The Generalized Nash equilibrium problem is an important model that has its roots in the economic sciences but is being fruitfully used in many different fields. In this survey paper we aim at discussing its main properties and solution algorithms, pointing out what could be useful topics for future research in the field. © 2007 Springer-Verlag

The Generalized Nash Equilibrium Problem is an important model that has its roots in the economic sciences but is being fruitfully used in many different fields. In this survey paper we aim at discussing its main properties and solution algorithms, pointing out what could be useful topics for future research in the field

Abstract-We propose a novel decomposition framework for the distributed optimization of general nonconvex sum-utility functions arising naturally in the system design of wireless multiuser interfering systems. Our main contributions are: i) the development of the first class of (inexact) Jacobi best-response algorithms with provable convergence, where all the users simultaneously and iteratively solve a suitably convexified version of the original sum-utility optimization problem; ii) the derivation of a general dynamic pricing mechanism that provides a unified view of existing pricing schemes that are based, instead, on heuristics; and iii) a framework that can be easily particularized to well-known applications, giving rise to very efficient practical (Jacobi or Gauss-Seidel) algorithms that outperform existing adhoc methods proposed for very specific problems. Interestingly, our framework contains as special cases well-known gradient algorithms for nonconvex sum-utility problems, and many blockcoordinate descent schemes for convex functions.

Abstract-In Part I of this paper, we proposed and analyzed a novel algorithmic framework for the minimization of a nonconvex (smooth) objective function, subject to nonconvex constraints, based on inner convex approximations. This Part II is devoted to the application of the framework to some resource allocation problems in communication networks. In particular, we consider two non-trivial case-study applications, namely: (generalizations of) i) the rate profile maximization in MIMO interference broadcast networks; and the ii) the max-min fair multicast multigroup beamforming problem in a multi-cell environment. We develop a new class of algorithms enjoying the following distinctive features: i) they are distributed across the base stations (with limited signaling) and lead to subproblems whose solutions are computable in closed form; and ii) differently from current relaxation-based schemes (e.g., semidefinite relaxation), they are proved to always converge to d-stationary solutions of the aforementioned class of nonconvex problems. Numerical results show that the proposed (distributed) schemes achieve larger worstcase rates (resp. signal-to-noise interference ratios) than state-ofthe-art centralized ones while having comparable computational complexity.

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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]. © 2010 IEEE

We propose a decomposition framework for the parallel optimization of the sum of a differentiable (possibly nonconvex) function and a (block) separable nonsmooth, convex one. The latter term is usually employed to enforce structure in the solution, typically sparsity. Our framework is very flexible and includes both fully parallel Jacobi schemes and Gauss–Seidel (i.e., sequential) ones, as well as virtually all possibilities “in between” with only a subset of variables updated at each iteration. Our theoretical convergence results improve on existing ones, and numerical results on LASSO, logistic regression, and some nonconvex quadratic problems show that the new method consistently outperforms existing algorithms