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 $$S_{{\mathcal {E}}}$$ .