Considering copositivity locally

Abstract

Let $A$ be an element of the copositive cone $coposn$. A zero $vu$ of $A$ is a nonnegative vector whose elements sum up to one and such that $vu^TAvu = 0$. The support of $vu$ is the index set $Suppvu subset 1,dots,n$ corresponding to the nonzero entries of $vu$. A zero $vu$ of $A$ is called minimal if there does not exist another zero $vv$ of $A$ such that its support $Suppvv$ is a strict subset of $Suppvu$. Our main result is a characterization of the cone of feasible directions at $A$, i.e., the convex cone $VarKA$ of real symmetric $n times n$ matrices $B$ such that there exists $delta > 0$ satisfying $A + delta B in coposn$. 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 $coposn$, and of the minimal exposed face of $A$ in $coposn$, 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 $coposn$ 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 $A - delta C notin coposn$.