Skip to main content

Extremality of convex sets with some applications

Optimization Letters volume 11, pages 1201–1215 (2017)Cite this article

Abstract

In this paper we introduce an enhanced notion of extremal systems for sets in locally convex topological vector spaces and obtain efficient conditions for set extremality in the convex case. Then we apply this machinery to deriving new calculus results on intersection rules for normal cones to convex sets and on infimal convolutions of support functions.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

39,95 €

Price includes VAT (Finland)

References

  1. Attouch, H., Baillon, J.-B., Théra, M.: Variational sum of monotone operators. J. Convex Anal. 1, 1–29 (1994)

    MathSciNet  MATH  Google Scholar 

  2. Attouch, H., Brézis, H.: Duality of the sum of convex functions in general Banach spaces. In: Barroso, J.A. (ed.) Aspects of Mathematics and Its Applications, vol. 34, pp. 125–133. North-Holland, Amsterdam (1986)

    Chapter  Google Scholar 

  3. Bauschke, H.H., Combettes, P.L.: Convex Analysis and Monotone Operator Theory in Hilbert Spaces. Springer, New York (2011)

    Book  MATH  Google Scholar 

  4. Borwein, J.M., Lewis, A.S.: Convex Analysis and Nonlinear Optimization, 2nd edn. Springer, New York (2006)

    Book  MATH  Google Scholar 

  5. Borwein, J.M., Lucet, Y., Mordukhovich, B.S.: Compactly epi-Lipschitzian convex sets and functions in normed spaces. J. Convex Anal. 7, 375–393 (2000)

    MathSciNet  MATH  Google Scholar 

  6. Borwein, J.M., Strójwas, H.: Tangential approximations. Nonlinear Anal. 9, 1347–1366 (1985)

    Article  MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  8. Boţ, R.I.: Conjugate Duality in Convex Optimization. Springer, Berlin (2010)

    MATH  Google Scholar 

  9. Clarke, F.H.: Optimization and Nonsmooth Analysis. Wiley, New York (1983)

    MATH  Google Scholar 

  10. Fabian, M., Mordukhovich, B.S.: Sequential normal compactness versus topological normal compactness in variational analysis. Nonlinear Anal. 54, 1057–1067 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hiriart-Urruty, J.-B., Lemaréchal, C.: Convex Analysis and Minimization Algorithms I, II. Springer, Berlin (1993)

    MATH  Google Scholar 

  12. Mordukhovich, B.S.: Variational Analysis and Generalized Differentiation, I: Basic Theory, II: Applications. Springer, Berlin (2006)

    Google Scholar 

  13. Mordukhovich, B.S., Nam, N.M.: An Easy Path to Convex Analysis and Applications. Morgan & Claypool Publisher, San Rafael (2014)

    MATH  Google Scholar 

  14. Mordukhovich, B.S., Nam, N.M.: Geometric approach to convex subdifferential calculus. Optimizaion (2016). doi:10.1080/02331934.2015.1105225

  15. Phelps, R.R.: Convex Functions, Monotone Operators and Differentiability, 2nd edn. Springer, Berlin (1993)

    MATH  Google Scholar 

  16. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  17. Rockafellar, R.T.: Conjugate Duality and Optimization. SIAM, Philadelphia (1974)

    Book  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

  19. Simons, S.: From Hahn-Banach to Monotonicity, 2nd edn. Springer, Berlin (2008)

    MATH  Google Scholar 

  20. Thibault, L.: Sequential convex subdifferential calculus and sequential Lagrange multipliers. SIAM J. Control Optim. 35, 1434–1444 (1997)

    Article  MathSciNet  MATH  Google Scholar 

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

    Book  MATH  Google Scholar 

Download references

Acknowledgements

B. S. Mordukhovich: was partly supported by the National Science Foundation under Grants DMS-1007132 and DMS-1512846 and by the Air Force Office of Scientific Research under Grant #15RT0462, Ministry of Education and Science of the Russian Federation (Agreement number 02.a03.21.0008 of 24 June 2016). N. M. Nam: was partly supported by the National Science Foundation under Grant DMS-1411817. The authors are grateful to both anonymous referees for their helpful comments that allowed us to improve the original presentation.

Author information

Authors and Affiliations

  1. Department of Mathematics, Wayne State University, Detroit, MI, 48202, USA

    Boris S. Mordukhovich

  2. RUDN University, Moscow, 117198, Russia

    Boris S. Mordukhovich

  3. Fariborz Maseeh Department of Mathematics and Statistics, Portland State University, Portland, OR, 97207, USA

    Nguyen Mau Nam

Authors
  1. Boris S. Mordukhovich
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nguyen Mau Nam
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Boris S. Mordukhovich.

Additional information

Dedicated to the memory of Jonathan Michael Borwein.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Mordukhovich, B.S., Nam, N.M. Extremality of convex sets with some applications. Optim Lett 11, 1201–1215 (2017). https://doi.org/10.1007/s11590-016-1106-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11590-016-1106-5

Keywords

  • Convex and variational analysis
  • Extremal systems of sets
  • Normals to convex sets
  • Normal intersection rules
  • Support functions
  • Infimal convolutions