Regularized Jacobi Iteration for Decentralized Convex Quadratic Optimization With Separable Constraints

Abstract

We consider 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 focus on a regularized variant of the so called Jacobi algorithm for decentralized computation in such problems. We first consider the case where the objective function is quadratic, and provide a fixed-point theoretic analysis showing that the algorithm converges to a minimizer of the centralized problem. Moreover, we quantify the potential benefits of such an iterative scheme by comparing it against a scaled projected gradient algorithm. We then consider the general case and show that all limit points of the proposed iteration are optimal solutions of the centralized problem. The efficacy of the proposed algorithm is illustrated 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.

On the connection between Nash equilibria and social optima in electric vehicle charging control games

“…On the same time, by Corollary 1 of Deori et al (2016b), this set of fixed-points coincides with the set of minimizers of P a , thus concluding the proof.…”

“…Moreover, Deori et al (2017) shows that in the case where the agents' heterogeneity parameters follow a discrete probability distribution, agents can be abstracted in homogeneous groups, while the effect of heterogeneity averages out as their number tends to infinity. It should be also noted that the established equivalence between Nash equilibria and social minimizers of an auxiliary problem, opens the road for the use of the regularized Jacobi algorithm, constructed in our earlier work for decentralized optimization Deori et al (2016b), for decentralized computation of Nash equilibria Deori et al (2017).…”

“…Problem P a is a centralized convex optimization program. By Corollary 1 of Deori et al (2016b), the set of minimizers of P a coincides with the set of fixed points of the mapping T a = T 0 a , . .…”

“…This allows to use the decentralized algorithm of Deori et al (2016b) to determine an element of N , i.e., a Nash equilibrium of the game of interest (see also Section 6 and Deori et al (2017)). Interestingly, in the limiting case of an infinite population of agents, the auxiliary optimization program tends to the cooperative social welfare optimization problem P : min…”

Price of anarchy in electric vehicle charging control games: When Nash equilibria achieve social welfare

“…An alternative proof for a result similar to Proposition 2 was provided in [11,Proposition 3], relying, however, on the additional assumption that the objective functions involved are differentiable. The following corollary is a direct consequence of Propositions 1 and 2.…”