On dimensions of maximal faces of completely positive cones

Because of the lack of characterizations of exposed extreme rays of the n× n copositive cone in general except for n≤ 6, by no means so far can we characterize all maximal faces of the n× n completely positive cone for n≥ 7. In this paper, we use the information of the maximal faces of lower order completely positive cones to study the dimensions of a class of maximal faces of higher order completely positive cones. Specifically, we establish a connection between the dimension of a maximal face of a lower order completely positive cone and the dimension of a maximal face of a higher order completely positive cone via a connection between exposed rays of a lower order copositive cone and a higher order copositive cone. Such a connection is used to find formulas for the dimensions of a certain class of maximal faces of higher order completely positive cones, which has not been studied in the related literature to the best of our knowledge.