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 recently developed optimization techniques in areas ranging 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 or developing suitable 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：数学优化现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。尽管优化在从1G到5G乃至未来6G的无线通信与网络技术革命性进展中扮演了关键角色，但无线技术的创新也极大地改变了系统设计所依赖的底层数学优化问题的本质，并推动了理解、分析和解决这些问题的方法论的重大创新。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同特征，并从优化视角探讨各类场景、用例及其关联的数学结构。随后概述近年发展的优化技术，涵盖非凸优化、全局优化、整数规划、分布式优化及学习型优化等领域。成功求解数学优化问题的关键在于审慎选择或开发能利用底层问题结构的合适算法（或神经网络架构）。最后，本文指出若干开放性研究挑战并展望未来研究方向。