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

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.