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

Andrey Afonin (), Roland Hildebrand () and Peter J. C. Dickinson ()
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Andrey Afonin: Moscow Institute of Physics and Technology
Roland Hildebrand: Univ. Grenoble Alpes
Peter J. C. Dickinson: RaboBank

Journal of Global Optimization, 2021, vol. 79, issue 1, No 6, 153-190

Abstract: 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.

Keywords: Copositive matrix; Extreme ray; Minimal zero; Non-convex optimization; 90C26; 15B48 (search for similar items in EconPapers)
Date: 2021
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Citations: View citations in EconPapers (1)

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DOI: 10.1007/s10898-020-00930-y

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