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

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 $𝒞$ of all completely positive symmetric n × n matrices (which can be factorized into ${\mathrm{FF}}^{\top }$ , where F is a rectangular matrix with no negative entry), and its dual cone ${𝒞}^{*}$ , 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.