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

Author

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  • 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
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    References listed on IDEAS

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

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