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About Extensions of the Extremal Principle

Vietnam Journal of Mathematics volume 46, pages 215–242 (2018)Cite this article

Abstract

In this paper, after recalling and discussing the conventional extremality, local extremality, stationarity and approximate stationarity properties of collections of sets, and the corresponding (extended) extremal principle, we focus on extensions of these properties and the corresponding dual conditions with the goal to refine the main arguments used in this type of results, clarify the relationships between different extensions, and expand the applicability of the generalized separation results. We introduce and study new more universal concepts of relative extremality and stationarity and formulate the relative extended extremal principle. Among other things, certain stability of the relative approximate stationarity is proved. Some links are established between the relative extremality and stationarity properties of collections of sets and (the absence of) certain regularity, lower semicontinuity, and Lipschitz-like properties of set-valued mappings.

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References

  1. Apetrii, M., Durea, M., Strugariu, R.: On subregularity properties of set-valued mappings. Set-Valued Var. Anal. 21, 93–126 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  2. Borwein, J. M., Jofré, A.: A nonconvex separation property in Banach spaces. Math. Methods Oper. Res. 48, 169–179 (1998)

    MathSciNet  Article  MATH  Google Scholar 

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

    MATH  Google Scholar 

  4. Cibulka, R., Fabian, M., Kruger, A. Y.: On semiregularity of mappings. arXiv:1711.04420 (2017)

  5. Dontchev, A. L., Rockafellar, R. T.: Implicit Functions and Solution Mappings: A View from Variational Analysis, 2nd edn. Springer Series in Operations Research and Financial Engineering. Springer, New York (2014)

    MATH  Google Scholar 

  6. Dubovitskii, A.Ya., Milyutin, A.A.: Extremum problems in the presence of restrictions. USSR Comp. Maths. Math. Phys. 5, 1–80 (1965)

    Article  MATH  Google Scholar 

  7. Fabian, M.: Subdifferentials, local ε-supports and Asplund spaces. J. Lond. Math. Soc. s2-34, 568–576 (1986)

    MathSciNet  Article  MATH  Google Scholar 

  8. Fabian, M.: Subdifferentiability and trustworthiness in the light of a new variational principle of Borwein and Preiss. Acta Univ. Carol. 30, 51–56 (1989)

    MathSciNet  MATH  Google Scholar 

  9. Ioffe, A. D.: Fuzzy principles and characterization of trustworthiness. Set-Valued Anal. 6, 265–276 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  10. Ioffe, A. D.: Metric regularity and subdifferential calculus. Russ. Math. Surveys 55, 501–558 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  11. Ioffe, A. D.: Transversality in variational analysis. J. Optim. Theory Appl. 174, 343–366 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  12. Klatte, D., Kummer, B.: Nonsmooth Equations in Optimization: Regularity, Calculus, Methods and Applications Nonconvex. Optimization and its Applications, vol. 60. Kluwer Academic Publishers, Dordrecht (2002)

    MATH  Google Scholar 

  13. Kruger, A.Y.: Generalized differentials of nonsmooth functions. VINITI no. 1332-81. Minsk, 67 pp. (In Russian). Available at: https://asterius.ballarat.edu.au/akruger/research/publications.html (1981)

  14. Kruger, A.Y.: ε-semidifferentials and ε-normal elements. VINITI no. 1331-81. Minsk, 76 pp. (In Russian). Available at https://asterius.ballarat.edu.au/akruger/research/publications.html (1981)

  15. Kruger, A. Y.: Generalized differentials of nonsmooth functions and necessary conditions for an extremum. Sibirsk. Mat. Zh. 26, 78–90 (1985). In Russian; English transl.: Sib. Math. J. 26, 370–379 (1985)

    MathSciNet  MATH  Google Scholar 

  16. Kruger, A.Y.: About extremality of systems of sets. Dokl. Nats. Akad. Nauk Belarusi 42, 24–28 (1998). In Russian. Available at https://asterius.ballarat.edu.au/akruger/research/publications.html

    MathSciNet  MATH  Google Scholar 

  17. Kruger, A.Y.: Strict (ε,δ)-semidifferentials and extremality of sets and functions. Dokl. Nats. Akad. Nauk Belarusi 44, 19–22 (2000). In Russian. Available at https://asterius.ballarat.edu.au/akruger/research/publications.html

    MathSciNet  MATH  Google Scholar 

  18. Kruger, A. Y.: Strict (ε,δ)-subdifferentials and extremality conditions. Optimization 51, 539–554 (2002)

    MathSciNet  Article  MATH  Google Scholar 

  19. Kruger, A. Y.: On Fréchet subdifferentials. J. Math. Sci. 116, 3325–3358 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  20. Kruger, A. Y.: Weak stationarity: Eliminating the gap between necessary and sufficient conditions. Optimization 53, 147–164 (2004)

    MathSciNet  Article  MATH  Google Scholar 

  21. Kruger, A. Y.: Stationarity and regularity of set systems. Pac. J. Optim. 1, 101–126 (2005)

    MathSciNet  MATH  Google Scholar 

  22. Kruger, A. Y.: About regularity of collections of sets. Set-Valued Anal. 14, 187–206 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  23. Kruger, A. Y.: About stationarity and regularity in variational analysis. Taiwan. J. Math. 13, 1737–1785 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  24. Kruger, A. Y., López, M. A.: Stationarity and regularity of infinite collections of sets. J. Optim. Theory Appl. 154, 339–369 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  25. Kruger, A. Y., Luke, D. R., Thao, N. H.: About subtransversality of collections of sets. Set-Valued Var. Anal. 25, 701–729 (2017)

    MathSciNet  Article  MATH  Google Scholar 

  26. Kruger, A.Y., Mordukhovich, B.S.: Extremal points and the Euler equation in nonsmooth optimization problems. Dokl. Akad. Nauk BSSR 24(8), 684–687 (1980). In Russian. Available at https://asterius.ballarat.edu.au/akruger/research/publications.html

    MathSciNet  MATH  Google Scholar 

  27. Kruger, A.Y., Mordukhovich, B.S.: Generalized normals and derivatives and necessary conditions for an extremum in problems of nondifferentiable programming. II. VINITI no. 494-80, 60 pp. Minsk (1980). In Russian. Available at https://asterius.ballarat.edu.au/akruger/research/publications.html

  28. Kruger, A. Y., Thao, N. H.: About uniform regularity of collections of sets. Serdica Math. J. 39, 287–312 (2013)

    MathSciNet  MATH  Google Scholar 

  29. Kruger, A. Y., Thao, N. H.: Quantitative characterizations of regularity properties of collections of sets. J. Optim. Theory Appl. 164, 41–67 (2015)

    MathSciNet  Article  MATH  Google Scholar 

  30. Mordukhovich, B. S.: Variational Analysis and Generalized Differentiation. I: Basic Theory. Grundlehren der Mathematischen Wissenschaften, vol. 330. Springer, Berlin Heidelberg (2006)

    Google Scholar 

  31. Mordukhovich, B. S., Shao, Y.: Extremal characterizations of Asplund spaces. Proc. Am. Math. Soc. 124, 197–205 (1996)

    MathSciNet  Article  MATH  Google Scholar 

  32. Phelps, R. R.: Convex Functions, Monotone Operators and Differentiability. Lecture Notes in Mathematics, 2nd edn., vol. 1364. Springer-Verlag, Berlin Heidelberg (1993)

  33. Rockafellar, R. T., Wets, R. J. -B.: Variational Analysis. Springer, Berlin Heidelberg (1998)

    Book  MATH  Google Scholar 

  34. Pierra, G.: Decomposition through formalization in a product space. Math. Program. 28, 96–115 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  35. Zheng, X. Y., Ng, K. F.: The Lagrange multiplier rule for multifunctions in Banach spaces. SIAM J. Optim. 17, 1154–1175 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  36. Zheng, X. Y., Ng, K. F.: A unified separation theorem for closed sets in a Banach space and optimality conditions for vector optimization. SIAM J. Optim. 21, 886–911 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  37. Zheng, X. Y., Yang, Z., Zou, J.: Exact separation theorem for closed sets in Asplund spaces. Optimization 66, 1065–1077 (2017)

    MathSciNet  Article  MATH  Google Scholar 

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Acknowledgements

The authors thank the referees for careful reading of the manuscript and their constructive comments and suggestions.

Funding

The research was partially supported by the Australian Research Council, project DP160100854. Hoa T. Bui is supported by an Australian Government Research Training Program (RTP) Stipend and RTP Fee-Offset Scholarship through Federation University Australia.

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Authors and Affiliations

  1. Centre for Informatics and Applied Optimization, Faculty of Science and Technology, Federation University Australia, POB 663, Ballarat, VIC, 3350, Australia

    Hoa T. Bui & Alexander Y. Kruger

Authors
  1. Hoa T. Bui
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  2. Alexander Y. Kruger
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Corresponding author

Correspondence to Alexander Y. Kruger.

Additional information

Dedicated to Professor Michel Théra on the occasion of his 70th birthday.

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Bui, H.T., Kruger, A.Y. About Extensions of the Extremal Principle. Vietnam J. Math. 46, 215–242 (2018). https://doi.org/10.1007/s10013-018-0278-y

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  • DOI: https://doi.org/10.1007/s10013-018-0278-y

Keywords

  • Extremality
  • Stationarity
  • Transversality
  • Regularity
  • Separability
  • Extremal principle
  • Ekeland variational principle

Mathematics Subject Classification (2010)

  • Primary: 49J52
  • 49J53
  • Secondary: 49K40
  • 90C30
  • 90C46