LAUSR

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