Non-exposed polyhedral faces of the completely positive cone

Abstract

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 C P ( 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.