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Abstract. We describe the main open problems which are currently of interest in the theory of copositive and completely positive matrices. We give motivation as to why these questions are relevant and provide a brief description of the state of the art in each open problem.

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,

Copositive and completely positive matrices play an increasingly important role in Applied Mathematics, namely as a key concept for approximating NP-hard optimization problems. The cone of copositive matrices of a given order and the cone of completely positive matrices of the same order are dual to each other with respect to the standard scalar product on the space of symmetric matrices. This paper establishes some new relations between orthogonal pairs of such matrices lying on the boundary of either cone. As a consequence, we can establish an improvement on the upper bound of the cp-rank of completely positive matrices of general order, and a further improvement for such matrices of order six.