### Abstract

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 |

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### Keywords

- Copositive matrix
- Extreme rays
- Irreducibility
- Face

### Cite this

*Journal of mathematical analysis and applications*,

*437*(2), 1184-1195. https://doi.org/10.1016/j.jmaa.2016.01.063

}

*Journal of mathematical analysis and applications*, vol. 437, no. 2, pp. 1184-1195. https://doi.org/10.1016/j.jmaa.2016.01.063

**Considering copositivity locally.** / Dickinson, Peter James Clair; Hildebrand, Roland.

Research output: Contribution to journal › Article › Academic › peer-review

TY - JOUR

T1 - Considering copositivity locally

AU - Dickinson, Peter James Clair

AU - Hildebrand, Roland

N1 - eemcs-eprint-26916

PY - 2016/5/15

Y1 - 2016/5/15

N2 - 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 $C$.

AB - 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 $C$.

KW - Copositive matrix

KW - Extreme rays

KW - Irreducibility

KW - Face

U2 - 10.1016/j.jmaa.2016.01.063

DO - 10.1016/j.jmaa.2016.01.063

M3 - Article

VL - 437

SP - 1184

EP - 1195

JO - Journal of mathematical analysis and applications

JF - Journal of mathematical analysis and applications

SN - 0022-247X

IS - 2

ER -