## Abstract

Let p _{n} denote the maximal cp-rank attained by completely positive n × n matrices. Only lower and upper bounds for p _{n} are known, when n ≥ 6, but it is known that (Formula Presented), and the difference of the current best upper and lower bounds for p _{n} is of order O(n ^{3/2}). In this paper, that gap is reduced to O(n log log n). 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 dth 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.

Original language | English |
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Pages (from-to) | 519-541 |

Number of pages | 23 |

Journal | The Electronic Journal of Linear Algebra |

Volume | 36 |

Issue number | 1 |

DOIs | |

Publication status | Published - 11 Aug 2020 |

## Austrian Fields of Science 2012

- 101016 Optimisation
- 101015 Operations research

## Keywords

- completely positive matrices
- completely positive tensors
- cp-rank
- copositive optimization
- . Completely positive matrices
- Cp-rank
- Completely positive tensors
- Copositive optimization