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Generalized Bregman Envelopes and Proximity Operators

Author

Listed:
  • Regina S. Burachik

    (University of South Australia)

  • Minh N. Dao

    (Federation University Australia)

  • Scott B. Lindstrom

    (Hong Kong Polytechnic University)

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.

Suggested Citation

  • Regina S. Burachik & Minh N. Dao & Scott B. Lindstrom, 2021. "Generalized Bregman Envelopes and Proximity Operators," Journal of Optimization Theory and Applications, Springer, vol. 190(3), pages 744-778, September.
    • Handle: RePEc:spr:joptap:v:190:y:2021:i:3:d:10.1007_s10957-021-01895-y
    DOI: 10.1007/s10957-021-01895-y
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    References listed on IDEAS

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    1. Charles Byrne & Yair Censor, 2001. "Proximity Function Minimization Using Multiple Bregman Projections, with Applications to Split Feasibility and Kullback–Leibler Distance Minimization," Annals of Operations Research, Springer, vol. 105(1), pages 77-98, July.
    2. Regina S. Burachik & Alfredo N. Iusem, 2008. "Set-Valued Mappings and Enlargements of Monotone Operators," Springer Optimization and Its Applications, Springer, number 978-0-387-69757-4, September.
    3. Regina S. Burachik & Alfredo N. Iusem, 2008. "Enlargements of Monotone Operators," Springer Optimization and Its Applications, in: Set-Valued Mappings and Enlargements of Monotone Operators, chapter 0, pages 161-220, Springer.
    4. Jonathan Eckstein, 1993. "Nonlinear Proximal Point Algorithms Using Bregman Functions, with Applications to Convex Programming," Mathematics of Operations Research, INFORMS, vol. 18(1), pages 202-226, February.
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