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Subdifferential of the Supremum via Compactification of the Index Set

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Abstract

We give new characterizations for the subdifferential of the supremum of an arbitrary family of convex functions, dropping out the standard assumptions of compactness of the index set and upper semi-continuity of the functions with respect to the index (J. Convex Anal. 26, 299–324, 2019). We develop an approach based on the compactification of the index set, giving rise to an appropriate enlargement of the original family. Moreover, in contrast to the previous results in the literature, our characterizations are formulated exclusively in terms of exact subdifferentials at the nominal point. Fritz–John and KKT conditions are derived for convex semi-infinite programming.

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Acknowledgements

Research supported by CONICYT (Fondecyt 1190012 and 1190110), Proyecto/Grant PIA AFB-170001, MICIU of Spain and Universidad de Alicante (Grant Beatriz Galindo BEA- GAL 18/00205), and Research Project PGC2018-097960-B-C21 from MICINN, Spain. The research of the third author is also supported by the Australian ARC - Discovery Projects DP 180100602.

The authors wish to thank the referee for the valuable comments and suggestions which have contributed to improve the first version of this paper.

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Authors and Affiliations

  1. Universidad de O’Higgins, Rancagua, Chile

    R. Correa

  2. DIM-CMM of Universidad de Chile, Santiago, Chile

    R. Correa

  3. Center for Mathematical Modeling (CMM), Universidad de Chile, Santiago, Chile

    A. Hantoute

  4. Universidad de Alicante, Alicante, Spain

    A. Hantoute & M. A. López

  5. CIAO, Federation University, Ballarat, Australia

    M. A. López

Authors
  1. R. Correa
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  2. A. Hantoute
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  3. M. A. López
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Correspondence to A. Hantoute.

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Dedicated by his coauthors to Prof. Marco A. López on his 70th birthday.

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Correa, R., Hantoute, A. & López, M.A. Subdifferential of the Supremum via Compactification of the Index Set. Vietnam J. Math. 48, 569–588 (2020). https://doi.org/10.1007/s10013-020-00403-5

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  • DOI: https://doi.org/10.1007/s10013-020-00403-5

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