- Considering Copositivity Locally Peter J.C. Dickinson (p.j.c.dickinsonutwente.nl) Roland Hildebrand (Roland.Hildebrandimag.fr) Abstract: Let $A$ be an element of the copositive cone $\mathcal{COP}^n$. A zero $\mathbf{u}$ of $A$ is a nonnegative vector whose elements sum up to one and such that $\mathbf{u}^TA\mathbf{u} = 0$. The support of $\mathbf{u}$ is the index set $\mathrm{supp}\mathbf{u} \subset \{1,\dots,n\}$ corresponding to the nonzero entries of $\mathbf{u}$. A zero $\mathbf{u}$ of $A$ is called minimal if there does not exist another zero $\mathbf{v}$ of $A$ such that its support $\mathrm{supp}\mathbf{v}$ is a strict subset of $\mathrm{supp}\mathbf{u}$. Our main result is a characterization of the cone of feasible directions at $A$, i.e., the convex cone $\mathcal{K}^A$ of real symmetric $n \times n$ matrices $B$ such that there exists $\delta > 0$ satisfying $A + \delta B \in \mathcal{COP}^n$. This cone is described by a set of linear inequalities on the elements of $B$ constructed from the set of zeros of $A$ and their supports. This characterization furnishes descriptions of the minimal face of $A$ in $\mathcal{COP}^n$, and of the minimal exposed face of $A$ in $\mathcal{COP}^n$, by sets of linear equalities and inequalities constructed from the set of minimal zeros of $A$ and their supports. In particular, we can check whether $A$ lies on an extreme ray of $\mathcal{COP}^n$ by examining the solution set of a system of linear equations. In addition, we deduce a simple necessary and sufficient condition on the irreducibility of $A$ with respect to a copositive matrix $C$. Here $A$ is called irreducible with respect to $C$ if for all $\delta > 0$ we have $A - \delta C \not\in \mathcal{COP}^n$. Keywords: copositive matrix, face, irreducibility, extreme rays Category 1: Linear, Cone and Semidefinite Programming (Other ) Citation: Submitted Download: [PDF]Entry Submitted: 04/14/2014Entry Accepted: 04/14/2014Entry Last Modified: 02/01/2016Modify/Update this entry Visitors Authors More about us Links Subscribe, Unsubscribe Digest Archive Search, Browse the Repository Submit Update Policies Coordinator's Board Classification Scheme Credits Give us feedback Optimization Journals, Sites, Societies Optimization Online is supported by the Mathematical Optmization Society.