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Mathematics > Optimization and Control

arXiv:2108.11928 (math)
[Submitted on 26 Aug 2021 (v1), last revised 1 Sep 2021 (this version, v2)]

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

Authors:Max Pfeffer, Jose Alejandro Samper
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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.
Comments: Comments are welcome
Subjects: Optimization and Control (math.OC); Algebraic Geometry (math.AG); Numerical Analysis (math.NA)
Cite as: arXiv:2108.11928 [math.OC]
  (or arXiv:2108.11928v2 [math.OC] for this version)
  https://doi.org/10.48550/arXiv.2108.11928

Submission history

From: Jose Alejandro Samper [view email]
[v1] Thu, 26 Aug 2021 17:32:19 UTC (23 KB)
[v2] Wed, 1 Sep 2021 14:07:48 UTC (23 KB)
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