ADMM for Distributed Dynamic Beamforming

This paper shows the capability of the alternating direction method of multipliers (ADMM) to track, in a distributed manner, the optimal down-link beam-forming solution in a multiple-input single-output multicell network given a dynamic channel. Each time the channel changes, ADMM is allowed to perform one algorithm iteration. In order to implement the proposed scheme, the base stations are not required to exchange channel state information. They will, however, be required to exchange interference values once. We show ADMM's tracking ability in terms of the algorithm's Lyapunov function. This is shown given that the primal and dual solutions to the convex optimization problem at hand can be understood as a continuous mapping from the problem's parameters. We show that this holds true even considering that the problem loses strong convexity when it is made distributed. We then show that these requirements hold for the down-link, and consequently the up-link, beam-forming case. Numerical examples corroborating the theoretical findings are also provided.