  Reconfigurable intelligent surfaces (RISs) represent a new technology that
can shape the radio wave propagation and thus offers a great variety of
possible performance and implementation gains. Motivated by this, we
investigate the achievable sum-rate optimization in a broadcast channel (BC)
that is equipped with an RIS. We exploit the well-known duality between the
Gaussian multiple-input multiple-output (MIMO) BC and multiple-access channel
(MAC) to derive an alternating optimization (AO) algorithm which optimizes the
users' covariance matrices and the RIS phase shifts in the dual MAC. The
optimal users' covariance matrices are obtained by a dual decomposition method
in which each iteration is solved in closed-form. The optimal RIS phase shifts
are also computed using a derived closed-form expression. Furthermore, we
present a computational complexity analysis for the proposed AO algorithm.
Simulation results show that the proposed AO algorithm can provide significant
achievable sum-rate gains in a BC.
