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

  • Authors:

  • Immanuel M. Bomze;Werner Schachinger;Gabriele Uchida

  • Affiliations:

  • ISOR, University of Vienna, Vienna, Austria;ISOR, University of Vienna, Vienna, Austria;DAC, University of Vienna, Vienna, Austria

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2012

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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 脳 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.