  This paper concentrates on a robust transmit optimization problem for the
multiuser multi-input single-output (MISO) downlink scenario and under
inaccurate channel state information (CSI). This robust problem deals with a
general-rank transmit covariance design, and it follows a safe rate-constrained
formulation under spherically bounded CSI uncertainties. Curiously, simulation
results in previous works suggested that the robust problem admits rank-one
optimal transmit covariances in most cases. Such a numerical finding is
appealing because transmission with rank-one covariances can be easily realized
by single-stream transmit beamforming. This gives rise to a fundamentally
important question, namely, whether we can theoretically identify conditions
under which the robust problem admits a rank-one solution. In this paper, we
identify one such condition. Simply speaking, we show that the robust problem
is guaranteed to admit a rank-one solution if the CSI uncertainties are not too
large and the multiuser channel is not too poorly conditioned. To establish the
aforementioned condition, we develop a novel duality framework, through which
an intimate relationship between the robust problem and a related maximin
problem is revealed. Our condition involves only a simple expression with
respect to the multiuser channel and other system parameters. In particular,
unlike other sufficient rank-one conditions that have appeared in the
literature, ours is verifiable. The application of our analysis framework to
several other CSI uncertainty models is also discussed.
