  In this paper, we investigate an intelligent reflecting surface (IRS)
assisted cell-free multiple input multiple output (MIMO) communication system,
where distributed multiple IRSs are dedicated deployed to assist distributed
multiple base stations (BSs) for cooperative transmission. Our objective is to
maximize the achievable sum-rate of the cell-free system by jointly optimizing
the active transmit beamforming matrices at BSs and the passive reflecting
beamforming matrices at IRSs, while the constraints on the maximum transmit
power of each BS and the phase shift of each IRS element are satisfied. We
propose an efficient framework to jointly design the BSs, the IRSs, and the
user equipment (UEs). As a compromise approach, we first transform the
non-convex problem into an equivalent form based on the fractional programming
methods and then decompose the reformulated problem into two subproblems and
solve them alternately. Particularly, we propose a Lagrangian dual sub-gradient
based algorithm to solve the subproblem of optimizing the active transmit
beamforming with nearly closed-form solutions. We reformulate the subproblem of
optimizing the passive reflecting beamforming as a constant modulus constrained
quadratic programming (CMC-QP) problem. We first provide two feasible solutions
by proposing a pair of relaxation-based algorithms. We also develop a
low-complexity alternating sequential optimization (ASO) algorithm to obtain
closed-form solutions. All three algorithms are guaranteed to converge to
locally optimal solutions. Simulation results demonstrate that the proposed
algorithms achieve considerable performance improvements compared with the
benchmark schemes.
