  Weighted sum-rate (WSR) maximization plays a critical role in communication
system design. This paper examines three optimization methods for WSR
maximization, which ensure convergence to stationary points: two block
coordinate ascent (BCA) algorithms, namely, weighted sum-minimum mean-square
error (WMMSE) and WSR maximization via fractional programming (WSR-FP), along
with a minorization-maximization (MM) algorithm, WSR maximization via MM
(WSR-MM). Our contributions are threefold. Firstly, we delineate the exact
relationships among WMMSE, WSR-FP, and WSR-MM, which, despite their extensive
use in the literature, lack a comprehensive comparative study. By probing the
theoretical underpinnings linking the BCA and MM algorithmic frameworks, we
reveal the direct correlations between the equivalent transformation
techniques, essential to the development of WMMSE and WSR-FP, and the surrogate
functions pivotal to WSR-MM. Secondly, we propose a novel algorithm, WSR-MM+,
harnessing the flexibility of selecting surrogate functions in MM framework. By
circumventing the repeated matrix inversions in the search for optimal Lagrange
multipliers in existing algorithms, WSR-MM+ significantly reduces the
computational load per iteration and accelerates convergence. Thirdly, we
reconceptualize WSR-MM+ within the BCA framework, introducing a new equivalent
transform, which gives rise to an enhanced version of WSR-FP, named as WSR-FP+.
We further demonstrate that WSR-MM+ can be construed as the basic gradient
projection method. This perspective yields a deeper understanding into its
computational intricacies. Numerical simulations corroborate the connections
between WMMSE, WSR-FP, and WSR-MM and confirm the efficacy of the proposed
WSR-MM+ and WSR-FP+ algorithms.
