The Computation of the Capacity Region of the Discrete MAC is a Rank-One Non-Convex Optimization Problem

The computation of the channel capacity of discrete memoryless channels is a convex problem that can be efficiently solved using the Arimoto-Blahut (AB) iterative algorithm. However, the extension of this algorithm to the computation of capacity regions of multiterminal networks is not straightforward since its computation gives rise to non-convex problems. In this context, the AB algorithm has been only successfully extended to the calculation of the sum-capacity of the discrete memoryless multiple-access channel. However, the computation of the capacity region still requires the use of computationally demanding random search algorithms or brute force (full search) methods. In this paper, we first give an alternative reformulation of the problem that identifies the non-convexity as a rank-one constraint. We then propose an efficient algorithm to compute outer and inner bounds on the capacity region by relaxing the original problem and then by projecting the relaxed solution onto the original space variable via a minimum divergence criterion. There exists a class of channels for which the proposed algorithm can be shown to compute exactly the capacity region. As an illustration, we analyze two particular channels, the binary adder MAC and the binary switching MAC, in detail. In the general case, the algorithm is able to compute very tight bounds as shown by simulation.