The cone of $5\times 5$ completely positive matrices

TL;DR

This paper investigates the structure and boundary of the cone of 5x5 completely positive matrices, providing characterizations, a numerical algorithm for cp-factorization, and insights into the complexity of the problem.

Contribution

It offers the first detailed analysis of the 5x5 cp cone, including boundary characterization and a fast cp-factorization algorithm for boundary matrices.

Findings

Characterized the boundary loci of the 5x5 cp cone

Developed a fast numerical cp-factorization algorithm

Illustrated the complexity of cp-factorization with numerical experiments

Abstract

We study the cone of completely positive (cp) matrices for the first interesting case $n = 5$. This is a semialgebraic set, which means that the polynomial equalities and inequlities that define its boundary can be derived. We characterize the different loci of this boundary and we examine the two open sets with cp-rank 5 or 6. A numerical algorithm is presented that is fast and able to compute the cp-factorization even for matrices in the boundary. With our results, many new example cases can be produced and several insightful numerical experiments are performed that illustrate the difficulty of the cp-factorization problem.