Solvability of coupled downlink beamforming problems

We address the problem of joint downlink beamforming and power control in interference limited cellular wireless systems. The goal is to support each user with a certain signal-to-interference level at the mobile. Several iterative algorithms have recently been proposed in the literature to solve this non-linear optimization problem. Obviously, no solution exists if the requirements are too demanding. This explains situations where the algorithms do not converge. On the other hand, the observation of convergence is only a necessary condition for the existence of a solution. We investigate the analytical structure of the problem and formulate a unifying framework. In particular, we provide a necessary and sufficient condition which can be used to judge whether or not a solution can be considered as valid. The results give new insights and may prove useful for the design of new fast convergent algorithms.