  The optimization of cooperative beamforming vectors in cell-free massive MIMO
(mMIMO) systems is presented where multi-antenna access points (APs) support
downlink data transmission of multiple users. Albeit the successes of the
weighted minimum mean squared error (WMMSE) algorithm and their variants, they
lack careful investigations about computational complexity that scales with the
number of antennas and APs. We propose a generalized and reduced WMMSE
(G-R-WMMSE) approach whose complexity is significantly lower than conventional
WMMSE. We partition the set of beamforming coefficients into subvectors, with
each subvector corresponding to a specific AP. Such a partitioning approach
decomposes the original WMMSE problem across individual APs. By leveraging the
Lagrange duality analysis, a closed-form solution can be derived for each
subproblem, which substantially reduces the computation burden. Additionally,
we present a parallel execution of the proposed G-R-WMMSE with adaptive step
sizes, aiming at further reducing the time complexity. Numerical results
validate that the proposed G-R-WMMSE schemes achieve over 99% complexity
savings compared to the conventional WMMSE scheme while maintaining almost the
same performance.
