We propose a reduced-complexity weighted minimum mean squared error (R-WMMSE) framework for cell-free massive MIMO (CF-mMIMO) systems with finite-capacity fronthaul links. To maximize the system capacity, we need to jointly design beamforming vectors at distributed access points (APs) and fronthaul quantization noise covariances. Such a task would be addressed using conventional WMMSE algorithms and their variants. However, existing WMMSE-based methods either assume ideal infinite-capacity fronthaul links or incur high computational complexity when jointly optimizing the network-wide beamforming and fronthaul compression variables. To this end, the proposed method jointly optimizes beamforming and fronthaul compression by leveraging a sequential per-AP optimization strategy. Since each subproblem is efficiently solved via Lagrangian duality with closed-form updates, the overall computational complexity can be significantly reduced compared to the conventional WMMSE method. To further reduce hardware cost, we extend the R-WMMSE framework to hybrid analog-digital beamforming systems, tackling the joint design of digital beamforming, fronthaul compression, and analog beamforming. Numerical results show that the proposed scheme achieves performance nearly identical to that of the conventional WMMSE while reducing the runtime by over 99%.

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