Maximizing the Partial Decode-and-Forward Rate in the Gaussian MIMO Relay Channel

It is known that circularly symmetric Gaussian signals are the optimal input signals for the partial decode-and-forward (PDF) coding scheme in the Gaussian multiple-input multiple-output (MIMO) relay channel, but there is currently no method to find the optimal covariance matrices nor to compute the optimal achievable PDF rate since the optimization is a non-convex problem in its original formulation. In this paper, we show that it is possible to find a convex reformulation of the problem by means of an approximation and a primal decomposition. We derive an explicit solution for the inner problem as well as an explicit gradient for the outer problem, so that the efficient cutting-plane method can be applied for solving the outer problem. Based on these ingredients, we propose a convergent algorithm whose output is an approximation of the globally optimal solution along with a certificate of accuracy in terms of a maximum distance to the true global optimum. In numerical simulations, this distance converged to values close to zero in all considered instances of the problem, showing that the proposed method manages to find the global optimum in all these instances.