TL;DR: A decentralized algorithm for multi-agent, convex optimization programs, subject to separable constraints, where the constraint function of each agent involves only its local decision vector, while the decision vectors of all agents are coupled via a common objective function is developed.

Abstract: We develop a decentralized algorithm for multi-agent, convex optimization programs, subject to separable constraints, where the constraint function of each agent involves only its local decision vector, while the decision vectors of all agents are coupled via a common objective function. We construct a variant of the so called Jacobi algorithm and show that, when the objective function is quadratic, convergence to some minimizer of the centralized problem counterpart is achieved. Our algorithm serves then as an effective alternative to gradient based methodologies. We illustrate its efficacy by applying it to the problem of optimal charging of electric vehicles, where, as opposed to earlier approaches, we show convergence to an optimal charging scheme for a finite, possibly large, number of vehicles.

Abstract: We consider the problem of optimal charging of heterogeneous plug-in electric vehicles (PEVs). We approach the problem as a multi-agent game in the presence of constraints and formulate an auxiliary minimization program whose solution is shown to be the unique Nash equilibrium of the PEV charging control game, for any finite number of possibly heterogeneous agents. Assuming that the parameters defining the constraints of each vehicle are drawn randomly from a given distribution, we show that, as the number of agents tends to infinity, the value of the game achieved by the Nash equilibrium and the social optimum of the cooperative counterpart of the problem under study coincide for almost any choice of the random heterogeneity parameters. To the best of our knowledge, this result quantifies for the first time the asymptotic behaviour of the price of anarchy for this class of games. A numerical investigation to support our result is also provided.

Abstract: We consider the problem of optimal charging of plug-in electric vehicles (PEVs). We treat this problem as a multi-agent game, where vehicles/agents are heterogeneous since they are subject to possibly different constraints. Under the assumption that electricity price is affine in total demand, we show that, for any finite number of heterogeneous agents, the PEV charging control game admits a unique Nash equilibrium, which is the optimizer of an auxiliary minimization program. We are also able to quantify the asymptotic behaviour of the price of anarchy for this class of games. More precisely, we prove that if the parameters defining the constraints of each vehicle are drawn randomly from a given distribution, then, the value of the game converges almost surely to the optimum of the cooperative problem counterpart as the number of agents tends to infinity. In the case of a discrete probability distribution, we provide a systematic way to abstract agents in homogeneous groups and show that, as the number of agents tends to infinity, the value of the game tends to a deterministic quantity.

TL;DR: The convergence analysis of the regularized Jacobi algorithm is revisited and it is shown that it also converges in iterates under very mild conditions on the objective function and achieves a linear convergence rate.

Abstract: In this paper, we consider the regularized version of the Jacobi algorithm, a block coordinate descent method for convex optimization with an objective function consisting of the sum of a differentiable function and a block-separable function. Under certain regularity assumptions on the objective function, this algorithm has been shown to satisfy the so-called sufficient decrease condition, and consequently, to converge in objective function value. In this paper, we revisit the convergence analysis of the regularized Jacobi algorithm and show that it also converges in iterates under very mild conditions on the objective function. Moreover, we establish conditions under which the algorithm achieves a linear convergence rate.

Abstract: We consider a resource allocation problem over an undirected network of agents, where edges of the network define communication links. The goal is to minimize the sum of agent-specific convex objective functions, while the agents' decisions are coupled via a convex conic constraint. We derive two methods by applying the alternating direction method of multipliers (ADMM) for decentralized consensus optimization to the dual of our resource allocation problem. Both methods are fully parallelizable and decentralized in the sense that each agent exchanges information only with its neighbors in the network and requires only its own data for updating its decision. We prove convergence of the proposed methods and demonstrate their effectiveness with a numerical example.

Abstract: We consider multiagent, convex quadratic optimization programs subject to separable constraints, where the constraint function of each agent involves only its local decision vector, while the decision vectors of all agents are coupled via a common objective function. We focus on a regularized variant of the so-called Jacobi algorithm for decentralized computation in such problems. We provide a fixed-point theoretic analysis showing that the algorithm converges to a minimizer of the centralized problem under more relaxed conditions on the regularization coefficient from those available in the literature, and in particular with respect to scaled projected gradient algorithms. The efficacy of the proposed algorithm is illustrated by applying it to the problem of optimal charging of electric vehicles.

TL;DR: This work discusses parallel and distributed architectures, complexity measures, and communication and synchronization issues, and it presents both Jacobi and Gauss-Seidel iterations, which serve as algorithms of reference for many of the computational approaches addressed later.

Abstract: gineering, computer science, operations research, and applied mathematics. It is essentially a self-contained work, with the development of the material occurring in the main body of the text and excellent appendices on linear algebra and analysis, graph theory, duality theory, and probability theory and Markov chains supporting it. The introduction discusses parallel and distributed architectures, complexity measures, and communication and synchronization issues, and it presents both Jacobi and Gauss-Seidel iterations, which serve as algorithms of reference for many of the computational approaches addressed later. After the introduction, the text is organized in two parts: synchronous algorithms and asynchronous algorithms. The discussion of synchronous algorithms comprises four chapters, with Chapter 2 presenting both direct methods (converging to the exact solution within a finite number of steps) and iterative methods for linear

5,430 citations

"On decentralized convex optimizatio..." refers background or methods in this paper

...3 in [1]), with 1/c playing the role of the gradient step-size and with (Qd + Ic) (inverse of the Hessian of the objective function in step 6 of Algorithm 1) being the scaling matrix....

[...]

...In this framework also the Gauss-Seidel algorithm which however is not of parallelizable nature unless a coloring scheme is adopted (see [1]), and block coordinate descent methods [6] can be considered....

[...]

...It should be noted that the condition on c in Proposition 4 is related to the requirement imposed in [1] (Proposition 3....

[...]

...625 in [1]), P admits at least one optimal solution....

[...]

...Algorithms for the decentralized solution to convex optimization problems with separable constraints can be found in [1], [2], and references therein....

TL;DR: This book provides a largely self-contained account of the main results of convex analysis and optimization in Hilbert space, and a concise exposition of related constructive fixed point theory that allows for a wide range of algorithms to construct solutions to problems in optimization, equilibrium theory, monotone inclusions, variational inequalities, and convex feasibility.

Abstract: This book provides a largely self-contained account of the main results of convex analysis and optimization in Hilbert space. A concise exposition of related constructive fixed point theory is presented, that allows for a wide range of algorithms to construct solutions to problems in optimization, equilibrium theory, monotone inclusions, variational inequalities, best approximation theory, and convex feasibility. The book is accessible to a broad audience, and reaches out in particular to applied scientists and engineers, to whom these tools have become indispensable.

TL;DR: The many different interpretations of proximal operators and algorithms are discussed, their connections to many other topics in optimization and applied mathematics are described, some popular algorithms are surveyed, and a large number of examples of proxiesimal operators that commonly arise in practice are provided.

Abstract: This monograph is about a class of optimization algorithms called proximal algorithms. Much like Newton's method is a standard tool for solving unconstrained smooth optimization problems of modest size, proximal algorithms can be viewed as an analogous tool for nonsmooth, constrained, large-scale, or distributed versions of these problems. They are very generally applicable, but are especially well-suited to problems of substantial recent interest involving large or high-dimensional datasets. Proximal methods sit at a higher level of abstraction than classical algorithms like Newton's method: the base operation is evaluating the proximal operator of a function, which itself involves solving a small convex optimization problem. These subproblems, which generalize the problem of projecting a point onto a convex set, often admit closed-form solutions or can be solved very quickly with standard or simple specialized methods. Here, we discuss the many different interpretations of proximal operators and algorithms, describe their connections to many other topics in optimization and applied mathematics, survey some popular algorithms, and provide a large number of examples of proximal operators that commonly arise in practice.

3,174 citations

"On decentralized convex optimizatio..." refers background in this paper

...This result extends the equivalence between proximal operators and gradient algorithms observed in [2] for the single-agent case, to the multi-agent setting....

[...]

...Proof: Step 6 of Algorithm 1 corresponds to the so called Krasnoselskij iteration [15] (referred to as averaged operator in [2]),...

[...]

...Algorithms for the decentralized solution to convex optimization problems with separable constraints can be found in [1], [2], and references therein....

Abstract: We survey here some recent studies concerning what we call mean-field models by analogy with Statistical Mechanics and Physics. More precisely, we present three examples of our mean-field approach to modelling in Economics and Finance (or other related subjects...). Roughly speaking, we are concerned with situations that involve a very large number of “rational players” with a limited information (or visibility) on the “game”. Each player chooses his optimal strategy in view of the global (or macroscopic) informations that are available to him and that result from the actions of all players. In the three examples we mention here, we derive a mean-field problem which consists in nonlinear differential equations. These equations are of a new type and our main goal here is to study them and establish their links with various fields of Analysis. We show in particular that these nonlinear problems are essentially well-posed problems i.e., have unique solutions. In addition, we give various limiting cases, examples and possible extensions. And we mention many open problems.

1,836 citations

"On decentralized convex optimizatio..." refers background in this paper

...A complete theoretical characterization for the stochastic, continuous-time variant of the problem, but in the absence of constraints, is provided in [7], [8]....

TL;DR: A state aggregation technique is developed to obtain a set of decentralized control laws for the individuals which possesses an epsiv-Nash equilibrium property and a stability property of the mass behavior is established.

Abstract: We consider linear quadratic Gaussian (LQG) games in large population systems where the agents evolve according to nonuniform dynamics and are coupled via their individual costs. A state aggregation technique is developed to obtain a set of decentralized control laws for the individuals which possesses an epsiv-Nash equilibrium property. A stability property of the mass behavior is established, and the effect of inaccurate population statistics on an isolated agent is also analyzed by variational techniques.