The cone of 5× 5 completely positive matrices

2021/08/26 by Pfeffer, Max, Samper, Jose Alejandro
#Algebraic Geometry (math.AG) #FOS: Mathematics #Numerical Analysis (math.NA) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2108.11928

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.

Related

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• Interiors of completely positive cones
• Factorization of CP-rank-3 completely positive matrices
• The CP-matrix Approximation Problem
• Inner approximating the completely positive cone via the cone of scaled diagonally dominant matrices