Practical Aspects of the Moreau--Yosida Regularization: Theoretical Preliminaries

Abstract

When computing the infimal convolution of a convex function f with the squared norm, the so-called Moreau--Yosida regularization of f is obtained. Among other things, this function has a Lipschitzian gradient. We investigate some more of its properties, relevant for optimization. The most important part of our study concerns second-order differentiability: existence of a second-order development of f implies that its regularization has a Hessian. For the converse, we disclose the importance of the decomposition of ${\Bbb R}^N$ along $\cal U$ (the subspace where f is "smooth") and $\cal V$ (the subspace parallel to the subdifferential of f).