New approximations for the cone of copositive matrices and its dual

We provide convergent hierarchies for the cone C of copositive matrices and its dual, the cone of completely positive matrices. In both cases the corresponding hierarchy consists of nested spectrahedra and provide outer (resp. inner) approximations for C (resp. for its dual), thus complementing previous inner (resp. outer) approximations for C (for the dual). In particular, both inner and outer approximations have a very simple interpretation. Finally, extension to K-copositivity and K-complete positivity for a closed convex cone K, is straightforward.