  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 ADMM's tracking
ability in terms of the algorithm's 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 problem's 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.
