IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v76y2020i2d10.1007_s10898-019-00861-3.html
   My bibliography  Save this article

Inner approximating the completely positive cone via the cone of scaled diagonally dominant matrices

Author

Listed:
  • João Gouveia

    (University of Coimbra)

  • Ting Kei Pong

    (The Hong Kong Polytechnic University)

  • Mina Saee

    (University of Coimbra)

Abstract

Motivated by the expressive power of completely positive programming to encode hard optimization problems, many approximation schemes for the completely positive cone have been proposed and successfully used. Most schemes are based on outer approximations, with the only inner approximations available being a linear programming based method proposed by Bundfuss and Dür (SIAM J Optim 20(1):30–53, 2009) and also Yıldırım (Optim Methods Softw 27(1):155–173, 2012), and a semidefinite programming based method proposed by Lasserre (Math Program 144(1):265–276, 2014). In this paper, we propose the use of the cone of nonnegative scaled diagonally dominant matrices as a natural inner approximation to the completely positive cone. Using projections of this cone we derive new graph-based second-order cone approximation schemes for completely positive programming, leading to both uniform and problem-dependent hierarchies. This offers a compromise between the expressive power of semidefinite programming and the speed of linear programming based approaches. Numerical results on random problems, standard quadratic programs and the stable set problem are presented to illustrate the effectiveness of our approach.

Suggested Citation

  • João Gouveia & Ting Kei Pong & Mina Saee, 2020. "Inner approximating the completely positive cone via the cone of scaled diagonally dominant matrices," Journal of Global Optimization, Springer, vol. 76(2), pages 383-405, February.
    • Handle: RePEc:spr:jglopt:v:76:y:2020:i:2:d:10.1007_s10898-019-00861-3
    DOI: 10.1007/s10898-019-00861-3
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-019-00861-3
    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/s10898-019-00861-3?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. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. E. Alper Yıldırım, 2022. "An alternative perspective on copositive and convex relaxations of nonconvex quadratic programs," Journal of Global Optimization, Springer, vol. 82(1), pages 1-20, January.

    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. 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.
    2. 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.
    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. 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.
    5. 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.
    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. 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.
    8. 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.
    9. 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.
    10. 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.

    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:jglopt:v:76:y:2020:i:2:d:10.1007_s10898-019-00861-3. 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.