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

Abstract We investigate the relation between the cone Cn of n×n copositive matrices and the approximating cone K1 n introduced by Parrilo. While these cones are known to be equal for n ≤ 4, we show that for n ≥ 5 they are unequal. This result is based on the fact that K1 n is not invariant under diagonal scaling. We show in fact 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 K1 n . For the 5× 5 case, we show the more surprising result that we can scale any copositive matrix X into K1 5 and in fact that any scaling D such that (DXD)ii ∈ {0, 1} for all i yields DXD ∈ K1 5. From this we are able to use the cone K1 5 to check if any order 5 matrix is copositive. Another consequence of this is a complete characterisation of C5 in terms of K1 5 . We end the paper by formulating several conjectures.