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Bregman Circumcenters: Basic Theory

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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.

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Notes

  1. Here and elsewhere, we use the convention that \(0 \ln (0) =0\).

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Acknowledgements

H. Ouyang was supported by the NSERC Discovery Grants of HB and XW. XW was partially supported by the NSERC Discovery Grant. The authors would like to thank H. H. Bauschke for his suggestions, and two anonymous referees for useful comments.

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Authors and Affiliations

  1. Department of Mathematics, University of British Columbia, Kelowna, BC, V1V 1V7, Canada

    Hui Ouyang & Xianfu Wang

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  1. Hui Ouyang
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  2. Xianfu Wang
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Correspondence to Xianfu Wang.

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Communicated by Boris S. Mordukhovich.

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Ouyang, H., Wang, X. Bregman Circumcenters: Basic Theory. J Optim Theory Appl 191, 252–280 (2021). https://doi.org/10.1007/s10957-021-01937-5

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  • DOI: https://doi.org/10.1007/s10957-021-01937-5

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