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Representative functions of maximally monotone operators and bifunctions

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Abstract

The aim of this paper is to show that every representative function of a maximally monotone operator is the Fitzpatrick transform of a bifunction corresponding to the operator. In fact, for each representative function \(\varphi \) of the operator, there is a family of equivalent saddle functions (i.e., bifunctions which are concave in the first and convex in the second argument) each of which has \(\varphi \) as Fitzpatrick transform. In the special case where \(\varphi \) is the Fitzpatrick function of the operator, the family of equivalent saddle functions is explicitly constructed. In this way we exhibit the relation between the recent theory of representative functions, and the much older theory of saddle functions initiated by Rockafellar.

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Acknowledgments

The authors would like to thank the referees for their suggestions that led to the improvement of the paper.

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Authors and Affiliations

  1. Università Cattolica del Sacro Cuore, Milan, Italy

    Monica Bianchi

  2. King Fahd University of Petroleum and Minerals, Dhahran, Kingdom of Saudi Arabia

    Nicolas Hadjisavvas

  3. Università degli Studi di Milano–Bicocca, Milan, Italy

    Rita Pini

Authors
  1. Monica Bianchi
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  2. Nicolas Hadjisavvas
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  3. Rita Pini
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Corresponding author

Correspondence to Rita Pini.

Additional information

Part of this work was done when Nicolas Hadjisavvas was visiting the Università Cattolica del Sacro Cuore, and the Università degli Studi di Milano–Bicocca, Italy. The author wishes to thank the Universities for their hospitality. Nicolas Hadjisavvas was supported by the startup research Grant No. SR141001 of the King Fahd University of Petroleum and Minerals.

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Bianchi, M., Hadjisavvas, N. & Pini, R. Representative functions of maximally monotone operators and bifunctions. Math. Program. 168, 433–448 (2018). https://doi.org/10.1007/s10107-016-1020-8

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  • DOI: https://doi.org/10.1007/s10107-016-1020-8

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