An Enhanced Baillon–Haddad Theorem for Convex Functions Defined on Convex Sets

Abstract

The Baillon–Haddad theorem establishes that the gradient of a convex and continuously differentiable function defined in a Hilbert space is \(\beta \)-Lipschitz if and only if it is \(1/\beta \)-cocoercive. In this paper, we extend this theorem to Gâteaux differentiable and lower semicontinuous convex functions defined on an open convex set of a Hilbert space. Finally, we give a characterization of \(C^{1,+}\) convex functions in terms of local cocoercivity.