Advertisement

Set-Valued and Variational Analysis

, Volume 24, Issue 2, pp 253–275 | Cite as

Complete Duality for Quasiconvex and Convex Set-Valued Functions

  • Samuel Drapeau
  • Andreas H. Hamel
  • Michael Kupper
Article
  • 153 Downloads

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.

Keywords

Set-valued functions Quasiconvexity Dual representation Increasing functions Fenchel-Moreau 

Mathematics Subject Classification (2010)

49N15 46A20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Benoist, J., Popovici, N.: Characterizations of convex and quasiconvex set-valued maps. Math. Meth. Oper. Res. 57(3), 427–435 (2003)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Benoist, J., Borwein, J., Popovici, N.: A characterization of quasiconvex vector-valued functions. Proc. Am. Math. Soc. 131(4), 1109–1113 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Uncertainty averse preferences. J. Econ. Taxon. Bot. 146(4), 1275–1330 (2011a)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Cerreia-Vioglio, S., Maccheroni, F., Marinacci, M., Montrucchio, L.: Complete monotone quasiconcave duality. Math. Oper. Res. 36(2), 321–339 (2011b)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Drapeau, S., Kupper, M.: Risk preferences and their robust representation. Math. Oper. Res. 28(1), 28–62 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Hamel, A.: A duality theory for set-valued functions I: Fenchel conjugation theory. Set-Valued and Variational Analysis 17, 153–182 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Hamel, A.H., Heyde, F.: Duality for set-valued measures of risk. SIAM Journal of Financial Mathematics 1(1), 66–95 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Hamel, A.H., Heyde, F., Rudloff, B.: Set-valued risk measures for conical market models. Math. Finan. Econ. 5(1), 1–28 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Hamel, A.H., Löhne, A.: Lagrange duality in set optimization. J. Optim. Theory Appl. 161(2), 368–397 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Jouini, E., Meddeb, M., Touzi, N.: Vector-valued coherent risk measures. Finance Stochast. 8(4), 531–552 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Löhne, A.: Vector optimization with infimum and supremum. Springer (2011)Google Scholar
  12. 12.
    Luc, D.T.: Connectedness of the efficient point sets in quasiconcave vector maximization. J. Math. Anal. Appl. 122(2), 346–354 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Penot, J.-P., Volle, M.: On quasi-convex duality. Math. Oper. Res. 15, 597–625 (1990a)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Penot, J.-P., Volle, M.: Inversion of real-valued functions and applications. Math. Meth. Oper. Res. 34(2), 117–141 (1990b)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Shephard, R.W.: Theory of cost and production functions. Princeton University Press, Princeton (1970)zbMATHGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  • Samuel Drapeau
    • 1
  • Andreas H. Hamel
    • 2
  • Michael Kupper
    • 3
  1. 1.SAIF/CAFR and Mathematics DepartmentShanghai Jiao Tong UniversityShanghaiPeople’s Republic of China
  2. 2.Free University of BolzanoBrunicoItaly
  3. 3.Konstanz UniversityKonstanzGermany

Personalised recommendations