  This paper tackles the problem of joint active and passive beamforming
optimization for an intelligent reflective surface (IRS)-assisted multi-user
downlink multiple-input multiple-output (MIMO) communication system. We aim to
maximize spectral efficiency of the users by minimizing the mean square error
(MSE) of the received symbol. For this, a joint optimization problem is
formulated under the minimum mean square error (MMSE) criterion. First, block
coordinate descent (BCD) is used to decouple the joint optimization into two
sub-optimization problems to separately find the optimal active precoder at the
base station (BS) and the optimal matrix of phase shifters for the IRS. While
the MMSE active precoder is obtained in a closed form, the optimal phase
shifters are found iteratively using a modified version (also introduced in
this paper) of the vector approximate message passing (VAMP) algorithm. We
solve the joint optimization problem for two different models for IRS phase
shifts. First, we determine the optimal phase matrix under a unimodular
constraint on the reflection coefficients, and then under the constraint when
the IRS reflection coefficients are modeled by a reactive load, thereby
validating the robustness of the proposed solution. Numerical results are
presented to illustrate the performance of the proposed method using multiple
channel configurations. The results validate the superiority of the proposed
solution as it achieves higher throughput compared to state-of-the-art
techniques.
