Considering copositivity locally

Peter James Clair Dickinson, Roland Hildebrand

Research output: Contribution to journalArticleAcademicpeer-review

4 Citations (Scopus)
32 Downloads (Pure)

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 languageEnglish
Pages (from-to)1184-1195
Number of pages11
JournalJournal of mathematical analysis and applications
Volume437
Issue number2
DOIs
Publication statusPublished - 15 May 2016

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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.

Research output: Contribution to journalArticleAcademicpeer-review

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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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