IDEAS home Printed from https://ideas.repec.org/a/spr/coopap/v83y2022i3d10.1007_s10589-022-00417-4.html
   My bibliography  Save this article

Completely positive factorization by a Riemannian smoothing method

Author

Listed:
  • Zhijian Lai

    (University of Tsukuba)

  • Akiko Yoshise

    (University of Tsukuba)

Abstract

Copositive optimization is a special case of convex conic programming, and it consists of optimizing a linear function over the cone of all completely positive matrices under linear constraints. Copositive optimization provides powerful relaxations of NP-hard quadratic problems or combinatorial problems, but there are still many open problems regarding copositive or completely positive matrices. In this paper, we focus on one such problem; finding a completely positive (CP) factorization for a given completely positive matrix. We treat it as a nonsmooth Riemannian optimization problem, i.e., a minimization problem of a nonsmooth function over a Riemannian manifold. To solve this problem, we present a general smoothing framework for solving nonsmooth Riemannian optimization problems and show convergence to a stationary point of the original problem. An advantage is that we can implement it quickly with minimal effort by directly using the existing standard smooth Riemannian solvers, such as Manopt. Numerical experiments show the efficiency of our method especially for large-scale CP factorizations.

Suggested Citation

  • Zhijian Lai & Akiko Yoshise, 2022. "Completely positive factorization by a Riemannian smoothing method," Computational Optimization and Applications, Springer, vol. 83(3), pages 933-966, December.
    • Handle: RePEc:spr:coopap:v:83:y:2022:i:3:d:10.1007_s10589-022-00417-4
    DOI: 10.1007/s10589-022-00417-4
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10589-022-00417-4
    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/s10589-022-00417-4?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. Jasmina Hasanhodzic & Andrew Lo & Emanuele Viola, 2011. "A computational view of market efficiency," Quantitative Finance, Taylor & Francis Journals, vol. 11(7), pages 1043-1050.
    2. Chen Chen & Ting Kei Pong & Lulin Tan & Liaoyuan Zeng, 2020. "A difference-of-convex approach for split feasibility with applications to matrix factorizations and outlier detection," Journal of Global Optimization, Springer, vol. 78(1), pages 107-136, September.
    3. Garms, Marco A. & Andrade, Marco T.C. & Caldas, Iberê L., 2009. "Fuzzy computational control for real Chua circuit," Chaos, Solitons & Fractals, Elsevier, vol. 39(5), pages 2169-2178.
    4. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
    5. Glaydston Carvalho Bento & João Xavier Cruz Neto & Paulo Roberto Oliveira, 2016. "A New Approach to the Proximal Point Method: Convergence on General Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 168(3), pages 743-755, March.
    6. Immanuel Bomze & Werner Schachinger & Gabriele Uchida, 2012. "Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization," Journal of Global Optimization, Springer, vol. 52(3), pages 423-445, March.
    7. Peter Dickinson & Luuk Gijben, 2014. "On the computational complexity of membership problems for the completely positive cone and its dual," Computational Optimization and Applications, Springer, vol. 57(2), pages 403-415, March.
    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. Bomze, Immanuel M. & Gabl, Markus, 2023. "Optimization under uncertainty and risk: Quadratic and copositive approaches," European Journal of Operational Research, Elsevier, vol. 310(2), pages 449-476.
    2. Andrey Afonin & Roland Hildebrand & Peter J. C. Dickinson, 2021. "The extreme rays of the $$6\times 6$$ 6 × 6 copositive cone," Journal of Global Optimization, Springer, vol. 79(1), pages 153-190, January.
    3. Immanuel M. Bomze & Bo Peng, 2023. "Conic formulation of QPCCs applied to truly sparse QPs," Computational Optimization and Applications, Springer, vol. 84(3), pages 703-735, April.
    4. Anwa Zhou & Jinyan Fan, 2015. "Interiors of completely positive cones," Journal of Global Optimization, Springer, vol. 63(4), pages 653-675, December.
    5. Akihiro Tanaka & Akiko Yoshise, 2018. "LP-based tractable subcones of the semidefinite plus nonnegative cone," Annals of Operations Research, Springer, vol. 265(1), pages 155-182, June.
    6. Peter Dickinson & Luuk Gijben, 2014. "On the computational complexity of membership problems for the completely positive cone and its dual," Computational Optimization and Applications, Springer, vol. 57(2), pages 403-415, March.
    7. Gabriele Eichfelder & Patrick Groetzner, 2022. "A note on completely positive relaxations of quadratic problems in a multiobjective framework," Journal of Global Optimization, Springer, vol. 82(3), pages 615-626, March.
    8. Abdeljelil Baccari & Mourad Naffouti, 2016. "Copositivity and Sparsity Relations Using Spectral Properties," Journal of Optimization Theory and Applications, Springer, vol. 171(3), pages 998-1007, December.
    9. Alexander Engau & Miguel Anjos & Immanuel Bomze, 2013. "Constraint selection in a build-up interior-point cutting-plane method for solving relaxations of the stable-set problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 78(1), pages 35-59, August.
    10. Immanuel M. Bomze & Jianqiang Cheng & Peter J. C. Dickinson & Abdel Lisser & Jia Liu, 2019. "Notoriously hard (mixed-)binary QPs: empirical evidence on new completely positive approaches," Computational Management Science, Springer, vol. 16(4), pages 593-619, October.
    11. Carmo Brás & Gabriele Eichfelder & Joaquim Júdice, 2016. "Copositivity tests based on the linear complementarity problem," Computational Optimization and Applications, Springer, vol. 63(2), pages 461-493, March.
    12. Jinyan Fan & Anwa Zhou, 2016. "Computing the distance between the linear matrix pencil and the completely positive cone," Computational Optimization and Applications, Springer, vol. 64(3), pages 647-670, July.
    13. Jinyan Fan & Jiawang Nie & Anwa Zhou, 2019. "Completely Positive Binary Tensors," Mathematics of Operations Research, INFORMS, vol. 44(3), pages 1087-1100, August.
    14. Badenbroek, Riley & de Klerk, Etienne, 2022. "An analytic center cutting plane method to determine complete positivity of a matrix," Other publications TiSEM 088da653-b943-4ed0-9720-6, Tilburg University, School of Economics and Management.
    15. Immanuel M. Bomze & Vaithilingam Jeyakumar & Guoyin Li, 2018. "Extended trust-region problems with one or two balls: exact copositive and Lagrangian relaxations," Journal of Global Optimization, Springer, vol. 71(3), pages 551-569, July.
    16. Faizan Ahmed & Mirjam Dür & Georg Still, 2013. "Copositive Programming via Semi-Infinite Optimization," Journal of Optimization Theory and Applications, Springer, vol. 159(2), pages 322-340, November.
    17. Glaydston C. Bento & Orizon P. Ferreira & Jefferson G. Melo, 2017. "Iteration-Complexity of Gradient, Subgradient and Proximal Point Methods on Riemannian Manifolds," Journal of Optimization Theory and Applications, Springer, vol. 173(2), pages 548-562, May.
    18. Glaydston Carvalho Bento & João Xavier Cruz Neto & Antoine Soubeyran & Valdinês Leite Sousa Júnior, 2016. "Dual Descent Methods as Tension Reduction Systems," Journal of Optimization Theory and Applications, Springer, vol. 171(1), pages 209-227, October.
    19. Deng, Zhibin & Fang, Shu-Cherng & Jin, Qingwei & Xing, Wenxun, 2013. "Detecting copositivity of a symmetric matrix by an adaptive ellipsoid-based approximation scheme," European Journal of Operational Research, Elsevier, vol. 229(1), pages 21-28.
    20. Immanuel M. Bomze & Michael Kahr & Markus Leitner, 2021. "Trust Your Data or Not—StQP Remains StQP: Community Detection via Robust Standard Quadratic Optimization," Mathematics of Operations Research, INFORMS, vol. 46(1), pages 301-316, February.

    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:coopap:v:83:y:2022:i:3:d:10.1007_s10589-022-00417-4. 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.