AI 中文总结
研究非凸二次问题，基于凸优化提出求解方法，该方法可应对信号处理、通信等多领域中由非凸QCQP带来的计算挑战
AI 中文摘要
数学优化在信号处理和无线通信中起着基础性作用，是现代系统系统设计的重要框架。这些领域以及其他领域的许多设计挑战可自然地表述为优化问题。多年来，信号处理应用的进展显著改变了这些优化问题的结构和复杂性。二次优化问题是现代工程系统中最重要的优化问题类别之一。在信号处理和通信中，二次形式在对功率、能量、协方差矩阵和欧几里得距离等建模时自然出现。虽然凸二次约束二次规划（QCQP）可通过多项式时间算法有效求解，但一般的非凸QCQP在计算上仍具有挑战性。非凸QCQP问题出现在广泛的信号处理、通信、控制、机器学习和网络优化应用中。
英文摘要
Mathematical optimization plays a fundamental role in signal processing and wireless communications, serving as an essential framework for the systematic design of modern systems. Many design challenges in these fields, as well as in many others, can naturally be formulated as optimization problems. Over the years, the advancements in signal processing applications have significantly changed the structure and complexity of these optimization problems, creating new challenges in their analysis, understanding, and solution \cite{liu2024survey}. Consequently, the rapid development of sophisticated optimization theories and algorithms tailored to the demands of next-generation systems is crucial. Quadratic optimization problems constitute one of the most important classes of optimization problems in modern engineering systems. In signal processing and communications, quadratic forms naturally emerge when modeling power, energy, covariance matrices, and Euclidean distances, to name a few examples. Consequently, a broad family of practical design problems can be represented using quadratically constrained quadratic programs (QCQPs), where both the objective function and the constraints are quadratic functions of the optimization variables. While convex QCQPs can be solved efficiently using polynomial-time algorithms, the general non-convex QCQP remains computationally challenging. Specifically, indefinite quadratic forms and rank constraints often induce NP-hardness. Non-convex QCQP problems arise in a broad range of signal processing, communications, control, machine learning, and network optimization applications.
Comments18 pages, submitted to IEEE Signal Processing Magazine
