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This paper considers a scenario where multiple antenna base stations use transmit beamforming to communicate with single antenna mobiles with minimum receive signal-to-interference ratio (SIR) constraints. In this system, minimizing the total power expenditure requires solving a non-convex optimization problem. A semi-definite program (SDP) relaxation can be used to convert the non-convex problem to a centralized SDP that can be solved efficiently and optimally with convex optimization techniques. Using the properties of the dual of this SDP relaxation, a two-stage iterative algorithm is proposed to solve the decentralized version of the problem. In the first stage, limited amount of centralized feedback is used to find the optimum beamforming directions through distributed eigenvalue computations. The second stage uses the distributed autonomous power control technique proposed by Foschini and Miljanic for single antenna systems.
Ekbal et al. (Tue,) studied this question.