Lorentz-positive mapswith applications to robust MISO downlink beamforming

Consider a unicast downlink beamforming optimization problem with robust signal-to-interference-plus-noise ratio constraints to account for non-perfect channel state information at the base station. The convexity of the robust beamforming problem remains unknown. A slightly conservative version of the robust beamforming problem is thus studied herein as a compromise. It is in the form of a semi-infinite second-order cone program (SOCP), and more importantly, it possesses an equivalent and explicit convex reformulation, due to an linear matrix inequality description of the cone of Lorentz-positive maps. Hence the robust beamforming problem can be efficiently solved by an optimization solver. The simulation results show that the conservativeness of the robust form of semi-infinite SOCP is appropriate in terms of problem feasibility rate and the average transmission power.