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

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 C of all completely positive symmetric n×n matrices (which can be factorized into F F ⊤ , 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.

Considering copositivity locally

“…The set of copositive matrices forms a convex cone, the copositive cone, COP n . This matrix cone is of interest for combinatorial optimization, for surveys see [7,10,12,17]. It is a classical result by Diananda [8,Theorem 2] that for n ≤ 4 the copositive cone can be described as the sum of the cone of positive semi-definite matrices S n + and the cone of element-wise nonnegative symmetric matrices N n .…”

Copositive optimization – Recent developments and applications

“…A complete picture of possible attainability/duality gap constellations in primaldual pairs of copositive optimization problems is provided in [26], which also lists some elementary algebraic properties and counterexamples illustrating the difference between the semidefinite cone P and the copositive/completely positive cone C * /C. This is important for many copositivity detection procedures, and as we saw in (3), the feasibility constraint incorporates most of the hardness in copositive optimization.…”

“…Recent book chapters with some character of a survey on copositivity from an optimization viewpoint are [17,Section 1.4] and [34]. Finally, [26] offers a rough literature review by clustering a considerable part of copositivity-related publications.…”

“…Before doing that, let us briefly mention other topics (and group related references) which cannot be extensively covered here; for a more extensive, clustered list of references we again refer to [26].…”

Copositivity and constrained fractional quadratic problems

“…The success of this topic is due, not only to the elegance of the theory, but also to the good results obtained in tighter semidefinite relaxations for hard combinatorial optimization problems. For recent papers with a survey flavor see, e.g., [10,19,24], and for a clustered bibliography [15].…”