In this work, we propose ultra-low-complexity design solutions for multi-group multicast beamforming in large-scale systems. For the quality-of-service (QoS) problem, by utilizing the optimal multicast beamforming structure obtained recently in [2], we convert the original problem into a non-convex weight optimization problem of a lower dimension and propose two fast first-order algorithms to solve it. Both algorithms are based on successive convex approximation (SCA) and provide fast iterative updates to solve each SCA subproblem. The first algorithm uses a saddle point reformulation in the dual domain and applies the extragradient method with an adaptive step-size procedure to find the saddle point with simple closed-form updates. The second algorithm adopts the alternating direction method of multipliers (ADMM) method by converting each SCA subproblem into a favorable ADMM structure. The structure leads to simple closed-form ADMM updates, where the problem in each update block can be further decomposed into parallel subproblems of small sizes, for which closed-form solutions are obtained. We also propose efficient initialization methods to obtain favorable initial points that facilitate fast convergence. Furthermore, taking advantage of the proposed fast algorithms, for the max-min fair (MMF) problem, we propose a simple closed-form scaling scheme that directly uses the solution obtained from the QoS problem, avoiding the conventional computationally expensive method that iteratively solves the inverse QoS problem. We further develop lower and upper bounds on the performance of this scaling scheme. Simulation results show that the proposed algorithms offer near-optimal performance with substantially lower computational complexity than the state-of-the-art algorithms for large-scale systems.

翻译：本文提出面向大规模系统中多组多播波束成形的超低复杂度设计解决方案。针对服务质量（QoS）问题，通过利用文献[2]近期得到的最优多播波束成形结构，我们将原问题转化为低维度的非凸权重优化问题，并提出两种快速一阶算法进行求解。两种算法均基于逐次凸近似（SCA）方法，并提供快速的迭代更新以求解每个SCA子问题。第一种算法采用对偶域的鞍点重构形式，并应用自适应步长过程的极梯度法，以简洁闭式更新求解鞍点。第二种算法采用交替方向乘子法（ADMM），通过将每个SCA子问题转换为有利的ADMM结构，进而导出简洁的ADMM闭式更新。其中每个更新块中的问题可进一步分解为易于求解的小规模并行子问题，并得到闭式解。我们同时提出高效的初始化方法以获取有利于快速收敛的初始点。此外，利用所提出的快速算法，针对最大最小公平（MMF）问题，我们提出采用简单闭式缩放方案直接利用QoS问题的解，规避了传统方法中迭代求解逆QoS问题的高计算代价。我们进一步推导了该缩放方案性能的下界与上界。仿真结果表明，与现有最优算法相比，所提算法在面向大规模系统时能以显著降低的计算复杂度实现接近最优的性能。