- Title
- The generalized bregman distance
- Creator
- Burachik, Regina; Dao, Minh; Lindstrom, Scott
- Date
- 2021
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/176308
- Identifier
- vital:15097
- Identifier
-
https://doi.org/10.1137/19M1288140
- Identifier
- ISBN:1052-6234 (ISSN)
- Abstract
- Recently, a new kind of distance has been introduced for the graphs of two point-to-set operators, one of which is maximally monotone. When both operators are the subdifferential of a proper lower semicontinuous convex function, this kind of distance specializes under modest assumptions to the classical Bregman distance. We name this new kind of distance the generalized Bregman distance, and we shed light on it with examples that utilize the other two most natural representative functions: the Fitzpatrick function and its conjugate. We provide sufficient conditions for convexity, coercivity, and supercoercivity: properties which are essential for implementation in proximal point type algorithms. We establish these results for both the left and right variants of this new kind of distance. We construct examples closely related to the Kullback-Leibler divergence, which was previously considered in the context of Bregman distances and whose importance in information theory is well known. In so doing, we demonstrate how to compute a difficult Fitzpatrick conjugate function, and we discover natural occurrences of the Lambert \scrW function, whose importance in optimization is of growing interest. © 2021 Society for Industrial and Applied Mathematics
- Publisher
- Society for Industrial and Applied Mathematics Publications
- Relation
- SIAM Journal on Optimization Vol. 31, no. 1 (2021), p. 404-424
- Rights
- All metadata describing materials held in, or linked to, the repository is freely available under a CC0 licence
- Rights
- Copyright © 2021 Society for Industrial and Applied Mathematics
- Rights
- Open Access
- Subject
- 0102 Applied Mathematics; 0103 Numerical and Computational Mathematics; Bregman distance; Convex function; Fitzpatrick distance; Fitzpatrick function; Generalized Bregman distance; Regularization; Representative function
- Full Text
- Reviewed
- Funder
- The second author was partially supported by the Australian Research Council (ARC) Discovery Project DP160101537. The third author was supported by an Australian Mathematical Society Lift-Off Fellowship and Hong Kong Research Grants Council PolyU153085/16p. \dagger Mathematics, UniSA STEM, University of South Australia, Mawson Lakes, SA 5095, Australia (regina.burachik@unisa.edu.au). \ddagger School of Engineering, Information Technology and Physical Sciences, Federation University Australia, Ballarat, VIC 3353, Australia (m.dao@federation.edu.au). \S Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong (lindstrom. scott@gmail.com).
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