# Considering copositivity locally

Peter James Clair Dickinson, Roland Hildebrand

5 Citations (Scopus)

### 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 1184-1195 11 Journal of mathematical analysis and applications 437 2 https://doi.org/10.1016/j.jmaa.2016.01.063 Published - 15 May 2016

### Fingerprint

Cones
Cone
Symmetric matrix
Irreducibility
Convex Cone
System of Linear Equations
Solution Set
Half line
Linear Inequalities
Deduce
Extremes
Linear equations
Non-negative
Face
Necessary Conditions
Sufficient Conditions
Zero

### Keywords

• Copositive matrix
• Extreme rays
• Irreducibility
• Face

### Cite this

Dickinson, Peter James Clair ; Hildebrand, Roland. / Considering copositivity locally. In: Journal of mathematical analysis and applications. 2016 ; Vol. 437, No. 2. pp. 1184-1195.
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Considering copositivity locally. / Dickinson, Peter James Clair; Hildebrand, Roland.

In: Journal of mathematical analysis and applications, Vol. 437, No. 2, 15.05.2016, p. 1184-1195.

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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 $C$.

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KW - Extreme rays

KW - Irreducibility

KW - Face

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