Minimax Solution to the Partial Decode-and-Forward Rate Maximization in the MIMO Relay Channel

We propose a novel method for optimizing the achievable rate of the partial decode-and-forward (PDF) protocol in the multiple-input multiple-output (MIMO) relay channel via a reformulation to a convex-concave game. To this end, we show that the cut-set upper bound, which is loose in general, can be turned into a tight bound to the achievable PDF rate by adding a worst-case noise minimization. A solution to the PDF rate maximization problem can thus be obtained by solving the resulting minimax problem after reformulating it as a nested optimization consisting of a scalar convex outer problem and a convex-concave game as inner problem. As an efficient algorithmic solution to the latter, we propose a combination of the barrier method and the infeasible start Newton method while the former can be solved using bisection.