## Abstract

Let p
_{n} denote the largest possible cp-rank of an n x n completely positive matrix. This matrix parameter has its significance both in theory and applications, as it sheds light on the geometry and structure of the solution set of hard optimization problems in their completely positive formulation. Known bounds for p
_{n} are s
_{n} = (
_{2}
^{n+1}) - 4, the current best upper bound, and the Drew-Johnson-Loewy (DJL) lower bound d
_{n} = [n
^{2}/4]. The famous DJL conjecture (1994) states that p
_{n} = d
_{n}. Here we show p
_{n} = n
^{2}/2 + φ(n
^{3/2}) = 2d
_{n} + φ(n
^{3/2})p
_{n} = n
^{2}/2 + φ(n
^{3/2}) = 2d
_{n} + φ(n
^{3/2}), and construct counterexamples to the DJL conjecture for all n ≥ 12 (for orders seven through eleven counterexamples were already given in [I. M. Bomze, W. Schachinger, and R. Ullrich, Linear Algebra Appl., 459 (2014), pp. 208-221].

Original language | English |
---|---|

Pages (from-to) | 20-37 |

Number of pages | 18 |

Journal | SIAM Journal on Matrix Analysis and Applications |

Volume | 36 |

Issue number | 1 |

Early online date | 22 Jan 2015 |

DOIs | |

Publication status | Published - 2015 |

## Austrian Fields of Science 2012

- 101015 Operations research

## Keywords

- COPOSITIVE OPTIMIZATION
- POSITIVE MATRICES
- completely positive matrices
- copositive optimization
- nonnegative factorization
- Nonnegative factorization
- Completely positive matrices
- Copositive optimization