  This paper considers a reconfigurable intelligent surface (RIS)-aided
millimeter wave (mmWave) downlink communication system where hybrid
analog-digital beamforming is employed at the base station (BS). We formulate a
power minimization problem by jointly optimizing hybrid beamforming at the BS
and the response matrix at the RIS, under signal-to-interference-plus-noise
ratio (SINR) constraints. The problem is highly challenging due to the
non-convex SINR constraints as well as the non-convex unit-modulus constraints
for both the phase shifts at the RIS and the analog beamforming at the BS. A
penalty-based algorithm in conjunction with the manifold optimization technique
is proposed to handle the problem, followed by an individual optimization
method with much lower complexity. Simulation results show that the proposed
algorithm outperforms the state-of-art algorithm. Results also show that the
joint optimization of RIS response matrix and BS hybrid beamforming is much
superior to individual optimization.
