Abstract

The Baillon-Haddad Theorem asserts that, if the gradient operator of a convex and Fréchet differentiable function on a Hilbert space is nonexpansive, then it is firmly nonexpansive. This theorem plays an important role in iterative optimization. In this note we present a short, elementary proof of a generalization of the Baillon-Haddad Theorem.

Author information

Contact details are reproduced from the original publication and may be historical.

Charles L. Byrne

Department of Mathematical Sciences, University of Massachusetts, Lowell, MA 01854, U.S.A.

Charles_Byrne@uml.edu

Keywords

  • Bregman distance
  • convex function
  • firmly nonexpansive
  • gradient
  • nonexpansive
  • Baillon-Haddad Theorem
  • Krasnosel'skii-Mann Theorem

Mathematics Subject Classification

47H09, 90C25; 26A51, 26B25

References

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Suggested citation

C. L. Byrne. “On a Generalized Baillon-Haddad Theorem for Convex Functions on Hilbert Space.” Journal of Convex Analysis 22 (2015), No. 4, 963–967. https://doi.org/10.68381/jca22051

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.