Scaling relationship between the copositive cone and Parrilo's first level approximation
Résumé
We investigate the relation between the cone $C^n$ of $n \times n$ copositive matrices and the approximating cone $K^1_n$ introduced by Parrilo. While these cones are known to be equal for $n \leq 4$, we show that for $n \geq 5$ they are not equal. This result is based on the fact that $K^1_n$ 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 $K^1_n$. In fact, we show that if all scaled versions of a matrix are contained in $K^1_n$ for some fixed $r$, then the matrix must be in $K^0_n$. For the 5 × 5 case, we show the more surprising result that we can scale any copositive matrix X into $K^1_5$ and in fact that any scaling D such that $(DXD)_{ii}\in\{0,1\}$ for all $i$ yields $DXD\in K^1_5$. From this we are able to use the cone $K^1_5$ to check if any order 5 matrix is copositive. Another consequence of this is a complete characterisation of $C^5$ in terms of $K^1_5$. We end the paper by formulating several conjectures.