Factorizations of completely positive matrices

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

SIAM Journal on Matrix Analysis and Applications ◽

10.1137/120885759 ◽

2013 ◽

Vol 34 (2) ◽

pp. 355-368 ◽

Cited By ~ 17

Author(s):

Naomi Shaked-Monderer ◽

Immanuel M. Bomze ◽

Florian Jarre ◽

Werner Schachinger

Keyword(s):

Positive Matrix ◽

Completely Positive ◽

Completely Positive Matrix

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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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