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An Enhanced Baillon–Haddad Theorem for Convex Functions Defined on Convex Sets

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Abstract

The Baillon–Haddad theorem establishes that the gradient of a convex and continuously differentiable function defined in a Hilbert space is \(\beta \)-Lipschitz if and only if it is \(1/\beta \)-cocoercive. In this paper, we extend this theorem to Gâteaux differentiable and lower semicontinuous convex functions defined on an open convex set of a Hilbert space. Finally, we give a characterization of \(C^{1,+}\) convex functions in terms of local cocoercivity.

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Acknowledgements

The authors wish to thank the referees for providing several helpful suggestions.

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Authors and Affiliations

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

    Pedro Pérez-Aros & Emilio Vilches

  2. Instituto de Ciencias de la Educación, Universidad de O’Higgins, Rancagua, Chile

    Emilio Vilches

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  1. Pedro Pérez-Aros
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  2. Emilio Vilches
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Correspondence to Emilio Vilches.

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E. Vilches was partially funded by CONICYT Chile under grant Fondecyt de Iniciación 11180098, P. Pérez-Aros was partially supported by CONICYT Chile under grant Fondecyt regular 1190110.

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Pérez-Aros, P., Vilches, E. An Enhanced Baillon–Haddad Theorem for Convex Functions Defined on Convex Sets. Appl Math Optim 83, 2241–2252 (2021). https://doi.org/10.1007/s00245-019-09626-6

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  • DOI: https://doi.org/10.1007/s00245-019-09626-6

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