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Duality for Closed Convex Functions and Evenly Convex Functions

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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.

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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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Authors and Affiliations

  1. Laboratoire de Mathématiques d’Avignon (EA 2151), Avignon University, 84000, Avignon, France

    M. Volle

  2. Departament d’Economia i d’Història Econòmica, Universitat Autònoma de Barcelona, 08193, Bellaterra, Spain

    J. E. Martínez-Legaz

  3. Department of Applied Mathematics, University of New South Wales, 2052, Sydney, Australia

    J. Vicente-Pérez

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  1. M. Volle
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  2. J. E. Martínez-Legaz
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  3. J. Vicente-Pérez
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Correspondence to J. E. Martínez-Legaz.

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Volle, M., Martínez-Legaz, J.E. & Vicente-Pérez, J. Duality for Closed Convex Functions and Evenly Convex Functions. J Optim Theory Appl 167, 985–997 (2015). https://doi.org/10.1007/s10957-013-0395-4

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  • DOI: https://doi.org/10.1007/s10957-013-0395-4

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