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On the cp-rank and minimal cp factorizations of a completely positive matrix

Naomi Shaked-Monderer (nomi***at***techunix.technion.ac.il)
Immanuel M. Bomze (immanuel.bomze***at***univie.ac.at)
Florian Jarre (jarre***at***opt.uni-duesseldorf.de)
Werner Schachinger (werner.schachinger***at***univie.ac.at)

Abstract: We show that the maximal cp-rank of $n\times n$ completely positive matrices is attained at a positive-definite matrix on the boundary of the cone of $n\times n$ completely positive matrices, thus answering a long standing question. We also show that the maximal cp-rank of $5\times 5$ matrices equals six, which proves the famous Drew-Johnson-Loewy conjecture (1994) 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.

Keywords: Copositive optimization, nonnegative factorization

Category 1: Linear, Cone and Semidefinite Programming (Other )

Citation: appeared in SIAM J. Matrix Analysis Appl., http://epubs.siam.org/doi/abs/10.1137/120885759

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Entry Submitted: 06/27/2012
Entry Accepted: 06/27/2012
Entry Last Modified: 05/07/2013

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