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New approximations for the cone of copositive matrices and its dual

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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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Acknowledgments

The author wishes to thank two anonymous referees for their very helpful remarks and suggestions to improve the initial version of this note.

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  1. LAAS-CNRS and Institute of Mathematics, University of Toulouse, LAAS, 7 Avenue du Colonel Roche, BP 54200, 31031, Toulouse cédex 4, France

    Jean B. Lasserre

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  1. Jean B. Lasserre
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Correspondence to Jean B. Lasserre.

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Lasserre, J.B. New approximations for the cone of copositive matrices and its dual. Math. Program. 144, 265–276 (2014). https://doi.org/10.1007/s10107-013-0632-5

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  • DOI: https://doi.org/10.1007/s10107-013-0632-5

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