Regularization in Banach Spaces with Respect to the Bregman Distance

Abstract

The Moreau envelope, also known as Moreau–Yosida regularization, and the associated proximal mapping have been widely used in Hilbert and Banach spaces. They have been objects of great interest for optimizers since their conception more than half a century ago. They were generalized by the notion of the D-Moreau envelope and D-proximal mapping by replacing the usual square of the Euclidean distance with the conception of Bregman distance for a convex function. Recently, the D-Moreau envelope has been developed in a very general setting. In this article, we present a regularizing and smoothing technique for convex functions defined in Banach spaces. We also investigate several properties of the D-Moreau envelope function and its related D-proximal mapping in Banach spaces. For technical reasons, we restrict our attention to the Lipschitz continuity property of the D-proximal mapping and differentiability properties of the D-Moreau envelope function. In particular, we prove the Frechet differentiability property of the envelope and the Lipschitz continuity property of its derivative.