  This paper considers a formulation of the robust adaptive beamforming (RAB)
problem based on worst-case signal-to-interference-plus-noise ratio (SINR)
maximization with a nonconvex uncertainty set for the steering vectors. The
uncertainty set consists of a similarity constraint and a (nonconvex)
double-sided ball constraint. The worst-case SINR maximization problem is
turned into a quadratic matrix inequality (QMI) problem using the strong
duality of semidefinite programming. Then a linear matrix inequality (LMI)
relaxation for the QMI problem is proposed, with an additional valid linear
constraint. Necessary and sufficient conditions for the tightened LMI
relaxation problem to have a rank-one solution are established. When the
tightened LMI relaxation problem still has a high-rank solution, the LMI
relaxation problem is further restricted to become a bilinear matrix inequality
(BLMI) problem. We then propose an iterative algorithm to solve the BLMI
problem that finds an optimal/suboptimal solution for the original RAB problem
by solving the BLMI formulations. To validate our results, simulation examples
are presented to demonstrate the improved array output SINR of the proposed
robust beamformer.
