This paper shows the capability the alternating direction method of multipliers (ADMM) has to track, in a distributed manner, the optimal down-link beam-forming solution in a multiple input multiple output (MISO) multi-cell 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 (CSI), but will require to exchange interference values once. We show ADMMs tracking ability in terms of the algorithms Lyapunov function given that the primal and dual solutions to the convex optimization problem at hand can be understood as a continuous mapping from the problems parameters. We show that this holds true even considering that the problem looses strong convexity when it is made distributed. We then show that these requirements hold for the down-link, and consequently up-link, beam-forming case. Numerical examples corroborating the theoretical findings are also provided.
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