  This paper considers the design of the beamformers for a multiple-input
single-output (MISO) downlink system that seeks to mitigate the impact of the
imperfections in the channel state information (CSI) that is available at the
base station (BS). The goal of the design is to minimize the outage probability
of specified signal-to-interference-and-noise ratio (SINR) targets, while
satisfying per-antenna power constraints (PAPCs), and to do so at a low
computational cost. Based on insights from the offset maximization technique
for robust beamforming, and observations regarding the structure of the
optimality conditions, low-complexity iterative algorithms that involve the
evaluation of closed-form expressions are developed. To further reduce the
computational cost, algorithms are developed for per-antenna power-constrained
variants of the zero-forcing (ZF) and maximum ratio transmission (MRT)
beamforming directions. In the MRT case, our low-complexity version for systems
with a large number of antennas may be of independent interest. The proposed
algorithms are extended to systems with both PAPCs and a total power
constraint. Simulation results show that the proposed robust designs can
provide substantial gains in the outage probability while satisfying the PAPCs.
