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The extreme rays of the \(6\times 6\) copositive cone

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Abstract

We provide a complete classification of the extreme rays of the \(6 \times 6\) copositive cone \(\mathcal {COP}^{6}\). We proceed via a coarse intermediate classification of the possible minimal zero support set of an exceptional extremal matrix \(A \in \mathcal {COP}^{6}\). To each such minimal zero support set we construct a stratified semi-algebraic manifold in the space of real symmetric \(6 \times 6\) matrices \({\mathcal {S}}^{6}\), parameterized in a semi-trigonometric way, which consists of all exceptional extremal matrices \(A \in \mathcal {COP}^{6}\) 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.

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Notes

  1. P.J.C. Dickinson, R. de Zeeuw. Generating irreducible copositive matrices using the stable set problem. In press in Discrete Appl. Math., 2020.

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Acknowledgements

The authors would like to thank the anonymous referees for a very thorough reading of the paper and for suggesting valuable improvements.

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Authors and Affiliations

  1. Moscow Institute of Physics and Technology, 9 Institutskiy per., Dolgoprudny, Moscow Region, Russian Federation, 141700

    Andrey Afonin

  2. CNRS, Grenoble INP, LJK, Univ. Grenoble Alpes, 38000, Grenoble, France

    Roland Hildebrand

  3. RaboBank, Utrecht, The Netherlands

    Peter J. C. Dickinson

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  1. Andrey Afonin
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  2. Roland Hildebrand
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  3. Peter J. C. Dickinson
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Correspondence to Roland Hildebrand.

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Afonin, A., Hildebrand, R. & Dickinson, P.J.C. The extreme rays of the \(6\times 6\) copositive cone. J Glob Optim 79, 153–190 (2021). https://doi.org/10.1007/s10898-020-00930-y

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  • DOI: https://doi.org/10.1007/s10898-020-00930-y

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