
Journal of Convex Analysis 22 (2015), No. 4, 963967 Copyright Heldermann Verlag 2015 On a Generalized BaillonHaddad Theorem for Convex Functions on Hilbert Space Charles L. Byrne Department of Mathematical Sciences, University of Massachusetts, Lowell, MA 01854, U.S.A. Charles_Byrne@uml.edu The BaillonHaddad 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 BaillonHaddad Theorem. Keywords: Bregman distance, convex function, firmly nonexpansive, gradient, nonexpansive, BaillonHaddad Theorem, Krasnosel'skiiMann Theorem. MSC: 47H09, 90C25; 26A51, 26B25 [ Fulltextpdf (78 KB)] for subscribers only. 