Strugariu, R.: Chain rules for linear openness in general Banach spaces. SIAM J. Optim. (accepted)Google Scholar
  21. 21.
    Durea, M., Strugariu, R.: Chain rules for linear openness in metric spaces and applications. Applications to parametric variational systems (submitted)Google Scholar
  22. 22.
    Durea, M., Tammer, C.: Fuzzy necessary optimality conditions for vector optimization problems. Optimization 58, 449–467 (2009)MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Henrion, R., Outrata, J.V.: Calmness of constraint systems with applications. Math. Program. Serie B 104, 437–464 (2005)MathSciNetMATHCrossRefGoogle Scholar
  24. 24.
    Ioffe, A.D.: Towards variational analysis in metric spaces: metric regularity and fixed points. Math. Program. Serie B 123, 241–252 (2010)MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Li, G., Mordukhovich, B.S.: Hölder metric subregularity with applications to proximal point method. (preprint, 2012)Google Scholar
  26. 26.
    Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, vol. I: Basic Theory, vol. II: Applications, Springer, Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), vols. 330 and 331. Berlin (2006)Google Scholar
  27. 27.
    Ngai, H.V., Nguyen, H.T., Théra, M.: Implicit multifunction theorems in complete metric spaces. Math. Program. Serie B (accepted)Google Scholar
  28. 28.
    Ngai, H.V., Nguyen, H.T., Théra, M.: Metric regularity of the sum of multifunctions and applications. Available at http://www.optimization-online.org/DB_HTML/2011/12/3291.html
  29. 29.
    Ngai, H.V., Théra, M.: Error bounds in metric spaces and application to the perturbation stability of metric regularity. SIAM J. Optim. 19, 1–20 (2008)MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 130–143 (1976)MathSciNetMATHCrossRefGoogle Scholar
  31. 31.
    Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)MathSciNetMATHCrossRefGoogle Scholar
  32. 32.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press (1970)Google Scholar
  33. 33.
    Rockafellar, R.T., Wets, R.: Variational Analysis. Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), vol. 317. Springer, Berlin (1998)Google Scholar
  34. 34.
    Ursescu, C.: Inherited openness. Rev. Roumaine Math. Pures Appl. 41, 5–6, 401–416 (1996)MathSciNetGoogle Scholar
  35. 35.
    Ye, J.J., Ye, X.Y.: Necessary optimality conditions for optimization problems with variational inequality constraints. Math. Oper. Res. 22, 977–997 (1997)MathSciNetMATHCrossRefGoogle Scholar
  36. 36.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, Singapore (2002)MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  1. 1.Faculty of Mathematics“Al. I. Cuza” UniversityIaşiRomania
  2. 2.Department of Mathematics“Gh. Asachi” Technical UniversityIaşiRomania

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