  In this work, two problems associated with a downlink multi-user system are
considered with the aid of intelligent reflecting surface (IRS): weighted
sum-rate maximization and weighted minimal-rate maximization. For the first
problem, a novel DOuble Manifold ALternating Optimization (DOMALO) algorithm is
proposed by exploiting the matrix manifold theory and introducing the
beamforming matrix and reflection vector using complex sphere manifold and
complex oblique manifold, respectively, which incorporate the inherent
geometrical structure and the required constraint. A smooth double manifold
alternating optimization (S-DOMALO) algorithm is then developed based on the
Dinkelbach-type algorithm and smooth exponential penalty function for the
second problem. Finally, possible cooperative beamforming gain between IRSs and
the IRS phase shift with limited resolution is studied, providing a reference
for practical implementation. Numerical results show that our proposed
algorithms can significantly outperform the benchmark schemes.
