Optimum SIR balancing using extended 1-norm beamforming optimization

We address the problem of joint downlink beamforming and power control in a wireless cellular system, where several users share the same time slot and the same frequency band. In order to enable a fair sharing of resources among all users, the SIR levels should be balance by minimizing the ∞-norm of a certain vector. To this end, Gerlach and Paulraj [1] proposed an iterative algorithm, which is based on a 1-norm cost function. It has been observed that this algorithm approximates the optimum ∞-norm solution. In this paper we characterize the case where this algorithm is optimal. The analytical results are used to develop a new algorithm, which always converges to the optimum ∞-norm solution, regardless of the chosen initialization. A proof of convergence is given.