  In this letter, we investigate a robust beamforming problem for
localization-aided millimeter wave (mmWave) communication systems. To handle
this problem, we propose a novel restriction and relaxation (R&R) method. The
proposed R&R method aims at minimizing the total transmit power while the
positioning error follows a Gaussian distribution. Specifically, in the
restriction phase of R&R, the probabilistic constraint is transformed into the
deterministic form by using the Bernsteintype inequality. In the relaxation
phase of R&R, the non-convex optimization problem is reformulated into a convex
semidefinite program (SDP) by using semidefinite relaxation (SDR) and
firstorder Taylor expansion methods. To the best of our knowledge, we first
consider the impact of the distribution of the positioning error on the channel
state information (CSI), which further influences the data rate. Numerical
results present the trade-off of the beamforming between the communication and
positioning.
