  


Irreducible elements of the copositive cone
Peter J.C. Dickinson (peter.dickinsoncantab.net) Abstract: An element $A$ of the $n \times n$ copositive cone $\copos{n}$ is called irreducible with respect to the nonnegative cone~$\NNM{n}$ if it cannot be written as a nontrivial sum $A = C+N$ of a copositive matrix $C$ and an elementwise nonnegative matrix $N$. This property was studied by Baumert~\cite{Baumert65} who gave a characterisation of irreducible matrices. We demonstrate here that Baumert's characterisation is incorrect and give a correct version of his theorem which establishes a necessary and sufficient condition for a copositive matrix to be irreducible. For the case of $5\times 5$ copositive matrices we give a complete characterisation of all irreducible matrices. We show that those irreducible matrices in $\copos{5}$ which are not positive semidefinite can be parameterized in a semitrigonometric way. Finally, we prove that every $5 \times 5$ copositive matrix which is not the sum of a nonnegative and a semidefinite matrix can be expressed as the sum of a nonnegative and a single irreducible matrix. Keywords: $5 \times 5$ copositive matrix, irreducibility, exceptional copositive matrix Category 1: Linear, Cone and Semidefinite Programming (Other ) Citation: Preprint, 2012 Download: [PDF] Entry Submitted: 03/08/2012 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  