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Determination of convex functions via subgradients of minimal norm

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Abstract

We show, in Hilbert space setting, that any two convex proper lower semicontinuous functions bounded from below, for which the norm of their minimal subgradients coincide, they coincide up to a constant. Moreover, under classic boundary conditions, we provide the same results when the functions are continuous and defined over an open convex domain. These results show that for convex functions bounded from below, the slopes provide sufficient first-order information to determine the function up to a constant, giving a positive answer to the conjecture posed in Boulmezaoud et al. (SIAM J Optim 28(3):2049–2066, 2018) .

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Acknowledgements

We would like to thank A. Daniilidis for presenting this problem to us and let us know about the unpublished ideas of J.-B. Baillon commented in the article. We thank the anonymous referees for the useful suggestions made during the review process of this manuscript.

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Authors and Affiliations

  1. Instituto de Ciencias de la Ingeniería, Universidad de O’Higgins, Av. Libertador Bernardo O’Higgins 611, Rancagua, Chile

    Pedro Pérez-Aros, David Salas & Emilio Vilches

  2. Centro de Modelamiento Matemático, CNRS-UMI 2807, Universidad de Chile, Av. Beauchef 851, Santiago, Chile

    David Salas

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  1. Pedro Pérez-Aros
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  2. David Salas
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  3. Emilio Vilches
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Correspondence to David Salas.

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The first author was partially supported by ANID Chile under grants Fondecyt Regular 1190110 and Fondecyt Regular 1200283. The second author was partially funded by ANID Chile under grant Fondecyt post-doctorado 3190229. The third author was partially funded by ANID Chile under grant Fondecyt de Iniciación 11180098.

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Pérez-Aros, P., Salas, D. & Vilches, E. Determination of convex functions via subgradients of minimal norm. Math. Program. 190, 561–583 (2021). https://doi.org/10.1007/s10107-020-01550-w

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  • DOI: https://doi.org/10.1007/s10107-020-01550-w

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