The past two decades have witnessed the onset of a surge of research in optimization. This includes theoretical aspects, algorithmic developments such as generalizations of interior-point methods to a rich class of convex-optimization problems, and many new engineering applications. The development of general-purpose software tools as well as the insight generated by the underlying theory have contributed to the emergence of convex optimization as a major signal-processing tool; this has made a significant impact on numerous problems previously considered intractable. Given this success of convex optimization, many new applications are continuously flourishing. This book aims at providing the reader with a series of tutorials on a wide variety of convex-optimization applications in signal processing and communications, written by worldwide leading experts, and contributing to the diffusion of these new developments within the signalprocessing community. The topics included are automatic code generation for real-time solvers, graphical models for autoregressive processes, gradient-based algorithms for signal-recovery applications, semidefinite programming (SDP) relaxation with worstcase approximation performance, radar waveform design via SDP, blind non-negative source separation for image processing, modern sampling theory, robust broadband beamforming techniques, distributed multiagent optimization for networked systems, cognitive radio systems via game theory, and the variational-inequality approach for Nash-equilibrium solutions.

Game theory is a field of applied mathematics that describes and analyzes scenarios with interactive decisions. In recent years, there has been a growing interest in adopting cooperative and non-cooperative game-theoretic approaches to model many communications and networking problems, such as power control and resource sharing in wireless/wired and peer-to-peer networks. In this chapter we show how many challenging unsolved resource-allocation problems in the emerging field of cognitive radio (CR) networks fit naturally in the game-theoretical paradigm. This provides us with the mathematical tools necessary to analyze the proposed equilibrium problems for CR systems (e.g., existence and uniqueness of the solution) and to devise distributed algorithms along with their convergence properties. The proposed algorithms differ in performance, level of protection of the primary users, computational effort and signaling among primary and secondary users, convergence analysis, and convergence speed; which makes them suitable for many different CR systems. We also propose a more general framework suitable for investigating and solving more sophisticated equilibrium problems in CR systems when classical game theory may fail, based on variation inequality (VI) that constitutes a very general class of problems in nonlinear analysis.

Introduction and motivation

In recent years, increasing demand of wireless services has made the radio spectrum a very scarce and precious resource. Moreover, most current wireless networks characterized by fixed-spectrum assignment policies are known to be very inefficient considering that licensed bandwidth demands are highly varying along the time or space dimensions (according to the Federal Communications Commission [FCC], only 15% to 85% of the licensed spectrum is utilized on average [1]).

The past two decades have witnessed the onset of a surge of research in optimization. This includes theoretical aspects, as well as algorithmic developments such as generalizations of interior-point methods to a rich class of convex-optimization problems. The development of general-purpose software tools together with insight generated by the underlying theory have substantially enlarged the set of engineering-design problems that can be reliably solved in an efficient manner. The engineering community has greatly benefited from these recent advances to the point where convex optimization has now emerged as a major signal-processing technique. On the other hand, innovative applications of convex optimization in signal processing combined with the need for robust and efficient methods that can operate in real time have motivated the optimization community to develop additional needed results and methods. The combined efforts in both the optimization and signal-processing communities have led to technical breakthroughs in a wide variety of topics due to the use of convex optimization. This includes solutions to numerous problems previously considered intractable; recognizing and solving convexoptimization problems that arise in applications of interest; utilizing the theory of convex optimization to characterize and gain insight into the optimal-solution structure and to derive performance bounds; formulating convex relaxations of difficult problems; and developing general purpose or application-driven specific algorithms, including those that enable large-scale optimization by exploiting the problem structure.