Convex transmit beamforming for downlink multicasting to multiple co-channel groups

We consider the problem of transmit beamforming to multiple co-channel multicast groups. Since the direct minimization of transmit power while guaranteeing a prescribed minimum signal to interference plus noise ratio (SINR) at each receiver is nonconvex and NP-hard, we present convex SDP relaxations of this problem and study when such relaxations are tight. Our results show that when the steering vectors for all receivers are of Vandermonde type (such as in the case of a uniform linear array and line-of-sight propagation), a globally optimum solution to the corresponding transmit beamforming problem can be obtained via an equivalent SDP reformulation. We also present various robust formulations for the problem of single-group multicasting, when the steering vectors are only approximately known. Simulation results are presented to illustrate the effectiveness of our SDP relaxations and reformulations.