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Bregman Proximal Mappings and Bregman-Moreau Envelopes under Relative Prox-Regularity
We systematically study the local single-valuedness of the Bregman proximal
mapping and local smoothness of the Bregman--Moreau envelope of a nonconvex
function under relative prox-regularity - an extension of prox-regularity -
which was originally introduced by Poliquin and Rockafellar. As Bregman
distances are asymmetric in general, in accordance with Bauschke et al., it is
natural to consider two variants of the Bregman proximal mapping, which,
depending on the order of the arguments, are called left and right Bregman
proximal mapping. We consider the left Bregman proximal mapping first. Then,
via translation result, we obtain analogue (and partially sharp) results for
the right Bregman proximal mapping. The class of relatively prox-regular
functions significantly extends the recently considered class of relatively
hypoconvex functions. In particular, relative prox-regularity allows for
functions with a possibly nonconvex domain. Moreover, as a main source of
examples and analogously to the classical setting, we introduce relatively
amenable functions, i.e. convexly composite functions, for which the inner
nonlinear mapping is component-wise smooth adaptable, a recently introduced
extension of Lipschitz differentiability. By way of example, we apply our
theory to locally interpret joint alternating Bregman minimization with
proximal regularization as a Bregman proximal gradient algorithm, applied to a
smooth adaptable function.Comment: This article is published in Journal of Optimization Theory and
Application