The worst-case robust adaptive beamforming problem for generalrank signal model is considered. This is a nonconvex problem, and an approximate version of it (by introducing a matrix decomposition on the presumed covariance matrix of the desired signal) has been studied in the literature. Herein the original robust adaptive beamforming problem is tackled. Resorting to the strong duality of a linear conic program, the robust beamforming problem is reformulated into a quadratic matrix inequality (QMI) problem. There is no general method for solving a QMI problem in the literature. Here- in, employing a linear matrix inequality (LMI) relaxation technique, the QMI problem is turned into a convex semidefinite programming problem. Due to the fact that there often is a positive gap between the QMI problem and its LMI relaxation, a deterministic approximate algorithm is proposed to solve the robust adaptive beamforming in the QMI form. Last but not the least, a sufficient optimality condition for the existence of an optimal solution for the QMI problem is derived. To validate our theoretical results, simulation examples are presented, which also demonstrate the improved performance of the new robust beamformer in terms of the output signal-to-interference- plus-noise ratio.