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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
Article

Abstract

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.

Keywords

Moreau conjugation Closed convex function Evenly convex function 

Notes

Acknowledgements

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.

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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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