Scaling relationship between the copositive cone and Parrilo’s first level approximation

Abstract

We investigate the relation between the cone ${\mathcal{C}^{n}}$ of n × n copositive matrices and the approximating cone ${\mathcal{K}_{n}^{1}}$ introduced by Parrilo. While these cones are known to be equal for n ≤ 4, we show that for n ≥ 5 they are not equal. This result is based on the fact that ${\mathcal{K}_{n}^{1}}$ is not invariant under diagonal scaling. We show that for any copositive matrix which is not the sum of a nonnegative and a positive semidefinite matrix we can find a scaling which is not in ${\mathcal{K}_{n}^{1}}$. In fact, we show that if all scaled versions of a matrix are contained in ${\mathcal{K}_{n}^{r}}$ for some fixed r, then the matrix must be in ${\mathcal{K}_{n}^{0}}$. For the 5 × 5 case, we show the more surprising result that we can scale any copositive matrix X into ${\mathcal{K}_{5}^{1}}$ and in fact that any scaling D such that ${(DXD)_{ii} \in \{0,1\}}$ for all i yields ${DXD \in \mathcal{K}_{5}^{1}}$. From this we are able to use the cone ${\mathcal{K}_{5}^{1}}$ to check if any order 5 matrix is copositive. Another consequence of this is a complete characterisation of ${\mathcal{C}^{5}}$ in terms of ${\mathcal{K}_{5}^{1}}$. We end the paper by formulating several conjectures.