Skip to main content
SpringerLink
Log in

Convexity Criteria for Set-Valued Maps

  • Published:
Set-Valued Analysis Aims and scope Submit manuscript

Abstract

The paper gives necessary and sufficient conditions for a set-valued map F between Banach spaces X and Y to be convex with respect to a convex cone K⊂Y These conditions are written in terms of the contingent derivative of the map \(\widehat F( \cdot ): = F( \cdot ) + K\)

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Aubin, J.-P. and Frankowska, H.: Set-Valued Analysis, Birkhäuser, Basel, 1990.

    Google Scholar 

  2. Borwein, J.: Weak tangent cone and optimization in Banach spaces, SIAM J. Control Optim. 16 (1978), 512–522.

    Google Scholar 

  3. Correa, R., Jofre, A. and Thibault, L.: Characterization of lower semicontinuous convex function, Proc. Amer. Math. Soc. 116 (1992), 67–72.

    Google Scholar 

  4. Ellaia, E. and Hassouni, H.: Characterization of nonsmooth functions through their generalized gradients, Optimization 22 (1991), 401–416.

    Google Scholar 

  5. Fabian, M.: Subdifferentials, local ε-supports and Asplund spaces, J. London Math. Soc. (2) 34 (1986), 568–576.

    Google Scholar 

  6. Fabian, M.: On classes of subdifferentiability spaces of Ioffe, Nonlinear Anal. Theory, Method Appl. 12 (1988), 63–74.

    Google Scholar 

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

    Google Scholar 

  8. Fabian, M. and Zivkov, N. V.: A characterization of Asplund spaces with help of local ε-support of Ekeland–Lebourg, C. R. Acad. Bulgare Sci. 38 (1985), 617–674.

    Google Scholar 

  9. Karamardian, S. and Schaible, S.: Seven kinds of monotone maps, J. Optim. Theory Appl. 66 (1990), 37–46.

    Google Scholar 

  10. Luc, D. T.: Theory of Vector Optimization, Lecture Notes in Econ. Math. Syst. 319, Springer-Verlag, New York, 1989. SVAN248.tex; 3/04/1997; 14:33; v.5; p.8

    Google Scholar 

  11. Luc, D. T. and Swaminathan, S.: A characterization of convex functions, Nonlinear Anal. Theory, Method Appl. 30 (1993), 697–701.

    Google Scholar 

  12. Luc, D. T.: Characterization of quasiconvex functions, Bull. Austral. Math. Soc. 48 (1993), 393–405.

    Google Scholar 

  13. Penot, J.-P. and Sach, P. H.: Generalized monotonicity of subdifferentials and generalized convexity, Université de Pau et des Pays de l'Adour, Preprint No. 94/16.

  14. Rudin, W.: Functional Analysis, McGraw-Hill, New York, 1973.

    Google Scholar 

Download references

Authors
  1. Pham Huu Sach
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nguyen Dong Yen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Huu Sach, P., Dong Yen, N. Convexity Criteria for Set-Valued Maps. Set-Valued Analysis 5, 37–45 (1997). https://doi.org/10.1023/A:1008657617548

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1008657617548

Search

Navigation