Journal of Optimization Theory and Applications

, Volume 167, Issue 3, pp 985–997 | Cite as

Duality for Closed Convex Functions and Evenly Convex Functions

  • M. Volle
  • J. E. Martínez-Legaz
  • J. Vicente-Pérez


We introduce two Moreau conjugacies for extended real-valued functions h on a separated locally convex space. In the first scheme, the biconjugate of h coincides with its closed convex hull, whereas, for the second scheme, the biconjugate of h is the evenly convex hull of h. In both cases, the biconjugate coincides with the supremum of the minorants of h that are either continuous affine or closed (respectively, open) halfspaces valley functions.


Moreau conjugation Closed convex function Evenly convex function 



The authors are grateful to the two anonymous referees and the editor for their constructive comments which have contributed to the final presentation of the paper. J.E. Martínez-Legaz has been supported by the MICINN of Spain, Grant MTM2011-29064-C03-01. He is affiliated to MOVE (Markets, Organizations and Votes in Economics). J. Vicente-Pérez has been supported by the MICINN of Spain, Grant MTM2011-29064-C03-02, and the Australian Research Council.


  1. 1.
    Moreau, J.J.: Inf-convolution, sous-additivité, convexité des fonctions numériques. J. Math. Pures Appl. 49, 109–154 (1970) MATHMathSciNetGoogle Scholar
  2. 2.
    Zălinescu, C.: Convex Analysis in General Vector Spaces. World Scientific, River Edge (2002) MATHCrossRefGoogle Scholar
  3. 3.
    Martínez-Legaz, J.E., Vicente-Pérez, J.: The e-support function of an e-convex set and conjugacy for e-convex functions. J. Math. Anal. Appl. 376, 602–612 (2011) MATHMathSciNetCrossRefGoogle Scholar
  4. 4.
    Martínez-Legaz, J.E.: Conjugation associated with a graph. In: Proceedings of the Ninth Spanish–Portuguese Conference on Mathematics, Vol. 2, Salamanca, 1982. Acta Salmanticensia. Ciencias, vol. 46, pp. 837–839. University of Salamanca, Salamanca (1982) (in Spanish) Google Scholar
  5. 5.
    Ekeland, I., Temam, R.: Analyse Convexe et Problèmes Variationnels. Collection Études Mathématiques. Dunod/Gauthier-Villars, Paris/Brussels (1974) Google Scholar
  6. 6.
    Penot, J.P.: What is quasiconvex analysis? Optimization 47, 35–110 (2000) MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Rodríguez, M.M.L., Vicente-Pérez, J.: On evenly convex functions. J. Convex Anal. 18, 721–736 (2011) MATHMathSciNetGoogle Scholar
  8. 8.
    Fenchel, W.: Convex Cones, Sets and Functions. Lecture Notes. Princeton University, Princeton (1953) Google Scholar
  9. 9.
    Goberna, M.A., Rodríguez, M.M.L.: Analyzing linear systems containing strict inequalities via evenly convex hulls. Eur. J. Oper. Res. 169, 1079–1095 (2006) MATHCrossRefGoogle Scholar
  10. 10.
    Crouzeix, J.P.: Contribution à l’étude des fonctions quasiconvexes. Thèse, Univ. Clermont-Ferrand II (1977) Google Scholar
  11. 11.
    Martínez-Legaz, J.E.: Exact quasiconvex conjugation. ZOR, Z. Oper.-Res. 27, A257–A266 (1983) Google Scholar
  12. 12.
    Passy, U., Prisman, E.Z.: Conjugacy in quasiconvex programming. Math. Program. 30, 121–146 (1984) MATHMathSciNetCrossRefGoogle Scholar
  13. 13.
    Penot, J.P., Volle, M.: On quasi-convex duality. Math. Oper. Res. 15, 597–625 (1990) MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Singer, I.: Abstract Convex Analysis. Canadian Mathematical Society Series of Monographs and Advanced Texts. Wiley, New York (1997) MATHGoogle Scholar
  15. 15.
    Volle, M.: Conjugaison par tranches. Ann. Mat. Pura Appl. 139, 279–311 (1985) MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • M. Volle
    • 1
  • J. E. Martínez-Legaz
    • 2
  • J. Vicente-Pérez
    • 3
  1. 1.Laboratoire de Mathématiques d’Avignon (EA 2151)Avignon UniversityAvignonFrance
  2. 2.Departament d’Economia i d’Història EconòmicaUniversitat Autònoma de BarcelonaBellaterraSpain
  3. 3.Department of Applied MathematicsUniversity of New South WalesSydneyAustralia

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