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