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Formulae for the Conjugate and the Subdifferential of the Supremum Function

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Abstract

This paper aims at providing some formulae for the subdifferential and the conjungate function of the supremum function over an arbitrary family of functions. The work is principally motivated by the case when data functions are lower semicontinuous proper and convex. Nevertheless, we explore the case when the family of functions is arbitrary, but satisfying that the biconjugate of the supremum functions is equal to the supremum of the biconjugate of the data functions. The study focuses its attention on functions defined in finite-dimensional spaces; in this case, the formulae can be simplified under certain qualification conditions. However, we show how to extend these results to arbitrary locally convex spaces, without any qualification condition.

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Acknowledgements

We would like to thank to Marco A. López Cérda for his nice discussions and great comments about this manuscript, which improved notably the quality of the presented manuscript. Also, we would like to thank the anonymous referees for their detailed comments and suggestions for the manuscript.

This work was supported in part by CONICYT-PCHA doctorado Nacional 2014-21140621.

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  1. Instituto de Ciencias de la Ingeniería, Universidad de O’higgins, Rancagua, Chile

    Pedro Pérez-Aros

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  1. Pedro Pérez-Aros
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Correspondence to Pedro Pérez-Aros.

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Communicated by Marco Antonio López-Cerdá.

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Pérez-Aros, P. Formulae for the Conjugate and the Subdifferential of the Supremum Function. J Optim Theory Appl 180, 397–427 (2019). https://doi.org/10.1007/s10957-018-1350-1

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