Bregman level proximal subdifferentials and new characterizations of Bregman proximal operators

Abstract

Classic subdifferentials in variational analysis may fail to fully represent the Bregman proximal operator in the absence of convexity. In this paper, we fill this gap by introducing the left and right \emph{Bregman level proximal subdifferentials} and investigate them systematically. Every Bregman proximal operator turns out to be the resolvent of a Bregman level proximal subdifferential under a standard range assumption, even without convexity. Aided by this pleasant feature, we establish new correspondences among useful properties of the Bregman proximal operator, the underlying function, and the (left) Bregman level proximal subdifferential, generalizing classical equivalences in the Euclidean case. Unlike the classical setting, asymmetry and duality gap emerge as natural consequences of the Bregman distance. Along the way, we improve results by Kan and Song and by Wang and Bauschke on Bregman proximal operators. We also characterize the existence and single-valuedness of the Bregman level proximal subdifferential, investigate coincidence results, and make an interesting connection to relative smoothness. Abundant examples are provided to justify the necessity of our assumptions. We also introduce \emph{anisotropic firm nonexpansiveness}, a new notion that is complementary to \emph{Bregman} firm nonexpansiveness and is shown to characterize relative smooth convex functions and convex envelopes via properties of gradient and proximal operators.