IDEAS home Printed from https://ideas.repec.org/a/gam/jmathe/v9y2021i21p2698-d663611.html
   My bibliography  Save this article

Structural Properties of Faces of the Cone of Copositive Matrices

Author

Listed:
  • Olga Kostyukova

    (Institute of Mathematics, National Academy of Sciences of Belarus, Surganov Str. 11, 220072 Minsk, Belarus
    These authors contributed equally to this work.)

  • Tatiana Tchemisova

    (Mathematical Department, Campus Universitario Santiago, University of Aveiro, 3810-193 Aveiro, Portugal
    These authors contributed equally to this work.)

Abstract

In this paper, we study the properties of faces and exposed faces of the cone of copositive matrices (copositive cone), paying special attention to issues related to their geometric structure. Based on the concepts of zero and minimal zero vectors, we obtain several explicit representations of faces of the copositive cone and compare them. Given a face of the cone of copositive matrices, we describe the subspace generated by that face and the minimal exposed face containing it. Summarizing the results obtained in the paper, we systematically show what information can be extracted about the given copositive face in the case of incomplete data. Several examples for illustrating the main findings of the paper and also for justifying the usefulness of the developed approach to the study of the facial structure of the copositive cone are discussed.

Suggested Citation

  • Olga Kostyukova & Tatiana Tchemisova, 2021. "Structural Properties of Faces of the Cone of Copositive Matrices," Mathematics, MDPI, vol. 9(21), pages 1-21, October.
    • Handle: RePEc:gam:jmathe:v:9:y:2021:i:21:p:2698-:d:663611
    as

    Download full text from publisher

    File URL: https://www.mdpi.com/2227-7390/9/21/2698/pdf
    Download Restriction: no
    File URL: https://www.mdpi.com/2227-7390/9/21/2698/
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
    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. Olga Kostyukova & Tatiana Tchemisova, 2017. "Optimality Conditions for Convex Semi-infinite Programming Problems with Finitely Representable Compact Index Sets," Journal of Optimization Theory and Applications, Springer, vol. 175(1), pages 76-103, October.
    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. Muhammad Faisal Iqbal & Faizan Ahmed, 2022. "Approximation Hierarchies for the Copositive Tensor Cone and Their Application to the Polynomial Optimization over the Simplex," Mathematics, MDPI, vol. 10(10), pages 1-17, May.

    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. 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.
    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. 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.
    5. 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.
    6. 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.
    7. 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.
    8. Orizon Pereira Ferreira & Yingchao Gao & Sándor Zoltán Németh & Petra Renáta Rigó, 2024. "Gradient projection method on the sphere, complementarity problems and copositivity," Journal of Global Optimization, Springer, vol. 90(1), pages 1-25, September.
    9. Anwa Zhou & Jinyan Fan, 2015. "Interiors of completely positive cones," Journal of Global Optimization, Springer, vol. 63(4), pages 653-675, December.
    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. 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.
    12. Cheng Lu & Zhibin Deng & Qingwei Jin, 2017. "An eigenvalue decomposition based branch-and-bound algorithm for nonconvex quadratic programming problems with convex quadratic constraints," Journal of Global Optimization, Springer, vol. 67(3), pages 475-493, March.
    13. 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.
    14. O. I. Kostyukova & T. V. Tchemisova & O. S. Dudina, 2024. "On the Uniform Duality in Copositive Optimization," Journal of Optimization Theory and Applications, Springer, vol. 203(2), pages 1940-1966, November.
    15. 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.
    16. 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.
    17. Riley Badenbroek & Etienne Klerk, 2022. "Simulated Annealing for Convex Optimization: Rigorous Complexity Analysis and Practical Perspectives," Journal of Optimization Theory and Applications, Springer, vol. 194(2), pages 465-491, August.
    18. 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.
    19. 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.
    20. 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.

    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:gam:jmathe:v:9:y:2021:i:21:p:2698-:d:663611. 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: MDPI Indexing Manager (email available below). General contact details of provider: https://www.mdpi.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.