  This paper proposes an elegant optimization framework consisting of a mix of
linear-matrix-inequality and second-order-cone constraints. The proposed
framework generalizes the semidefinite relaxation (SDR) enabled solution to the
typical transmit beamforming problems presented in the form of quadratically
constrained quadratic programs (QCQPs) in the literature. It is proved that the
optimization problems subsumed under the framework always admit a rank-one
optimal solution when they are feasible and their optimal solutions are not
trivial. This finding indicates that the relaxation is tight as the optimal
solution of the original beamforming QCQP can be straightforwardly obtained
from that of the SDR counterpart without any loss of optimality. Four
representative examples of transmit beamforming, i.e., transmit beamforming
with perfect channel state information (CSI), transmit beamforming with
imperfect CSI, chance-constraint approach for imperfect CSI, and
reconfigurable-intelligent-surface (RIS) aided beamforming, are shown to
demonstrate how the proposed optimization framework can be realized in deriving
the SDR counterparts for different beamforming designs.
