Lorentz-positive maps and quadratic matrix inequalities with applications to robust MISO transmit beamforming

Consider a unicast downlink beamforming optimization problem with robust signal-to-interference-plus-noise ratio constraints to account for imperfect channel state information at the base station in a multiple-input single-output (MISO) communication system. The convexity of this 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 a linear matrix inequality (LMI) description of the cone of Lorentz-positive maps. Hence, the conservative robust beamforming problem can be efficiently solved by an optimization solver. Additional robust shaping constraints can also be easily handled to control the amount of interference generated on other co-existing users such as in cognitive radio systems.