### Abstract

We say that a symmetric matrix $A$ is copositive if $\mathbf{v}^T A\mathbf{v}\geq0$ for all nonnegative vectors $\mathbf{v}$. The main result of this paper is a characterization of the cone of feasible directions at a copositive matrix $A$, i.e., the convex cone of symmetric matrices $B$ such that there exists $\delta > 0$ satisfying $A + \delta B$ being copositive. This cone is described by a set of linear inequalities on the elements of $B$ constructed from the so called set of (minimal) zeros of $A$. This characterization is used to furnish descriptions of the minimal (exposed) face of the copositive cone containing $A$ in a similar manner. In particular, we can check whether $A$ lies on an extreme ray of the copositive cone by examining the solution set of a system of linear equations. In addition, we deduce a simple necessary and sufficient condition for the irreducibility of $A$ with respect to a copositive matrix $C$.

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

Pages (from-to) | 1184-1195 |

Number of pages | 11 |

Journal | Journal of mathematical analysis and applications |

Volume | 437 |

Issue number | 2 |

DOIs | |

Publication status | Published - 15 May 2016 |

### Keywords

- Copositive matrix
- Extreme rays
- Irreducibility
- Face

## Fingerprint Dive into the research topics of 'Considering copositivity locally'. Together they form a unique fingerprint.

## Cite this

Dickinson, P. J. C., & Hildebrand, R. (2016). Considering copositivity locally.

*Journal of mathematical analysis and applications*,*437*(2), 1184-1195. https://doi.org/10.1016/j.jmaa.2016.01.063