Exterior Point Method for Completely Positive Factorization

TL;DR

This paper introduces an exterior point iteration method for completely positive factorization, transforming symmetric matrices to nonnegative form efficiently and accurately, outperforming existing algorithms especially in challenging cases.

Contribution

It proposes a novel exterior point iteration approach for CPF, with convergence analysis and superior performance over existing methods.

Findings

Outperforms other algorithms in efficiency and accuracy

Effective in hard CPF cases

Convergence to local or global optima

Abstract

Completely positive factorization (CPF) is a critical task with applications in many fields. This paper proposes a novel method for the CPF. Based on the idea of exterior point iteration, an optimization model is given, which aims to orthogonally transform a symmetric lower rank factor to be nonnegative. The optimization problem can be solved via a modified nonlinear conjugate gradient method iteratively. The iteration points locate on the exterior of the orthonormal manifold and the closed set whose transformed matrices are nonnegative before convergence generally. Convergence analysis is given for the local or global optimum of the objective function, together with the iteration algorithm. Some potential issues that may affect the CPF are explored numerically. The exterior point method performs much better than other algorithms, not only in the efficiency of computational cost or accuracy, but also in the ability to address the CPF in some hard cases.