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Let p n denote the largest possible cp-rank of an n × n completely positive matrix. This matrix parameter has its significance both in theory and applications, as it sheds light on the geometry and structure of the solution set of hard optimization problems in their completely positive formulation. Known bounds for p n are s n = n+1 2 − 4, the current best upper bound, and the Drew-Johnson-Loewy (DJL) lower bound d n = n 2 4. The famous DJL conjecture (1994) states that p n = d n. Here we show

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.

We show that the maximal cp-rank of n × n completely positive matrices is attained at a positive-definite matrix on the boundary of the cone of n × n completely positive matrices, thus answering a long-standing question. We also show that the maximal cp-rank of 5 × 5 matrices equals six, which proves the famous Drew-Johnson-Loewy conjecture [Linear Multilinear Algebra, 37 (1994), pp. 303-310] for matrices of this order. In addition we present a simple scheme for generating completely positive matrices of high cp-rank and investigate the structure of a minimal cp factorization. Introduction.In this article we consider completely positive matrices M and their cp-rank. An n × n matrix M is said to be completely positive if there exists a nonnegative (not necessarily square) matrix V such that M = V V . Typically, a completely positive matrix M may have many such factorizations, and the cp-rank of M , cpr M , is the minimum number of columns in such a nonnegative factor V . (For completeness, we define cpr M = 0 if M is a square zero matrix and cpr M = ∞ if M is not completely positive.) Completely positive matrices play an increasingly important role as they form a cone dual to the cone of copositive matrices. An n × n matrix A is said to be copositive if x Ax ≥ 0 for every nonnegative vector x ∈ R n + . Both cones are central in the rapidly evolving field of copositive optimization which links discrete and continuous optimization and has numerous real-world applications. For recent surveys and structured bibliographies, we refer to [8,9,10,17], and for a fundamental text book to [6].Determining the maximum possible cp-rank of n×n completely positive matrices,

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