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

Research paper by Peter J. C. Dickinson, Mirjam Dür, Luuk Gijben, Roland Hildebrand

Indexed on: 21 Jul '12Published on: 21 Jul '12Published in: Optimization Letters

#### 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.