  


On the algebraic structure of the copositive cone
Roland Hildebrand (roland.hildebrandunivgrenoblealpes.fr) Abstract: We decompose the copositive cone $\copos{n}$ into a disjoint union of a finite number of open subsets $S_{\cal E}$ of algebraic sets $Z_{\cal E}$. Each set $S_{\cal E}$ consists of interiors of faces of $\copos{n}$. On each irreducible component of $Z_{\cal E}$ these faces generically have the same dimension. Each algebraic set $Z_{\cal E}$ is characterized by a finite collection ${\cal E} = \{(I_{\alpha},J_{\alpha})\}_{\alpha = 1,\dots,{\cal E}}$ of pairs of index sets. Namely, $Z_{\cal E}$ is the set of symmetric matrices $A$ such that the submatrices $A_{I_{\alpha} \times J_{\alpha}}$ are rankdeficient for all $\alpha$. For every copositive matrix $A \in S_{\cal E}$, the index sets $I_{\alpha}$ are the minimal zero supports of $A$. If $u^{\alpha}$ is a corresponding minimal zero of $A$, 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 ${\cal E}$ the extended minimal zero support set of $A$. We provide some necessary conditions on ${\cal E}$ for $S_{\cal E}$ to be nonempty, and for a subset $S_{{\cal E}'}$ to intersect the boundary of another subset $S_{\cal E}$. Keywords: copositive matrix, minimal zero, facial structure, algebraic sets Category 1: Applications  OR and Management Sciences Category 2: Global Optimization (Theory ) Citation: Download: Entry Submitted: 01/31/2020 Modify/Update this entry  
Visitors  Authors  More about us  Links  
Subscribe, Unsubscribe Digest Archive Search, Browse the Repository

Submit Update Policies 
Coordinator's Board Classification Scheme Credits Give us feedback 
Optimization Journals, Sites, Societies  