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 problems 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. As the accuracy of this provably convergent algorithm might be impaired by the previous approximation, we additionally propose a modified algorithm, for which we cannot give a converge guarantee, but which provides rigorous upper and lower bounds to the optimum of the original rate maximization. In numerical simulations, these bounds become tight in all considered instances of the problem, showing that the proposed method manages to find the global optimum in all these instances.