The CP-matrix completion problem

TL;DR

This paper introduces a semidefinite algorithm to determine whether a partial symmetric matrix can be completed to a completely positive matrix, providing certificates of non-completability and solutions when possible.

Contribution

It proposes a novel semidefinite approach for the CP-completion problem, with theoretical guarantees and practical computational experiments.

Findings

Algorithm can certify non-CP-completability.

Algorithm almost always finds a CP-completion if it exists.

Computational experiments demonstrate effectiveness of the method.

Abstract

A symmetric matrix $C$ is completely positive (CP) if there exists an entrywise nonnegative matrix $B$ such that $C=BB^T$. The CP-completion problem is to study whether we can assign values to the missing entries of a partial matrix (i.e., a matrix having unknown entries) such that the completed matrix is completely positive. We propose a semidefinite algorithm for solving general CP-completion problems, and study its properties. When all the diagonal entries are given, the algorithm can give a certificate if a partial matrix is not CP-completable, and it almost always gives a CP-completion if it is CP-completable. When diagonal entries are partially given, similar properties hold. Computational experiments are also presented to show how CP-completion problems can be solved.