  This paper studies RIS-aided cell-free massive MIMO systems, where multiple
RISs are deployed to assist the communication between multiple access points
(APs) and multiple users, with either continuous or discrete phase shifts at
the RISs. We formulate the max-min fairness problem that maximizes the minimum
achievable rate among all users by jointly optimizing the transmit beamforming
at active APs and the phase shifts at passive RISs, subject to power
constraints at the APs. To address such a challenging problem, we first study
the special single-user scenario and propose an algorithm that can transform
the optimization problem into semidefinite program (SDP) or integer linear
program (ILP) for the cases of continuous and discrete phase shifts,
respectively. By solving the resulting SDP and ILP, we first obtain the optimal
phase shifts, and then design the optimal transmit beamforming accordingly. To
solve the optimization problem for the multi-user scenario and continuous phase
shifts at RISs, we extend the single-user algorithm and propose an alternating
optimization algorithm, which can first decompose the max-min fairness problem
into two subproblems related to transmit beamforming and phase shifts, and then
transform the two subproblems into second-order-cone program and SDP,
respectively. For the multi-user scenario and discrete phase shifts, the
max-min fairness problem is shown to be a mixed-integer non-linear program
(MINLP). To tackle it, we design a ZF-based successive refinement algorithm,
which can find a suboptimal transmit beamforming and phase shifts by means of
alternating optimization. Numerical results show that compared with benchmark
schemes of random phase shifts and without using RISs, the proposed algorithms
can significantly increase the minimum achievable rate among all users,
especially when the number of reflecting elements at each RIS is large.
