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Metric regularity of epigraphical multivalued mappings and applications to vector optimization

Abstract

In this work we combine in a meaningful way two techniques of variational analysis and nonsmooth optimization. On one hand, we use the error bound approach to study the metric regularity of some special types of multifunctions and, on the other hand, we exploit the incompatibility between the metric regularity and the Pareto minimality. This method allows us to present some \(\varepsilon \)-Fermat rules for set-valued optimization problem in the setting of general Banach spaces. Our results are comparable to several recent results in literature.

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References

  1. Aubin, J.P., Frankowska, H.: Set-Valued Analysis. Birkäuser, Basel (1990)

    MATH  Google Scholar 

  2. Azé, D.: A survey on error bounds for lower semicontinuous functions. In: Proceedings of the 2003 MODE-SMAI Conference, ESAIM Proc., 1, EDP Sci., Les Ulis, pp. 1–17 (2003)

  3. 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  MathSciNet  MATH  Google Scholar 

  4. Azé, D., Corvellec, J.-N., Luccheti, R.E.: Variational pairs and applications to stability in nonsmooth analysis. Nonlin. Anal. 49, 643–670 (2002)

    Article  MATH  Google Scholar 

  5. Bao, T.Q., Mordukhovich, B.S.: Relative Pareto minimizers for multiobjective problems: existence and optimality conditions. Math. Program. Ser. A 122, 301–347 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Borwein, J.M., Zhu, Q.J.: Techniques of Variational Analysis. Springer, New York (2005)

    MATH  Google Scholar 

  7. Borwein, J.M., Zhuang, D.M.: Verifiable necessary and sufficient conditions for openness and regularity of set-valued maps. J. Math. Anal. Appl. 134, 441–459 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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)

    MathSciNet  MATH  Google Scholar 

  9. Dmitruk, A.V.: On a nonlocal metric regularity of nonlinear operators. Control Cybern. 34, 723–746 (2005)

    MathSciNet  MATH  Google Scholar 

  10. Dmitruk, A.V., Milyutin, A.A., Osmolovskiĭ, N.P.: Lyusternik’s theorem and the theory of extrema. Uspekhi Mat. Nauk 35, 11–46 (1980) (in Russian); English translation in. Russian Math. Surv. 35, 11–51 (1980)

    Google Scholar 

  11. Dontchev, A.L., Frankowska, H.: Lyusternik-Graves theorem and fixed points. Proc. Am. Math. Soc. 139, 521–534 (2010)

    Article  MathSciNet  Google Scholar 

  12. Dontchev, A.L., Rockafellar, R.T.: Implicit Functions and Solution Mappings. Springer, Berlin (2009)

    Book  MATH  Google Scholar 

  13. Dontchev, A.L., Lewis, A.S., Rockafellar, R.T.: The radius of metric regularity. Trans. Am. Math. Soc. 355, 493–517 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  14. Durea, M., Strugariu, R.: On some Fermat rules for set-valued optimization problems. Optimization 60, 575–591 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  15. Durea, M., Strugariu, R.: Optimality conditions in terms of Bouligand derivatives for Pareto efficiency in set-valued optimization. Optim. Lett. 5, 141–151 (2010)

    Article  MathSciNet  Google Scholar 

  16. Ekeland, I.: On the variational principle. J. Math. Anal. Appl. 47, 324–353 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  17. Graves, L.M.: Some mapping theorems. Duke Math. J. 17, 111–114 (1950)

    Article  MathSciNet  MATH  Google Scholar 

  18. Halkin, H., Neustadt, L.W.: General necessary conditions for optimization problems. Proc. Natl. Acad. Sci. USA 56, 1066–1071 (1966)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ioffe, A.D.: Metric regularity and subdifferential calculus. Uspekhi Mat. Nauk 55, 103–162 (2000); English translation in Math. Surv. 55, 501–558 (2000)

  20. Ioffe, A.D.: Towards variational analysis in metric spaces: metric regularity and fixed points. Math. Programm. Ser. B 123, 241–252 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  21. Jourani, A., Thibault, L.: Metric regularity for strongly compactly Lipschitzian mappings. Nonlinear Anal. TMA 24, 229–240 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Jourani, A., Thibault, L.: Verifiable conditions for openness and metric regularity of multivalued mappings in Banach spaces. Trans. Am. Math. Soc. 347, 1255–1268 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  23. Jourani, A., Thibault, L.: Coderivatives of multivalued mappings, locally compact cones and metric regularity. Nonlinear Anal. TMA 35, 925–945 (1998)

    Article  MathSciNet  Google Scholar 

  24. Klatte, D., Kummer, B.: Nonsmooth equations in optimization. Regularity, calculus, methods and applications. Nonconvex Optim. Appl. 60 (2002) Kluwer Academic Publishers, Dordrecht

  25. Ledyaev, Y., Zhu, Q.: Implicit multifunction theorems. Set-Valued Anal. 7, 209–238 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  26. Lyusternik, L.A.: On conditional extrema of functionals. Math. Sbornik 41, 390–401 (1934). (in Russian)

    MATH  Google Scholar 

  27. Mordukhovich, B.S.: Metric approximations and necessary optimality conditions for general classes of extremal problems. Soviet Math. Dokl. 22, 526–530 (1980)

    MATH  Google Scholar 

  28. 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, 331, Berlin (2006)

  29. Mordukhovich, B.S., Shao, Y.: Differential characterizations of covering, metric regularity, and Lipschitzian properties of multifunctions between Banach spaces. Nonlinear Anal. TMA 25, 1401–1424 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  30. Mordukhovich, B.S., Shao, Y.: Nonsmooth sequential analysis in Asplund spaces. Trans. Am. Math. Soc. 348, 1235–1280 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  31. Mordukhovich, B.S., Shao, Y.: Stability of set-valued mappings in infinite dimensions: point criteria and applications. SIAM J. Control Optim. 35, 285–314 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  32. 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)

    Article  MathSciNet  MATH  Google Scholar 

  33. Ngai, H.V., Nguyen, H.T., Théra, M.: Implicit multifunction theorems in complete metric spaces. Math. Program. (to appear)

  34. Ng, K.F., Zheng, X.Y.: Error bound for lower semicontinuous functions in normed spaces. SIAM J. Optim. 12, 1–17 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  35. Penot, J.-P.: Regularity, openness and Lipschitzian behavior of multifunctions. Nonlinear Anal. 13, 629–643 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  36. Penot, J.-P.: Compactness properties, openness criteria and coderivatives. Set-Valued Anal. 6, 363–380 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  37. Robinson, S.M.: Extension of Newton’s method to nonlinear functions with values in a cone. Numer. Math. 19, 341–347 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  38. Robinson, S.M.: Regularity and stability for convex multivalued functions. Math. Oper. Res. 1, 130–143 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  39. Robinson, S.M.: Strongly regular generalized equations. Math. Oper. Res. 5, 43–62 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  40. Rockafellar, R.T., Wets, R.: Variational Analysis. Grundlehren der mathematischen Wissenschaften (A Series of Comprehensive Studies in Mathematics), vol. 317. Springer, Berlin (1998)

  41. Ursescu, C.: Multifunctions with closed convex graphs. Czech. Math. J. 25, 438–441 (1975)

    MathSciNet  Google Scholar 

  42. Ursescu, C.: Inherited openness. Revue Roumaine des Mathématiques Pures et Appliquées 41, 401–416 (1996)

    MathSciNet  MATH  Google Scholar 

  43. Wu, Z., Ye, J.: On error bounds for lower semicontinuous functions. Math. Program. 92, 301–314 (2002)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  45. Zheng, X.Y., Ng, K.F.: The Fermat rule for multifunctions on Banach spaces. Math. Program. Ser. A 104, 69–90 (2005)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

The second author thanks Professor Constantin Zălinescu for meaningful help and interesting discussions while he visited “Al. I. Cuza” University of Iaşi with a research grant offered by Romanian Government within Eugen Ionescu programme coordinated by Agence Universitaire de la Francophonie (AUF). Research of the second author was partially supported by a doctoral grant from Région Limousin and by the ECOS-SUD C\(10\)E\(08\) project. The first and the third authors were supported by a grant of the Romanian National Authority for Scientific Research, CNCS–UEFISCDI, project number PN-II-ID-PCE-2011-3-0084. All the authors are indebted to Professors Michel Théra and Huynh van Ngai for their interest and for many valuable discussions on this work.

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Correspondence to Marius Durea.

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Dedicated to Professor J. M. Borwein on the occasion of his sixtieth anniversary.

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Durea, M., Nguyen, H.T. & Strugariu, R. Metric regularity of epigraphical multivalued mappings and applications to vector optimization. Math. Program. 139, 139–159 (2013). https://doi.org/10.1007/s10107-013-0665-9

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  • DOI: https://doi.org/10.1007/s10107-013-0665-9

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