  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 the signal-to-interference-plus-noise
ratio (SINR) constraints at all users. The problem is highly challenging to
solve due to the non-convex SINR constraints as well as the unit-modulus phase
shift constraints for both the RIS reflection coefficients and the analog
beamformer. A two-layer penalty-based algorithm is proposed to decouple
variables in SINR constraints, and manifold optimization is adopted to handle
the non-convex unit-modulus constraints. {We also propose a low-complexity
sequential optimization method, which optimizes the RIS reflection
coefficients, the analog beamformer, and the digital beamformer sequentially
without iteration.} Furthermore, the relationship between the power
minimization problem and the max-min fairness (MMF) problem is discussed.
Simulation results show that the proposed penalty-based algorithm outperforms
the state-of-the-art semidefinite relaxation (SDR)-based algorithm. Results
also demonstrate that the RIS plays an important role in the power reduction.
