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We provide convergent hierarchies for the cone C of copositive matrices andits dual, the cone of completely positive matrices. In both cases thecorresponding hierarchy consists of nested spectrahedra and provide outer(resp. inner) approximations for C (resp. for its dual), thus complementingprevious inner (resp. outer) approximations for C (for the dual). Inparticular, both inner and outer approximations have a very simpleinterpretation. Finally, extension to K-copositivity and K-complete positivityfor a closed convex cone K, is straightforward.

New approximations for the cone of copositive matrices and its dual