Completely Positive Factorization by a Riemannian Smoothing Method

TL;DR

This paper introduces a novel Riemannian smoothing method for efficiently computing completely positive factorizations of matrices, addressing a key open problem in copositive optimization with promising large-scale results.

Contribution

It develops a general smoothing framework for nonsmooth Riemannian optimization problems and demonstrates its effectiveness for CP matrix factorization.

Findings

Efficiently computes CP factorizations for large matrices.

Converges to stationary points with minimal implementation effort.

Outperforms existing methods in numerical experiments.

Abstract

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.