On the algebraic structure of the copositive cone

Abstract

We decompose the copositive cone $\mathcal {COP}^{n}$ into a disjoint union of a finite number of open subsets $S_{{\mathcal {E}}}$ of algebraic sets $Z_{{\mathcal {E}}}$. Each set $S_{{\mathcal {E}}}$ consists of interiors of faces of $\mathcal {COP}^{n}$. On each irreducible component of $Z_{{\mathcal {E}}}$ these faces generically have the same dimension. Each algebraic set $Z_{{\mathcal {E}}}$ is characterized by a finite collection ${{\mathcal {E}}} = \{(I_{\alpha },J_{\alpha })\}_{\alpha = 1,\dots ,|\mathcal{E}|}$ of pairs of index sets. Namely, $Z_{{\mathcal {E}}}$ is the set of symmetric matrices A such that the submatrices $A_{J_{\alpha } \times I_{\alpha }}$ are rank-deficient for all $\alpha$. For every copositive matrix $A \in S_{{\mathcal {E}}}$, the index sets $I_{\alpha }$ are the minimal zero supports of A. If $u^{\alpha }$ is a corresponding minimal zero, then $J_{\alpha }$ is the set of indices j such that $(Au^{\alpha })_j = 0$. We call the pair $(I_{\alpha },J_{\alpha })$ the extended support of the zero $u^{\alpha }$, and ${{\mathcal {E}}}$ the extended minimal zero support set of A. We provide some necessary conditions on ${{\mathcal {E}}}$ for $S_{{\mathcal {E}}}$ to be non-empty, and for a subset $S_{{{\mathcal {E}}}'}$ to intersect the boundary of another subset $S_{{\mathcal {E}}}$.