IDEAS home Printed from https://ideas.repec.org/a/spr/joptap/v191y2021i1d10.1007_s10957-021-01937-5.html
   My bibliography  Save this article

Bregman Circumcenters: Basic Theory

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10957-021-01937-5
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10957-021-01937-5?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. Marc Teboulle, 1992. "Entropic Proximal Mappings with Applications to Nonlinear Programming," Mathematics of Operations Research, INFORMS, vol. 17(3), pages 670-690, August.
    2. 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.
    3. 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.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Papa Quiroz, E.A. & Roberto Oliveira, P., 2012. "An extension of proximal methods for quasiconvex minimization on the nonnegative orthant," European Journal of Operational Research, Elsevier, vol. 216(1), pages 26-32.
    2. Regina Sandra Burachik & B. F. Svaiter, 2001. "A Relative Error Tolerance for a Family of Generalized Proximal Point Methods," Mathematics of Operations Research, INFORMS, vol. 26(4), pages 816-831, November.
    3. Villacorta, Kely D.V. & Oliveira, P. Roberto, 2011. "An interior proximal method in vector optimization," European Journal of Operational Research, Elsevier, vol. 214(3), pages 485-492, November.
    4. Paul Tseng, 2004. "An Analysis of the EM Algorithm and Entropy-Like Proximal Point Methods," Mathematics of Operations Research, INFORMS, vol. 29(1), pages 27-44, February.
    5. 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.
    6. K. C. Kiwiel, 1998. "Subgradient Method with Entropic Projections for Convex Nondifferentiable Minimization," Journal of Optimization Theory and Applications, Springer, vol. 96(1), pages 159-173, January.
    7. Pinheiro, Ricardo B.N.M. & Lage, Guilherme G. & da Costa, Geraldo R.M., 2019. "A primal-dual integrated nonlinear rescaling approach applied to the optimal reactive dispatch problem," European Journal of Operational Research, Elsevier, vol. 276(3), pages 1137-1153.
    8. Jonathan Eckstein & Paulo Silva, 2010. "Proximal methods for nonlinear programming: double regularization and inexact subproblems," Computational Optimization and Applications, Springer, vol. 46(2), pages 279-304, June.
    9. S. H. Pan & J. S. Chen, 2008. "Proximal-Like Algorithm Using the Quasi D-Function for Convex Second-Order Cone Programming," Journal of Optimization Theory and Applications, Springer, vol. 138(1), pages 95-113, July.
    10. Volkovich, Vladimir & Kogan, Jacob & Nicholas, Charles, 2007. "Building initial partitions through sampling techniques," European Journal of Operational Research, Elsevier, vol. 183(3), pages 1097-1105, December.
    11. Arnaldo S. Brito & J. X. Cruz Neto & Jurandir O. Lopes & P. Roberto Oliveira, 2012. "Interior Proximal Algorithm for Quasiconvex Programming Problems and Variational Inequalities with Linear Constraints," Journal of Optimization Theory and Applications, Springer, vol. 154(1), pages 217-234, July.
    12. Jein-Shan Chen & Shaohua Pan, 2010. "An entropy-like proximal algorithm and the exponential multiplier method for convex symmetric cone programming," Computational Optimization and Applications, Springer, vol. 47(3), pages 477-499, November.
    13. H. Attouch & M. Teboulle, 2004. "Regularized Lotka-Volterra Dynamical System as Continuous Proximal-Like Method in Optimization," Journal of Optimization Theory and Applications, Springer, vol. 121(3), pages 541-570, June.
    14. Hong T. M. Chu & Ling Liang & Kim-Chuan Toh & Lei Yang, 2023. "An efficient implementable inexact entropic proximal point algorithm for a class of linear programming problems," Computational Optimization and Applications, Springer, vol. 85(1), pages 107-146, May.
    15. K. Kiwiel, 1994. "A Note on the Twice Differentiable Cubic Augmented Lagrangian," Working Papers wp94012, International Institute for Applied Systems Analysis.
    16. K. Kiwiel, 1995. "Proximal Minimization Methods with Generalized Bregman Functions," Working Papers wp95024, International Institute for Applied Systems Analysis.
    17. Heinz H. Bauschke & Jérôme Bolte & Marc Teboulle, 2017. "A Descent Lemma Beyond Lipschitz Gradient Continuity: First-Order Methods Revisited and Applications," Mathematics of Operations Research, INFORMS, vol. 42(2), pages 330-348, May.
    18. Y.M. Ermoliev & S.D. Flam, 1997. "Learning in Potential Games," Working Papers ir97022, International Institute for Applied Systems Analysis.
    19. K. Kiwiel, 1994. "Free-Steering Relaxation Methods for Problems with Strictly Convex Costs and Linear Constraints," Working Papers wp94089, International Institute for Applied Systems Analysis.
    20. Yi Zhou & Yingbin Liang & Lixin Shen, 2019. "A simple convergence analysis of Bregman proximal gradient algorithm," Computational Optimization and Applications, Springer, vol. 73(3), pages 903-912, July.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:joptap:v:191:y:2021:i:1:d:10.1007_s10957-021-01937-5. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.