Sidiropoulos Convex Optimization in Signal Processing

I n recent years, we have witnessed technical breakthroughs in a wide variety of topics where the key to success is the use of convex optimization. In fact, convex optimization has now emerged as a major signal processing tool that has made a significant impact on numerous problems previously considered intractable. Considering the foundational nature and potential impact of convex optimization in signal processing, we have put together this special issue that aims to provide tutorials of convex optimization techniques (including available software) and various successful signal processing applications. Our goal is not only to contribute to the diffusion of recent developments in this research area within the signal processing community, but also to spur further advances in and applications of convex optimization for signal processing. There was an enthusiastic response to our initial call for papers. More than 40 white papers were received, representing a broad mix of classical and contemporary signal processing topics where convex optimization has made a significant impact. Aiming for balanced coverage while respecting page limitations, we were only able to accommodate eight articles in this special issue. Collectively, these eight articles introduce convex optimization techniques, give insights into how convex optimization can make a difference, and showcase some notable successes. The first three articles in this special issue introduce various convex optimization methodologies that are especially useful in signal processing applications. The article by Luo et al. introduces semidefinite programming and the semidefinite relaxation (SDR) technique, and highlights their uses in a wide variety of problems, ranging from multiple-input, multiple-output (MIMO) detection, to transmit shimming in magnetic resonance imaging, and on to sensor network localization. This article also gives an overview of key theoretical results on SDR from a signal processing perspective. The article by Scutari et al. introduces game theoretic techniques for signal processing applications. In recent years, game theory has made headway in distributed signal processing, communications, and networking, e.g., to address fundamental issues in peer-to-peer wireless and emerging cognitive radio networks. This article starts at the confluence of convex optimization and game theory and takes us on to a fascinating (yet crisp and rigorous) tour-de-force of variational inequality theory and methods—a powerful framework that allows tackling not only classical optimization and equilibrium problems, but also pertinent extensions that arise in the aforementioned application areas and beyond. The article by Mattingley and Boyd reviews the present state of the art in realtime convex optimization-based signal processing. The authors describe how to leverage the enormous speed increases in computing hardware, advanced programming, and modern algorithms to solve moderately sized convex optimization problems in microsecondor millisecondtime scales, and with strict deadlines. The next five articles in this special issue showcase some notable successes of convex optimization in signal processing applications. The article by Gershman et al. presents a comprehensive tutorial survey on the use of convex optimization techniques to solve a wide variety of beamforming problems in wireless communications. The review focuses on three main classes of beamforming: receive beamforming, transmit beamforming, and distributed beamforming for relay networks. The authors discuss the importance of robust designs based on different criteria, such as worst-case and probabilistic analysis, and they illustrate the importance of convex optimization techniques in solving the resulting problem formulations. Sparse signal reconstruction is an other important application of convex optimization. The currently predominant algorithmic approach for solving this type of problem is based on mixed oneand two-norm optimization, which yields a convex problem. The article by Zibulevsky and Elad offers a concise review of recent developments towards acceleration of an important class of problem-specific solutions known as iterative shrinkage algorithms (ITAs), which is significantly more efficient than the generic convex optimization solvers. Design of finite impulse response (FIR) filters is one of the classic problems in digital signal processing. Effective filter design requires judicious tradeoffs between conflicting properties of the filter. The article by Davidson presents FIR filter design from the perspective of convex optimization. It shows how this approach can enrich the art of designing FIR filters, through both the direct design of filters and the efficient computation of appropriate tradeoffs between conflicting design criteria. In addition, it illustrates, with some simple examples, how new user interfaces to general-purpose convex optimization software have placed these tools at our fingertips. Dynamic spectrum access is an emerging area where convex optimization plays