  We consider the problem of power allocation for the single-cell multi-user
(MU) multiple-input single-output (MISO) downlink with quality-of-service (QoS)
constraints. The base station acquires an estimate of the channels and, for a
given beamforming structure, designs the power allocation so as to minimize the
total transmission power required to ensure that target
signal-to-interference-and-noise ratios at the receivers are met, subject to a
specified outage probability. We consider scenarios in which the errors in the
base station's channel estimates can be modelled as being zero-mean and
Gaussian. Such a model is particularly suitable for time division duplex (TDD)
systems with quasi-static channels, in which the base station estimates the
channel during the uplink phase. Under that model, we employ a precise
deterministic characterization of the outage probability to transform the
chance-constrained formulation to a deterministic one. Although that
deterministic formulation is not convex, we develop a coordinate descent
algorithm that can be shown to converge to a globally optimal solution when the
starting point is feasible. Insight into the structure of the deterministic
formulation yields approximations that result in coordinate update algorithms
with good performance and significantly lower computational cost. The proposed
algorithms provide better performance than existing robust power loading
algorithms that are based on tractable conservative approximations, and can
even provide better performance than robust precoding algorithms based on such
approximations.
