On equivalent representations and properties of faces of the cone of copositive matrice

The paper is devoted to a study of the cone $\cop$ of copositive matrices. Based on the known from semi-infinite optimization concept of immobile indices, we define zero and minimal zero vectors of a subset of the cone $\cop$ and use them to obtain different representations of faces of $\cop$ and the corresponding dual cones. We describe the minimal face of $\cop$ containing a given convex subset of this cone and prove some propositions that allow to obtain equivalent descriptions of the feasible sets of a copositive problems and may be useful for creating new numerical methods based on their regularization.