  Copositive optimization is a special case of convex conic programming, and it
consists of optimizing a linear function over the cone of all completely
positive matrices under linear constraints. Copositive optimization provides
powerful relaxations of NP-hard quadratic problems or combinatorial problems,
but there are still many open problems regarding copositive or completely
positive matrices. In this paper, we focus on one such problem; finding a
completely positive (CP) factorization for a given completely positive matrix.
We treat it as a nonsmooth Riemannian optimization problem, i.e., a
minimization problem of a nonsmooth function over a Riemannian manifold. To
solve this problem, we present a general smoothing framework for solving
nonsmooth Riemannian optimization problems and show convergence to a stationary
point of the original problem. An advantage is that we can implement it quickly
with minimal effort by directly using the existing standard smooth Riemannian
solvers, such as Manopt. Numerical experiments show the efficiency of our
method especially for large-scale CP factorizations.
