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.

翻译：加权和速率（WSR）最大化在通信系统设计中扮演着关键角色。本文研究了三种确保收敛到驻点的WSR最大化优化方法：两种块坐标上升（BCA）算法，即加权和最小均方误差（WMMSE）算法和基于分数规划的WSR最大化（WSR-FP）算法，以及一种极小化-极大化（MM）算法，即基于MM的WSR最大化（WSR-MM）算法。我们的贡献分为三点。首先，我们阐明了WMMSE、WSR-FP和WSR-MM之间的确切关系——尽管这些算法在文献中被广泛使用，但仍缺乏全面的比较研究。通过探究连接BCA和MM算法框架的理论基础，我们揭示了WMMSE和WSR-FP开发中关键的等效变换技术与WSR-MM中关键的替代函数之间的直接关联。其次，我们提出了一种新颖的算法WSR-MM+，该算法利用了MM框架中选择替代函数的灵活性。通过规避现有算法中寻找最优拉格朗日乘子时所需的重复矩阵求逆，WSR-MM+显著降低了每轮迭代的计算负载并加速了收敛。第三，我们在BCA框架下重新构建了WSR-MM+，引入了一种新的等效变换，由此衍生出WSR-FP的增强版本，命名为WSR-FP+。我们进一步证明，WSR-MM+可被解释为基本的梯度投影方法。这一视角深化了对其计算复杂性的理解。数值仿真验证了WMMSE、WSR-FP和WSR-MM之间的关联性，并确认了所提出的WSR-MM+和WSR-FP+算法的有效性。