Abstract

J.-B. Baillon and G. Haddad ["Quelque propriétés des opérateurs angle-bornés et n-cycliquement monotones", Israel J. Math. 26 (1977) 137–150] proved that if the gradient of a convex and continuously differentiable function is nonexpansive, then it is actually firmly nonexpansive. This result, which has become known as the Baillon-Haddad theorem, has found many applications in optimization and numerical functional analysis. In this note, we propose short alternative proofs of this result and strengthen its conclusion.

Author information

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

Heinz H. Bauschke

Dept. of Mathematics, University of British Columbia, Okanagan, Kelowna, B.C. V1V 1V7, Canada

heinz.bauschke@ubc.ca

Patrick L. Combettes

UPMC Université Paris 06, Lab. J.-L. Lions - UMR 7598, 75005 Paris, France

plc@math.jussieu.fr

Keywords

  • Backward-backward splitting
  • Bregman distance
  • cocoercivity
  • convex function
  • Dunn property
  • firmly nonexpansive
  • forward-backward splitting
  • gradient
  • inverse strongly monotone
  • Moreau envelope
  • proximal mapping
  • proximity operator

Mathematics Subject Classification

47H09, 90C25; 26A51, 26B25, 46C05, 47H05, 52A41

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Suggested citation

H. H. Bauschke, P. L. Combettes. “The Baillon-Haddad Theorem Revisited.” Journal of Convex Analysis 17 (2010), No. 3&4, 781–787. https://doi.org/10.68381/jca17051

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Published by Heldermann Verlag, 2010. Rights now held by Banach Press LLC.