# Optimal Solutions for Joint Beamforming and Antenna Selection: From Branch and Bound to Graph Neural Imitation Learning

arXiv: https://arxiv.org/abs/2206.05576 · HTML: https://mlsanity.com/paper/2206.05576/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 78 references that are in the map.
- First posted on arXiv 11 Jun 2022.

## Builds on (most cited first, 19 of 19)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Message Passing for Quantum Chemistry](https://mlsanity.com/paper/1704.01212/) (2017)
- [Spectrally-normalized margin bounds for neural networks](https://mlsanity.com/paper/1706.08498/) (2017)
- [CVXPY: A Python-Embedded Modeling Language for Convex Optimization](https://mlsanity.com/paper/1603.00943/) (2016)
- [Exact Combinatorial Optimization with Graph Convolutional Neural Networks](https://mlsanity.com/paper/1906.01629/) (2019)
- [Generalization and Representational Limits of Graph Neural Networks](https://mlsanity.com/paper/2002.06157/) (2020)
- [Solving Mixed Integer Programs Using Neural Networks](https://mlsanity.com/paper/2012.13349/) (2020)
- [Best Subset Selection via a Modern Optimization Lens](https://mlsanity.com/paper/1507.03133/) (2015)
- [Learning to Optimize: Training Deep Neural Networks for Wireless Resource Management](https://mlsanity.com/paper/1705.09412/) (2017)
- [On Generalization Bounds of a Family of Recurrent Neural Networks](https://mlsanity.com/paper/1910.12947/) (2019)
- [Learning in Non-convex Games with an Optimization Oracle](https://mlsanity.com/paper/1810.07362/) (2018)
- [Joint Antenna Selection and Hybrid Beamformer Design using Unquantized and Quantized Deep Learning Networks](https://mlsanity.com/paper/1905.03107/) (2019)
- [Hierarchical Bipartite Graph Convolution Networks](https://mlsanity.com/paper/1812.03813/) (2018)
- [Deep Learning based Antenna Selection and CSI Extrapolation in Massive MIMO Systems](https://mlsanity.com/paper/2101.06951/) (2021)
- [Feasible Point Pursuit and Successive Approximation of Non-convex QCQPs](https://mlsanity.com/paper/1410.2277/) (2014)
- [Optimal Transmit Antenna Selection Algorithm in Massive MIMOME Channels](https://mlsanity.com/paper/1810.08879/) (2018)
- [Fast Algorithms for Joint Multicast Beamforming and Antenna Selection in Massive MIMO](https://mlsanity.com/paper/2003.02240/) (2020)
- [MMSE Based Greedy Antenna Selection Scheme for AF MIMO Relay Systems](https://mlsanity.com/paper/2003.07538/) (2020)
- [Antenna Selection for Improving Energy Efficiency in XL-MIMO Systems](https://mlsanity.com/paper/2009.02542/) (2020)

## Abstract (arXiv)

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.