Non-exposed polyhedral faces of the completely positive cone

In this paper, we consider the cone of p×p completely positive matrices CP(p). Currently, some families of non-exposed faces of the 5×5 completely positive cone CP(5) were constructed. Inspired by these results, in this paper, we continue the study of the existence and properties of non-exposed faces of CP(p) for any p≥5. We prove a criterion for a polyhedral sub-cone of the cone CP(p) to be a non-exposed face of this cone. We also provide several types of sufficient conditions that can be easily checked numerically. For any p≥5 we give examples of non-exposed faces of CP(p). By doing this, we demonstrate that the existence of non-exposed faces is not an exclusive feature of only 5×5 completely positive cone.