Skip to main content
SpringerLink
Log in

Complete Duality for Quasiconvex and Convex Set-Valued Functions

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

Abstract

This paper provides a unique dual representation of set-valued lower semi-continuous quasiconvex and convex functions. The results are based on a duality result for increasing set-valued functions.

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. Benoist, J., Popovici, N.: Characterizations of convex and quasiconvex set-valued maps. Math. Meth. Oper. Res. 57(3), 427–435 (2003)

    MathSciNet  MATH  Google Scholar 

  2. Benoist, J., Borwein, J., Popovici, N.: A characterization of quasiconvex vector-valued functions. Proc. Am. Math. Soc. 131(4), 1109–1113 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  3. Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Uncertainty averse preferences. J. Econ. Taxon. Bot. 146(4), 1275–1330 (2011a)

    MathSciNet  MATH  Google Scholar 

  4. Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Complete monotone quasiconcave duality. Math. Oper. Res. 36(2), 321–339 (2011b)

    Article  MathSciNet  MATH  Google Scholar 

  5. Drapeau, S., Kupper, M.: Risk preferences and their robust representation. Math. Oper. Res. 28(1), 28–62 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  6. Hamel, A.: A duality theory for set-valued functions I: Fenchel conjugation theory. Set-Valued and Variational Analysis 17, 153–182 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Hamel, A.H., Heyde, F.: Duality for set-valued measures of risk. SIAM Journal of Financial Mathematics 1(1), 66–95 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Hamel, A.H., Heyde, F., Rudloff, B.: Set-valued risk measures for conical market models. Math. Finan. Econ. 5(1), 1–28 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Hamel, A.H., Löhne, A.: Lagrange duality in set optimization. J. Optim. Theory Appl. 161(2), 368–397 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  10. Jouini, E., Meddeb, M., Touzi, N.: Vector-valued coherent risk measures. Finance Stochast. 8(4), 531–552 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Löhne, A.: Vector optimization with infimum and supremum. Springer (2011)

  12. Luc, D.T.: Connectedness of the efficient point sets in quasiconcave vector maximization. J. Math. Anal. Appl. 122(2), 346–354 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  13. Penot, J.-P., Volle, M.: On quasi-convex duality. Math. Oper. Res. 15, 597–625 (1990a)

    Article  MathSciNet  MATH  Google Scholar 

  14. Penot, J.-P., Volle, M.: Inversion of real-valued functions and applications. Math. Meth. Oper. Res. 34(2), 117–141 (1990b)

    Article  MathSciNet  MATH  Google Scholar 

  15. Shephard, R.W.: Theory of cost and production functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. SAIF/CAFR and Mathematics Department, Shanghai Jiao Tong University, 211 West Huaihai Road, Shanghai, 200030, People’s Republic of China

    Samuel Drapeau

  2. Free University of Bolzano, Piazzetta dell’Università 1, 39031, Brunico, Italy

    Andreas H. Hamel

  3. Konstanz University, Universitätsstraße 10, Konstanz, 78464, Germany

    Michael Kupper

Authors
  1. Samuel Drapeau
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Andreas H. Hamel
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael Kupper
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Samuel Drapeau.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Drapeau, S., Hamel, A.H. & Kupper, M. Complete Duality for Quasiconvex and Convex Set-Valued Functions. Set-Valued Var. Anal 24, 253–275 (2016). https://doi.org/10.1007/s11228-015-0332-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11228-015-0332-9

Keywords

Mathematics Subject Classification (2010)

Search

Navigation