Distributionally robust chance-constrained transmit beamforming for multiuser MISO downlink

This paper considers robust transmit beamforming for multiuser multi-input single-output (MISO) downlink transmission, where imperfect channel state information (CSI) is assumed at the base station (BS). The imperfect CSI is captured by a moment-based random error model, in which the BS knows only the mean and covariance of each CSI error, but not the exact distribution. Under this error model, we formulate a distributionally robust beamforming (DRB) problem, in which the total transmit power at the BS is to be minimized, while each user´s SINR outage probability, evaluated w.r.t. any distribution with the given mean and covariance, is kept below a given threshold. The DRB problem is a semi-infinite chance-constrained problem. By employing recent results in distributionally robust optimization, we show that the DRB problem admits an explicit conic reformulation, which can be conveniently turned into a convex optimization problem after semidefinite relaxation (SDR). We also consider the case where the mean and covariance are not perfectly known. We show that the resulting DRB problem still admits a conic reformulation and can be approximately solved using SDR. The robustness of the proposed designs are demonstrated by numerical simulations.