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On the algebraic structure of the copositive cone
Optimization Letters ( IF 1.502 ) Pub Date : 2020-05-20 , DOI: 10.1007/s11590-020-01591-2
Roland Hildebrand

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 \(S_{{\mathcal {E}}}\).
更新日期:2020-05-20

 

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