Multi-group Multicast Beamforming: Optimal Structure and Efficient Algorithms
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This paper considers the multi-group multicast beamforming optimization problem, for which the optimal solution has been unknown due to its non-convex and NP-hard nature. By utilizing the successive convex approximation numerical method and Lagrangian duality, we obtain the optimal multicast beamforming solution in a semi-closed form for both the quality-of-service (QoS) problem and the max-min fair (MMF) problem. From the optimal beamforming structure obtained, we show that the notion of uplink-downlink duality can be generalized to the multicast beamforming problem. The optimal multicast beamformer is a weighted MMSE filter based on a group-channel direction – a generalized version of the optimal downlink multi-user unicast beamformer. We also show that there is an inherent low-dimensional structure in the optimal beamforming solution independent of the number of transmit antennas, leading to efficient numerical algorithm design, especially for systems with large antenna arrays. We propose efficient algorithms to compute the multicast beamformer based on the optimal beamforming structure. We characterize the asymptotic behavior of the beamformers through asymptotic analysis, and in turn, provide simple closed-form approximate multicast beamformers for both the QoS and MMF problems. The approximation offers practical multicast beamforming solutions with a near-optimal performance at very low computational complexity for large-scale antenna systems.
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