  This paper considers an artificial noise (AN) aided secrecy rate maximization
(SRM) problem for a multi-input single-output (MISO) channel overheard by
multiple single-antenna eavesdroppers. We assume that the transmitter has
perfect knowledge about the channel to the desired user but imperfect knowledge
about the channels to the eavesdroppers. Therefore, the resultant SRM problem
is formulated in the way that we maximize the worst-case secrecy rate by
jointly designing the signal covariance ${\bf W}$ and the AN covariance ${\bf
\Sigma}$. However, such a worst-case SRM problem turns out to be hard to
optimize, since it is nonconvex in ${\bf W}$ and ${\bf \Sigma}$ jointly.
Moreover, it falls into the class of semi-infinite optimization problems.
Through a careful reformulation, we show that the worst-case SRM problem can be
handled by performing a one-dimensional line search in which a sequence of
semidefinite programs (SDPs) are involved. Moreover, we also show that the
optimal ${\bf W}$ admits a rank-one structure, implying that transmit
beamforming is secrecy rate optimal under the considered scenario. Simulation
results are provided to demonstrate the robustness and effectiveness of the
proposed design compared to a non-robust AN design.
