Considering copositivity locally

Let A be an element of the copositive cone COP. A zero u of A is a nonnegative vector whose elements sum up to one and such that uTAu = 0. The support of u is the index set suppu ⊂ {1, . . . , n} corresponding to the nonzero entries of u. A zero u of A is called minimal if there does not exist another zero v of A such that its support suppv is a strict subset of suppu. Our main result is a characterization of the cone of feasible directions at A, i.e., the convex cone KA of real symmetric n × n matrices B such that there exists δ > 0 satisfying A + δB ∈ COP. 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 ofA in COP, and of the minimal exposed face of A in COP, by sets of linear equalities and inequalities constructed from the set of minimal zeros ofA and their supports. In particular, we can check whether A lies on an extreme ray of COP 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 δ > 0 we have A− δC 6∈ COP.