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

We investigate the relation between the cone ${𝒞}^n$ of n × n copositive matrices and the approximating cone ${𝒦}_n^1$ 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 ${𝒦}_n^1$ 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 ${𝒦}_n^1$ . In fact, we show that if all scaled versions of a matrix are contained in ${𝒦}_n^r$ for some fixed r, then the matrix must be in ${𝒦}_n^0$ . For the 5 × 5 case, we show the more surprising result that we can scale any copositive matrix X into ${𝒦}_5^1$ and in fact that any scaling D such that $(\mathrm{DXD}{)}_{\mathrm{ii}}\in \{\mathrm{0,1}\}$ for all i yields $\mathrm{DXD}\in {𝒦}_5^1$ . From this we are able to use the cone ${𝒦}_5^1$ to check if any order 5 matrix is copositive. Another consequence of this is a complete characterisation of ${𝒞}^5$ in terms of ${𝒦}_5^1$ . We end the paper by formulating several conjectures.