Generalized Bregman Envelopes and Proximity Operators

Regina S. Burachik (), Minh N. Dao () and Scott B. Lindstrom ()
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Regina S. Burachik: University of South Australia
Minh N. Dao: Federation University Australia
Scott B. Lindstrom: Hong Kong Polytechnic University

Journal of Optimization Theory and Applications, 2021, vol. 190, issue 3, No 2, 744-778

Abstract: 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.

Keywords: Convex function; Fitzpatrick function; Generalized Bregman distance; Maximally monotone operator; Moreau envelope; Proximity operator; Regularization; Representative function; 90C25; 26A51; 26B25; 47H05; 47H09 (search for similar items in EconPapers)
Date: 2021
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DOI: 10.1007/s10957-021-01895-y

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