irreducibility

Considering Copositivity Locally

  • Peter J.C. Dickinson
  • Roland Hildebrand
    • Categories Linear, Cone and Semidefinite Programming Tags , , ,

    Let $A$ be an element of the copositive cone $\mathcal{COP}^n$. A zero $\mathbf{u}$ of $A$ is a nonnegative vector whose elements sum up to one and such that $\mathbf{u}^TA\mathbf{u} = 0$. The support of $\mathbf{u}$ is the index set $\mathrm{supp}\mathbf{u} \subset \{1,\dots,n\}$ corresponding to the nonzero entries of $\mathbf{u}$. A zero $\mathbf{u}$ of $A$ is … Read more

    Irreducible elements of the copositive cone

  • Peter J.C. Dickinson
  • Mirjam Dür
  • Roland Hildebrand
  • Luuk Gijben
    • Categories Linear, Cone and Semidefinite Programming Tags , ,

    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 … Read more