In this work, we focus on solving non-smooth non-convex maximization problems in multi-group multicast transmission. Leveraging Karush-Kuhn-Tucker (KKT) optimality conditions and successive incumbent transcending (SIT) duality, we thoroughly analyze the optimal beamforming structure for a set of optimization problems characterized by a general utility-based objective function. By exploiting the identified optimal structure, we further unveil inherent low-dimensional beamforming structures within the problems, which are asymptotically optimal in various regimes of transmit signal-to-noise ratios (SNRs) or the number of transmit antennas. Building upon the discovered optimal and low-dimensional beamforming structures, we then propose highly efficient and toolbox-free optimization algorithms to solve a specific multi-group multicast optimization problem based on the weighted sum rate (WSR) utility function. The proposed algorithms first use the cyclic maximization (CM) framework to decompose the problem into multiple subproblems, each has an optimal or low-dimensional closed-form beamforming solution structure. Then, we propose the projected adaptive gradient descent (PAGD) algorithm to compute the optimal Lagrangian dual variables for each subproblem. Numerical results show that the proposed algorithms maintain comparable or improved WSR performance compared to baseline algorithms, while dramatically reducing the computational complexity. Notably, the proposed ultra-low-complexity algorithms based on low-dimensional beamforming structures achieve near optimal WSR performance with extremely low computational complexity. This complexity remains independent of the number of transmit antennas, making them promising and practical for extremely large multiple-input multiple-output (XL-MIMO) applications in 6G.

翻译：本文聚焦于解决多组多播传输中的非光滑非凸最大化问题。通过利用Karush-Kuhn-Tucker（KKT）最优性条件与逐次递归超越（SIT）对偶性，我们深入分析了基于通用效用目标函数的一类优化问题的最优波束成形结构。基于所揭示的最优结构，进一步揭示了问题中固有的低维波束成形结构，该结构在发射信噪比（SNR）或发射天线数的不同区间内具有渐近最优性。基于发现的最优及低维波束成形结构，我们提出了高效且无需工具包的优化算法，用于求解以加权和速率（WSR）为效用函数的特定多组多播优化问题。所提算法首先采用循环最大化（CM）框架将原问题分解为多个子问题，每个子问题均具有最优或低维的闭式波束成形解结构；进而提出投影自适应梯度下降（PAGD）算法计算各子问题的最优拉格朗日对偶变量。数值结果表明，与基线算法相比，所提算法在保持相当或更优WSR性能的同时，大幅降低了计算复杂度。值得注意的是，基于低维波束成形结构的超低复杂度算法能以极低计算量实现接近最优的WSR性能，其复杂度与发射天线数无关，使其成为第六代移动通信（6G）超大规模多输入多输出（XL-MIMO）应用中极具前景的实用方案。