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Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization

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Abstract

Copositive optimization is a quickly expanding scientific research domain with wide-spread applications ranging from global nonconvex problems in engineering to NP-hard combinatorial optimization. It falls into the category of conic programming (optimizing a linear functional over a convex cone subject to linear constraints), namely the cone \({\mathcal{C}}\) of all completely positive symmetric n × n matrices (which can be factorized into \({FF^\top}\) , where F is a rectangular matrix with no negative entry), and its dual cone \({\mathcal{C}^*}\) , which coincides with the cone of all copositive matrices (those which generate a quadratic form taking no negative value over the positive orthant). We provide structural algebraic properties of these cones, and numerous (counter-)examples which demonstrate that many relations familiar from semidefinite optimization may fail in the copositive context, illustrating the transition from polynomial-time to NP-hard worst-case behaviour. In course of this development we also present a systematic construction principle for non-attainability phenomena, which apparently has not been noted before in an explicit way. Last but not least, also seemingly for the first time, a somehow systematic clustering of the vast and scattered literature is attempted in this paper.

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    Immanuel M. Bomze & Werner Schachinger

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    Gabriele Uchida

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Bomze, I.M., Schachinger, W. & Uchida, G. Think co(mpletely)positive ! Matrix properties, examples and a clustered bibliography on copositive optimization. J Glob Optim 52, 423–445 (2012). https://doi.org/10.1007/s10898-011-9749-3

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