IDEAS home Printed from https://ideas.repec.org/a/spr/jglopt/v79y2021i1d10.1007_s10898-020-00930-y.html
   My bibliography  Save this article

The extreme rays of the $$6\times 6$$ 6 × 6 copositive cone

Author

Listed:
  • Andrey Afonin

    (Moscow Institute of Physics and Technology)

  • Roland Hildebrand

    (Univ. Grenoble Alpes)

  • Peter J. C. Dickinson

    (RaboBank)

Abstract

We provide a complete classification of the extreme rays of the $$6 \times 6$$ 6 × 6 copositive cone $$\mathcal {COP}^{6}$$ COP 6 . We proceed via a coarse intermediate classification of the possible minimal zero support set of an exceptional extremal matrix $$A \in \mathcal {COP}^{6}$$ A ∈ COP 6 . To each such minimal zero support set we construct a stratified semi-algebraic manifold in the space of real symmetric $$6 \times 6$$ 6 × 6 matrices $${\mathcal {S}}^{6}$$ S 6 , parameterized in a semi-trigonometric way, which consists of all exceptional extremal matrices $$A \in \mathcal {COP}^{6}$$ A ∈ COP 6 having this minimal zero support set. Each semi-algebraic stratum is characterized by the supports of the minimal zeros u as well as the supports of the corresponding matrix-vector products Au. The analysis uses recently and newly developed methods that are applicable to copositive matrices of arbitrary order.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:jglopt:v:79:y:2021:i:1:d:10.1007_s10898-020-00930-y
    DOI: 10.1007/s10898-020-00930-y
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10898-020-00930-y
    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/s10898-020-00930-y?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.
    2. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
    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. Olga Kostyukova & Tatiana Tchemisova, 2021. "Structural Properties of Faces of the Cone of Copositive Matrices," Mathematics, MDPI, vol. 9(21), pages 1-21, October.

    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. 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.
    3. 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.
    4. 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.
    5. 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.
    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. Wong, Man Hong & Zhang, Shuzhong, 2014. "On distributional robust probability functions and their computations," European Journal of Operational Research, Elsevier, vol. 233(1), pages 23-33.
    9. O. I. Kostyukova & T. V. Tchemisova, 2022. "On strong duality in linear copositive programming," Journal of Global Optimization, Springer, vol. 83(3), pages 457-480, July.
    10. 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.
    11. Li, Xiaobo & Natarajan, Karthik & Teo, Chung-Piaw & Zheng, Zhichao, 2014. "Distributionally robust mixed integer linear programs: Persistency models with applications," European Journal of Operational Research, Elsevier, vol. 233(3), pages 459-473.
    12. Immanuel Bomze & Markus Gabl, 2021. "Interplay of non-convex quadratically constrained problems with adjustable robust optimization," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 93(1), pages 115-151, February.
    13. Jean-Baptiste Hiriart-Urruty & Hai Le, 2013. "A variational approach of the rank function," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 21(2), pages 207-240, July.
    14. de Klerk, Etienne & Badenbroek, Riley, 2022. "Simulated annealing with hit-and-run for convex optimization: complexity analysis and practical perspectives," Other publications TiSEM 323b4588-65e0-4889-a555-9, Tilburg University, School of Economics and Management.
    15. Tiago Andrade & Fabricio Oliveira & Silvio Hamacher & Andrew Eberhard, 2019. "Enhancing the normalized multiparametric disaggregation technique for mixed-integer quadratic programming," Journal of Global Optimization, Springer, vol. 73(4), pages 701-722, April.
    16. Andreani, R. & Júdice, J.J. & Martínez, J.M. & Martini, T., 2016. "Feasibility problems with complementarity constraints," European Journal of Operational Research, Elsevier, vol. 249(1), pages 41-54.
    17. 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.
    18. Billionnet, Alain, 2013. "Mathematical optimization ideas for biodiversity conservation," European Journal of Operational Research, Elsevier, vol. 231(3), pages 514-534.
    19. 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.
    20. Abraham Berman & Naomi Shaked-Monderer, 2022. "Triangle-free graphs and completely positive matrices," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 30(3), pages 1093-1099, September.

    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:79:y:2021:i:1:d:10.1007_s10898-020-00930-y. 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.