On the cp-Rank and Minimal cp Factorizations of a Completely Positive Matrix

Abstract

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,

The Drew–Johnson–Loewy conjecture for matrices over max–min semirings

“…The conjecture was proved for triangle free graphs in [3], for graphs which contain no odd cycle on five or more vertices in [5], for all graphs on five vertices which are not the complete graph in [6], for nonnegative matrices with a positive semidefinite comparison matrix (and any graph) in [7] and for all 5 × 5 CP matrices in [8]. However, the general conjecture is still open.…”

The structure of completely positive matrices according to their CP-rank and CP-plus-rank

“…(2) and (3) were proven in [20], and (4) follows directly from (3) and Theorem 3.3. The leftmost inequality in (5) follows from (4).…”

“…This proof is an adaptation of one from [20]. In fact, we will prove the more general result that considers M ∈ N n \ {O} with a P-F vector x ∈ R n ++ .…”

“…While (6) and (7) are well known since long, see for example [4], the bounds in (8) and (9) were established quite recently, namely in [8] and in [19], respectively. For n = 5 we have p n = n 2 /4 [20]. It was conjectured in [12] that this equality holds for all n ≥ 5, however counter examples to this conjecture for n = 7, .…”

Interiors of completely positive cones

“…Shaked-Monderer, Bomze, Jarre & Schachinger [27] showed that the maximal CPrank of nˆn CP-matrices is attained at a positive definite matrix on bdpC n q. Denote R ǹ :" tx P R n | x ě 0u and R ǹ`: " tx P R n | x ą 0u. Dür & Still [15] characterized intpC n q as:…”