Available at arxiv.
Authors
Radu I. Boţ (Vienna U.), Minh N. Dao (RMIT), Tianxiang Liu (Institute of Science Tokyo), Bruno F. Lourenço (ISM) and Naoki Marumo (U. Tokyo)
Abstract
It is known that if a twice differentiable function has a Lipschitz continuous Hessian, then its gradients satisfy a Jensen-type inequality. In particular, this inequality is Hessian-free in the sense that the Hessian does not actually appear in the inequality. In this paper, we show that the converse holds in a generalized setting: if a continuous function from a Hilbert space to a reflexive Banach space satisfies such an inequality, then it is Fréchet differentiable and its derivative is Lipschitz continuous. Our proof relies on the Baillon-Haddad theorem.