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Journal of Convex Analysis 22 (2015), No. 4, 963--967
Copyright Heldermann Verlag 2015

On a Generalized Baillon-Haddad Theorem for Convex Functions on Hilbert Space

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

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.

Keywords: Bregman distance, convex function, firmly nonexpansive, gradient, nonexpansive, Baillon-Haddad Theorem, Krasnosel'skii-Mann Theorem.

MSC: 47H09, 90C25; 26A51, 26B25

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