Copositive optimization is a special case of convex conic programming, and it optimizes 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 of such open problems; finding a completely positive (CP) factorization for a given completely positive matrix. We treat it as a nonsmooth Riemannian optimization, i.e., a minimization of a nonsmooth function over the Riemannian manifolds. To solve this problem, we present a general smoothing framework for nonsmooth Riemannian optimization and guarantee 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.
Department of Policy and Planning Sciences Discussion Paper Series No. 1377, to appear in Computational Optimization and Applications