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，优化技术始终扮演着关键角色，但无线技术的创新也深刻改变了系统设计所依赖的基础数学优化问题的本质特征，并推动了理解、分析和求解这些问题的研究方法论的重大创新。本文对面向无线通信系统设计的数学优化理论与算法最新进展进行了全面综述。我们首先阐述了无线通信系统设计中数学优化问题的共性特征，从优化视角讨论了各类场景、用例及其关联的数学结构。继而系统介绍了近年来发展的优化技术，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谨慎选择或开发能够利用问题内在结构的合适算法（或神经网络架构）。最后，本文指出了若干开放研究挑战并展望了未来研究方向。