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This paper considers downlink multi-user transmission facilitated by areconfigurable intelligent surface (RIS). First, focusing on the multi-groupmulticast beamforming scenario, we develop a fast and scalable algorithm forthe joint base station (BS) and RIS beamforming optimization to minimize thetransmit power subject to the user quality-of-service (QoS) constraints. Byexploring the structure of this QoS problem, we show that the joint beamformingoptimization can be naturally decomposed into a BS multicast beamforming QoSproblem and an RIS passive multicast beamforming max-min-fair (MMF) problem. Wepropose an alternating multicast beamforming (AMBF) algorithm to solve the twosubproblems alternatingly. For the BS QoS subproblem, we utilize the optimalmulticast beamforming structure to obtain the BS beamformers efficiently.Furthermore, we reformulate the challenging RIS MMF subproblem and employ afirst-order projected subgradient algorithm (PSA), which yields closed-formupdates. The computational complexity of the AMBF algorithm grows linearly withthe number of RIS elements and BS antennas. We further show that the AMBFapproach is also an efficient method for the RIS-assisted downlink multi-userunicast beamforming problem, providing semi-closed-form updates. Next, we studythe MMF problem for the RIS-assisted downlink beamforming design and propose aPSA-based fast algorithm to compute the BS and RIS beamforming solutions withclosed-form updates per iteration, leading to a highly computationallyefficient solution. Simulation results show the efficacy of our proposedalgorithms in both performance and computational cost compared to otheralternative methods.

Efficient Design for Multi-user Downlink Beamforming with Reconfigurable Intelligent Surface