Convexity, monotonicity, and gradient processes in Hilbert space

Abstract

It is shown that in Hilbert spaces the gradient maps of convex functionals with uniformly bounded continuous second Frechét derivatives satisfy monotonicity conditions that insure that some convex combination of the identity, I, and I − ▽ f is either strictly contractive or at worst nonexpansive. This result leads to a complete resolution of the convergence question for a large class of associated gradient processes. In particular, weak convergence of the successive approximation sequence is established even in the singular case where f″ is not strictly positive at critical points of f.