Convex optimization, game theory, and variational inequality theory

The use of optimization methods is ubiquitous in communications and signal processing. In particular, convex optimization techniques have been widely used in the design and analysis of single user and multiuser communication systems and signal processing algorithms (e.g., [1] and [2]). Game theory is a field of applied mathematics that describes and analyzes scenarios with interactive decisions (e.g., [3] and [4]). Roughly speaking, a game can be represented as a set of coupled optimization problems. In recent years, there has been a growing interest in adopting cooperative and noncooperative 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 (e.g., [5][12]), cognitive radio systems (e.g., [13][17]), and distributed routing, flow, and congestion control in communication networks (e.g., [18] and [19] and references therein). Two recent special issues on the subject are [20] and [21]. A more general framework suitable for investigating and solving various optimization problems and equilibrium models, even when classical game theory may fail, is known to be the variation inequality (VI) problem that constitutes a very general class of problems in nonlinear analysis [22].