Mathematical optimization is now widely regarded as an indispensable modeling and solution tool for the design of wireless communications systems. While optimization has played a significant role in the revolutionary progress in wireless communication and networking technologies from 1G to 5G and onto the future 6G, the innovations in wireless technologies have also substantially transformed the nature of the underlying mathematical optimization problems upon which the system designs are based and have sparked significant innovations in the development of methodologies to understand, to analyze, and to solve those problems. In this paper, we provide a comprehensive survey of recent advances in mathematical optimization theory and algorithms for wireless communication system design. We begin by illustrating common features of mathematical optimization problems arising in wireless communication system design. We discuss various scenarios and use cases and their associated mathematical structures from an optimization perspective. We then provide an overview of recent advances in mathematical optimization theory and algorithms, from nonconvex optimization, global optimization, and integer programming, to distributed optimization and learning-based optimization. The key to successful solution of mathematical optimization problems is in carefully choosing and/or developing suitable optimization algorithms (or neural network architectures) that can exploit the underlying problem structure. We conclude the paper by identifying several open research challenges and outlining future research directions.

翻译：数学优化被广泛认为是无线通信系统设计中不可或缺的建模与求解工具。尽管从1G到5G乃至未来6G的无线通信与网络技术革命性进步中，优化始终扮演关键角色，但无线技术的创新也深刻改变了系统设计所依据的数学优化问题的本质，并催生了理解、分析与求解此类问题的方法论重大创新。本文全面综述了面向无线通信系统设计的数学优化理论与算法的最新进展。首先阐述无线通信系统设计中数学优化问题的共性特征，从优化视角探讨不同场景、用例及其对应的数学结构。进而系统概述数学优化理论与算法的最新进展，涵盖非凸优化、全局优化、整数规划、分布式优化以及基于学习的优化等领域。成功求解数学优化问题的关键在于审慎选择/开发能够利用问题底层结构的优化算法（或神经网络架构）。最后，本文指出若干开放研究挑战，并展望未来研究方向。