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

• Published in: Journal of global optimization Vol. 52; no. 3; pp. 423 - 445
• Main Authors: Bomze, Immanuel M., Schachinger, Werner, Uchida, Gabriele
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.03.2012; Springer Nature B.V
• Subjects: Algebra; Bibliographic literature; Computer Science; Game theory; Mathematics; Mathematics and Statistics; Operations Research/Decision Theory; Optimization; Queuing; Real Functions; Studies; Queueing; Congruence; Completely positive matrix; Tensor product; Switched system; Friction and contact problem; Traffic; Hadamard product; Network stability; Game theory; Conic optimization; Posynomial; Copositive matrix; Relaxation; Robust optimization; Optimal control; Attainability; Feasibility; Schur complement; Reliability; Copositive reformulation; Conic duality
• ISSN: 0925-5001; 1573-2916
• DOI: 10.1007/s10898-011-9749-3

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 of all completely positive symmetric n × n matrices (which can be factorized into , where F is a rectangular matrix with no negative entry), and its dual cone , 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.