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

摘要

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 Gateaux differentiable and lower semicontinuous convex functions defined on an open convex set of a Hilbert space. Finally, we give a characterization of C1,+ convex functions in terms of local cocoercivity.