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Let $p_n$ denote the maximal cp-rank attained by completely positive $n\times n$ matrices. Only lower and upper bounds for $p_n$ are known, when $n\ge6$, but it is known that $p_n=\frac{n^2}2\big(1+o(1)\big)$, and the difference of the current best upper and lower bounds for $p_n$ is of order $\mathcal{O}\big(n^{3/2}\big)$. In this paper, that gap is reduced to $\mathcal{O}\big(n\log\log n\big)$. To achieve this result, a sequence of generalized ranks of a given matrix A has to be introduced. Properties of that sequence and its generating function are investigated. For suitable A, the $d$th term of that sequence is the cp-rank of some completely positive tensor of order $d$. This allows the derivation of asymptotically matching lower and upper bounds for the maximal cp-rank of completely positive tensors of order $d>2$ as well.

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Zero-one completely positive matrices and the A(R, S) classes

Special Matrices ◽

10.1515/spma-2016-0024 ◽

2016 ◽

Vol 4 (1) ◽

Author(s):

G. Dahl ◽

T. A. Haufmann

Keyword(s):

Related Concept ◽

Completely Positive ◽

Completely Positive Matrices ◽

Positive Matrices

AbstractA matrix of the form A = BBT where B is nonnegative is called completely positive (CP). Berman and Xu (2005) investigated a subclass of CP-matrices, called f0, 1g-completely positive matrices. We introduce a related concept and show connections between the two notions. An important relation to the so-called cut cone is established. Some results are shown for f0, 1g-completely positive matrices with given graphs, and for {0,1}-completely positive matrices constructed from the classes of (0, 1)-matrices with fixed row and column sums.

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A note on completely positive relaxations of quadratic problems in a multiobjective framework

Journal of Global Optimization ◽

10.1007/s10898-021-01091-2 ◽

2021 ◽

Author(s):

Gabriele Eichfelder ◽

Patrick Groetzner

Keyword(s):

Positive Semidefinite ◽

Weighted Sum ◽

Quadratic Problem ◽

Completely Positive ◽

Nonconvex Problems ◽

Completely Positive Matrices ◽

Convex Problems ◽

Positive Matrices ◽

Nondominated Points ◽

Single Objective

AbstractIn a single-objective setting, nonconvex quadratic problems can equivalently be reformulated as convex problems over the cone of completely positive matrices. In small dimensions this cone equals the cone of matrices which are entrywise nonnegative and positive semidefinite, so the convex reformulation can be solved via SDP solvers. Considering multiobjective nonconvex quadratic problems, naturally the question arises, whether the advantage of convex reformulations extends to the multicriteria framework. In this note, we show that this approach only finds the supported nondominated points, which can already be found by using the weighted sum scalarization of the multiobjective quadratic problem, i.e. it is not suitable for multiobjective nonconvex problems.

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Factorizations of completely positive matrices

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100046405 ◽

1971 ◽

Vol 69 (1) ◽

pp. 53-58 ◽

Cited By ~ 27

Author(s):

Thomas L. Markham

Keyword(s):

Quadratic Form ◽

Positive Matrix ◽

Completely Positive ◽

Positive Form ◽

Completely Positive Matrices ◽

Completely Positive Matrix ◽

Positive Matrices ◽

Real Quadratic

1. Introduction. DEFINITION. If with aij = aji, is a real quadratic form inx1 …,xn, andwhere Lk = cklx1 + … + cknxn (ckj ≥ 0 for k = 1, …, t), then Q is called a completely positive form, and A = (aij) is called a completely positive matrix.

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