- Title
- Generalized bregman envelopes and proximity operators
- Creator
- Burachik, Regina; Dao, Minh; Lindstrom, Scott
- Date
- 2021
- Type
- Text; Journal article
- Identifier
- http://researchonline.federation.edu.au/vital/access/HandleResolver/1959.17/178552
- Identifier
- vital:15422
- Identifier
-
https://doi.org/10.1007/s10957-021-01895-y
- Identifier
- ISBN:0022-3239 (ISSN)
- Abstract
- Every maximally monotone operator can be associated with a family of convex functions, called the Fitzpatrick family or family of representative functions. Surprisingly, in 2017, Burachik and Martínez-Legaz showed that the well-known Bregman distance is a particular case of a general family of distances, each one induced by a specific maximally monotone operator and a specific choice of one of its representative functions. For the family of generalized Bregman distances, sufficient conditions for convexity, coercivity, and supercoercivity have recently been furnished. Motivated by these advances, we introduce in the present paper the generalized left and right envelopes and proximity operators, and we provide asymptotic results for parameters. Certain results extend readily from the more specific Bregman context, while others only extend for certain generalized cases. To illustrate, we construct examples from the Bregman generalizing case, together with the natural “extreme” cases that highlight the importance of which generalized Bregman distance is chosen. © 2021, The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature.
- Publisher
- Springer
- Relation
- Journal of Optimization Theory and Applications Vol. 190, no. 3 (2021), p. 744-778
- Rights
- All metadata describing materials held in, or linked to, the repository is freely available under a CC0 licence
- Rights
- Copyright @ The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2021
- Rights
- Open Access
- Subject
- 0102 Applied Mathematics; 0103 Numerical and Computational Mathematics; 0906 Electrical and Electronic Engineering; Convex function; Fitzpatrick function; Generalized Bregman distance; Maximally monotone operator; Moreau envelope; Proximity operator; Regularization; Representative function
- Full Text
- Reviewed
- Funder
- SBL was supported by an Australian Mathematical Society Lift-Off Fellowship and by Hong Kong Research Grants Council PolyU153085/16p.
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