  Consider a robust multiple-input single-output downlink beamforming
optimization problem in a frequency division duplexing system. The base station
(BS) sends training signals to the users, and every user estimates the channel
coefficients, quantizes the gain and the direction of the estimated channel and
sends them back to the BS. Suppose that the channel state information at the
transmitter is imperfectly known mainly due to the channel direction
quantization errors, channel estimation errors and outdated channel effects.
The actual channel is modeled as in an uncertainty set composed of two
inequality homogeneous and one equality inhomogeneous quadratic constraints, in
order to account for the aforementioned errors and effects. Then the transmit
power minimization problem is formulated subject to robust
signal-to-noise-plus-interference ratio constraints. Each robust constraint is
transformed equivalently into a quadratic matrix inequality (QMI) constraint
with respect to the beamforming vectors. The transformation is accomplished by
an equivalent phase rotation process and the strong duality result for a
quadratically constrained quadratic program. The minimization problem is
accordingly turned into a QMI problem, and the problem is solved by a
restricted linear matrix inequality relaxation with additional valid convex
constraints. Simulation results are presented to demonstrate the performance of
the proposed method, and show the efficiency of the restricted relaxation.
