  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.
