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Bregman Circumcenters: Basic Theory

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  • Hui Ouyang

    (University of British Columbia)

  • Xianfu Wang

    (University of British Columbia)

Abstract

Circumcenters play an important role in the design and analysis of accelerating various iterative methods in optimization. In this work, we propose Bregman (pseudo-)circumcenters associated with finite sets. We show the existence of and give explicit formulae for the unique backward and forward Bregman pseudo-circumcenters of finite sets. Moreover, we use duality to establish connections between backward and forward Bregman (pseudo-)circumcenters. Various examples are presented to illustrate the backward and forward Bregman (pseudo-)circumcenters of finite sets. Our general framework for circumcenters paves the way for the development of accelerating iterative methods by Bregman circumcenters.

Suggested Citation

  • Hui Ouyang & Xianfu Wang, 2021. "Bregman Circumcenters: Basic Theory," Journal of Optimization Theory and Applications, Springer, vol. 191(1), pages 252-280, October.
    • Handle: RePEc:spr:joptap:v:191:y:2021:i:1:d:10.1007_s10957-021-01937-5
    DOI: 10.1007/s10957-021-01937-5
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    References listed on IDEAS

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    1. Emanuel Laude & Peter Ochs & Daniel Cremers, 2020. "Bregman Proximal Mappings and Bregman–Moreau Envelopes Under Relative Prox-Regularity," Journal of Optimization Theory and Applications, Springer, vol. 184(3), pages 724-761, March.
    2. Reza Arefidamghani & Roger Behling & Yunier Bello-Cruz & Alfredo N. Iusem & Luiz-Rafael Santos, 2021. "The circumcentered-reflection method achieves better rates than alternating projections," Computational Optimization and Applications, Springer, vol. 79(2), pages 507-530, June.
    3. Marc Teboulle, 1992. "Entropic Proximal Mappings with Applications to Nonlinear Programming," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 670-690, August.
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