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In this paper, system design for a multiple-input–multiple-output (MIMO) relay broadcast channel with individual signal-to-interference-plus-noise ratio (SINR) constraints at mobile stations (MSs) is considered. By exploring the structure of downlink–uplink duality at either the base station (BS) or the relay station (RS), we propose two schemes of joint power allocation and beamforming design at the BS and the RS. The problem of the existence of feasible solutions under practical power constraints at the BS and the RS with given SINR targets is considered first. Then, the problem of sum power minimization is considered. Each design problem can be efficiently solved using optimal joint power allocation and beamforming under the framework of convex optimization. We also show that with subchannel pairing at the RS, the transmission power can be further reduced. Finally, a generalization to the multihop scenario is provided to further improve power efficiency. [ABSTRACT FROM PUBLISHER]