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

  • Pedro Pérez-Aros
  • Emilio VilchesEmail author
Article
  • 24 Downloads

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.

Keywords

Convex function Cocoercivity Lipschitz function Nonexpansive operator Baillon–Haddad Theorem 

Mathematics Subject Classification

47H05 47H09 47N10 49J50 90C25 

Notes

Acknowledgements

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

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Instituto de Ciencias de la IngenieríaUniversidad de O’HigginsRancaguaChile
  2. 2.Instituto de Ciencias de la EducaciónUniversidad de O’HigginsRancaguaChile

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