New approximations for the cone of copositive matrices and its dual

TL;DR

This paper introduces convergent hierarchies of spectrahedra that approximate the cone of copositive matrices and its dual, enhancing existing methods with simple, interpretable approximations and extensions to K-copositivity.

Contribution

It develops new convergent hierarchies of spectrahedra providing both inner and outer approximations for copositive and completely positive cones, with straightforward extensions to K-copositivity.

Findings

Hierarchies are convergent and nested

Approximations are simple to interpret

Extensions to K-copositivity are straightforward

Abstract

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.