Part I of this two-part paper focused on the formulation of percentile problems, complexity analysis, and development of power control algorithms via the quadratic fractional transform (QFT) and logarithmic fractional transform (LFT) for sum-least-qth-percentile (SLqP) rate maximization problems. In this second part, we first tackle the significantly more challenging problems of optimizing SLqP rate via beamforming in a multiuser, multiple-input multiple-output (MU- MIMO) network to maximize cell-edge throughput. To this end, we first propose an adaptation of the QFT algorithm presented in Part I that enables optimization of the complex-valued multidimensional beamforming weights for the SLqP rate utility function. We also introduce a new class of problems which we term as sum-greatest-qth-percentile weighted mean squared error (SGqP-WMSE) minimization. We show that this class subsumes the well-known sum-weighted mean squared error (WMMSE) minimization and max-WMSE minimization problems. We demonstrate an equivalence between this class of problems and the SLqP rate maximization problems, and show that this correspondence can be exploited to obtain stationary-point solutions for the aforementioned beamforming problem. Next, we develop extensions for the QFT and LFT algorithms from Part I to optimize ergodic long-term average or ergodic SLqP utility. Finally, we also consider related problems which can be solved using the proposed techniques, including hybrid utility functions targeting optimization at specific subsets of users within cellular networks.

翻译：本两部分论文的第二部分聚焦于百分位数问题的公式化、复杂度分析，以及通过二次分数变换（QFT）和对数分数变换（LFT）实现和最小百分位数（SLqP）速率最大化问题的功率控制算法。在第一部分中，我们已针对这些问题进行了研究。在本第二部分中，我们首先解决通过多用户多输入多输出（MU-MIMO）网络中的波束赋形优化SLqP速率这一更具挑战性的问题，以最大化小区边缘吞吐量。为此，我们首先提出对第一部分中QFT算法的改进，使其能够针对SLqP速率效用函数优化复值多维波束赋形权重。同时，我们引入一类新问题，称为和最大百分位数加权均方误差（SGqP-WMSE）最小化。我们证明该类问题包含了经典的和加权均方误差（WMMSE）最小化与最大WMS E最小化问题。我们展示了此类问题与SLqP速率最大化问题之间的等价性，并表明利用这种对应关系可以获得上述波束赋形问题的驻点解。接下来，我们扩展第一部分中的QFT和LFT算法，以优化遍历长期平均或遍历SLqP效用函数。最后，我们还考虑了可通过所提技术求解的相关问题，包括针对蜂窝网络中特定用户子集优化目标的混合效用函数。