Convex Optimization-Based Procedures for Non-Convex Quadratic Problems

Abstract

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.