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Structural Properties of Faces of the Cone of Copositive Matrices

Author

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  • 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
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    References listed on IDEAS

    as
    1. 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.
    2. Bomze, Immanuel M., 2012. "Copositive optimization – Recent developments and applications," European Journal of Operational Research, Elsevier, vol. 216(3), pages 509-520.
    3. 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.
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    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.

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