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 under relative prox-regularity, an extension of prox-regularity for nonconvex functions which has been originally introduced by Poliquin and Rockafellar. Although, we focus on the left Bregman proximal mapping, a translation result yields 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 possibly nonconvex domain. Moreover, as a main source of examples, in analogy to the classical setting, we introduce relatively amenable functions by invoking the recently proposed notion of smooth adaptability or relative smoothness. Exemplarily we apply our theory to interpret joint alternating Bregman minimization with proximal regularization, locally, as a Bregman proximal gradient algorithm.

@article{LOC20,
  title        = {Bregman Proximal Mappings and Bregman-Moreau Envelopes under Relative Prox-Regularity},
  author       = {E. Laude and P. Ochs and D. Cremers},
  year         = {2020},
  journal      = {Journal of Optimization Theory and Applications},
  number       = {3},
  volume       = {184},
  pages        = {724--761}
}