Optimization Letters

, Volume 7, Issue 8, pp 1669–1679 | Cite as

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

  • Peter J. C. Dickinson
  • Mirjam Dür
  • Luuk Gijben
  • Roland Hildebrand
Original Paper


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.


Copositive cone Parrilo’s approximations Sum-of-squares conditions 5 × 5 copositive matrices 

Mathematics Subject Classification (2000)

15A48 15A21 90C22 


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Copyright information

© Springer-Verlag 2012

Authors and Affiliations

  • Peter J. C. Dickinson
    • 1
  • Mirjam Dür
    • 2
  • Luuk Gijben
    • 1
  • Roland Hildebrand
    • 3
  1. 1.Johann Bernoulli Institute for Mathematics and Computer ScienceUniversity of GroningenGroningenThe Netherlands
  2. 2.Department of MathematicsUniversity of TrierTrierGermany
  3. 3.LJK, Université Grenoble 1/CNRS51 rue des MathématiquesGrenoble CedexFrance

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