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This work considers a Gaussian multi-antenna broadcast channel with individual power constraints on each antenna, rather than the usual sum power constraint over all antennas. Per-antenna power constraints are more realistic because in practical implementations each antenna has its own power amplifier. The main contribution of this paper is a new derivation of the duality result for this class of broadcast channels that allows the input optimization problem to be solved efficiently. Specifically, we show that uplink-downlink duality is equivalent to Lagrangian duality in minimax optimization, and the dual multiple-access problem has a much lower computational complexity than the original problem. This duality applies to the entire capacity region. Further, we derive a novel application of Newton's method for the dual minimax problem that finds an optimal search direction for both the minimization and the maximization problems at the same time. This new computational method is much more efficient than the previous iterative water-filling-based algorithms and it is applicable to the entire capacity region. Finally, we show that the previous QR-based precoding method can be easily modified to accommodate the per-antenna constraint.
Lan et al. (Tue,) studied this question.