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 vuTAvu=0. The support of vu is the index set Suppvusubset1,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 ntimesn matrices B such that there exists delta>0 satisfying A+deltaBincoposn. 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−deltaCnotincoposn.