Comparison of ℓ_∞-norm and ℓ₁-norm optimization criteria for SIR-balanced multi-user beamforming

We address the problem of joint downlink beamforming and power control in an interference limited wireless system under hard fairness constraints. The transmitting base station is equipped with an antenna array, whereas the mobiles are equipped with single antennas. Transmit channel state information is available. The design goal: control the SIR levels in order to fulfill individual target thresholds. This leads to an ℓ_∞-norm minimization problem, which has no obvious solution. Thus, Gerlach and Paulraj (Signal Processing 54 (1996) 59) proposed an alternative optimization strategy that is based on ℓ₁-norm minimization. In this paper, we study whether or not this approach leads to the optimal solution. We provide necessary and sufficient conditions for the equivalence of ℓ _∞ and ℓ₁ optimization. These conditions are not always fulfilled, as can be shown by counterexamples. An exceptional case, however, is the 2-user scenario, for which both concepts are indeed equivalent and the solution can be given in closed form. For arbitrary numbers of users, we propose a modified optimization scheme that alternately performs power control and beamforming optimization based on the ℓ₁ cost function. We prove the convergence of this iterative algorithm by showing that the iteration is a monotonic sequence. Each local optimum is a global optimum of the ℓ_∞-norm minimization problem.