Journal of Convex Analysis 24 (2017), No. 1, 135--148
Copyright Heldermann Verlag 2017
Regular Self-Proximal Distances are Bregman
Felipe Alvarez
Centro de Modelamiento Matemático, Dep. de Ingeniería Matemática, Universidad de Chile, Beauchef 851, Santiago, Chile
Rafael Correa
Centro de Modelamiento Matemático, Dep. de Ingeniería Matemática, Universidad de Chile, Beauchef 851, Santiago, Chile
Matthieu Marechal
Instituto de Ciencias Basicas, Facultad de Ingeniería, Universidad Diego Portales, Ejército 441, Santiago, Chile
matthieu.marechal@udp.cl
Bregman distances play a key role in generalized versions of the proximal algorithm. This paper proposes a new characterization of Bregman distances in terms of their gradient and Hessian matrix. Thanks to this characterization, we obtain two results: all the so called self-proximal distances are Bregman, and all the induced proximal distances, under some regularity assumptions, are Bregman functions.
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