Minimax duality of Gaussian vector broadcast channels

This paper establishes a connection between the uplink-downlink duality of the Gaussian vector multiple-access channel and broadcast channel and the Lagrangian duality in minimax optimization. This new minimax duality allows the optimal transmit covariance matrix and the least-favorable noise for the broadcast channel to be characterized in terms of the dual variables. Further, it allows uplink-downlink duality to be generalized to broadcast channels with arbitrary linear constraints. In particular, it shows that the dual of a broadcast channel with individual per-antenna power constraint is a multiple-access channel with a diagonal uncertain noise.