  Precoding design for maximizing weighted sum-rate (WSR) is a fundamental
problem for downlink of massive multi-user multiple-input multiple-output
(MU-MIMO) systems. It is well-known that this problem is generally NP-hard due
to the presence of multi-user interference. The weighted minimum mean-square
error (WMMSE) algorithm is a popular approach for WSR maximization. However,
its computational complexity is cubic in the number of base station (BS)
antennas, which is unaffordable when the BS is equipped with a large antenna
array. In this paper, we consider the WSR maximization problem with either a
sum-power constraint (SPC) or per-antenna power constraints (PAPCs). For the
former, we prove that any nontrivial stationary point must have a
low-dimensional subspace structure, and then propose a reduced-WMMSE (R-WMMSE)
with linear complexity by exploiting the solution structure. For the latter, we
propose a linear-complexity WMMSE approach, named PAPC-WMMSE, by using a novel
recursive design of the algorithm. Both R-WMMSE and PAPC-WMMSE have simple
closed-form updates and guaranteed convergence to stationary points. Simulation
results verify the efficacy of the proposed designs, especially the much lower
complexity as compared to the state-of-the-art approaches for massive MU-MIMO
systems.
