This work revisits the joint beamforming (BF) and antenna selection (AS) problem, as well as its robust beamforming (RBF) version under imperfect channel state information (CSI). Such problems arise due to various reasons, e.g., the costly nature of the radio frequency (RF) chains and energy/resource-saving considerations. The joint (R)BF\&AS problem is a mixed integer and nonlinear program, and thus finding {\it optimal solutions} is often costly, if not outright impossible. The vast majority of the prior works tackled these problems using techniques such as continuous approximations, greedy methods, and supervised machine learning -- yet these approaches do not ensure optimality or even feasibility of the solutions. The main contribution of this work is threefold. First, an effective {\it branch and bound} (B\&B) framework for solving the problems of interest is proposed. Leveraging existing BF and RBF solvers, it is shown that the B\&B framework guarantees global optimality of the considered problems. Second, to expedite the potentially costly B\&B algorithm, a machine learning (ML)-based scheme is proposed to help skip intermediate states of the B\&B search tree. The learning model features a {\it graph neural network} (GNN)-based design that is resilient to a commonly encountered challenge in wireless communications, namely, the change of problem size (e.g., the number of users) across the training and test stages. Third, comprehensive performance characterizations are presented, showing that the GNN-based method retains the global optimality of B\&B with provably reduced complexity, under reasonable conditions. Numerical simulations also show that the ML-based acceleration can often achieve an order-of-magnitude speedup relative to B\&B.

翻译：本文重新审视了联合波束成形（BF）与天线选择（AS）问题，以及其在非完美信道状态信息（CSI）下的鲁棒波束成形（RBF）版本。此类问题的产生源于多种原因，例如射频（RF）链路的高成本特性以及节能/资源节约等方面的考量。联合（R）BF\&AS问题属于混合整数非线性规划，因此求解其**最优解**往往代价高昂，甚至根本无法实现。先前绝大多数工作采用连续逼近、贪心方法和监督式机器学习等技术来处理这些问题——但这些方法无法确保解的最优性甚至可行性。本文的主要贡献体现在三个方面。首先，提出了一种有效的**分支定界**（B\&B）框架用于求解所关注的问题。通过利用现有的BF和RBF求解器，证明该B\&B框架能够保证所考虑问题的全局最优性。其次，为加速潜在代价高昂的B\&B算法，提出了一种基于机器学习（ML）的方案，用于帮助跳过B\&B搜索树的中间状态。该学习模型采用基于**图神经网络**（GNN）的设计，能够有效应对无线通信中常见的挑战，即训练与测试阶段问题规模（如用户数量）的变化。第三，给出了全面的性能表征，表明在合理条件下，基于GNN的方法能够在以可证明的降低复杂度前提下保持B\&B的全局最优性。数值仿真还表明，这种基于ML的加速方案相对于B\&B往往能实现一个数量级的加速。