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Journal of Convex Analysis 17 (2010), No. 3&4, 781--787
Copyright Heldermann Verlag 2010

The Baillon-Haddad Theorem Revisited

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

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

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 continously 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.

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.

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

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