Towards Time-Varying Proximal Dynamics in Multi-Agent Network Games

Abstract

In this paper, we study multi-agent network games subject to affine time-varying coupling constraints and a timevarying communication network. We focus on the class of games adopting proximal dynamics and study their convergence to a persistent equilibrium. The assumptions considered to solve the problem are discussed and motivated. We develop an iterative equilibrium seeking algorithm, using only local information, that converges to a special class of game equilibria. Its derivation is motivated by several examples, showing that the original game dynamics fail to converge. Finally, we apply the designed algorithm to solve a constrained consensus problem, validating the theoretical results.

Asynchronous and Time-Varying Proximal Type Dynamics in Multiagent Network Games

“…2) We prove global convergence of synchronous proximal dynamics in network games under time-invariant directed communication graph (hence described by a row-stochastic adjacency matrix). This extends the results in [2] and [7]. 3) We establish global convergence for asynchronous proximal dynamics and synchronous proximal dynamics over time-varying networks.…”

“…The former setup is considered here for the first time. The latter is studied in [2] for undirected communication graph (hence doubly stochastic adjacency matrix), and in [7] via a dwell-time restriction.…”

“…Remark 1: Definition 1 can be equivalently cast in terms of fixed points of the operator in the left-hand side of (7). In fact, a collective vector x is an NWE if and only if x ∈ fix(prox f • A).…”

“…Solutions for noncooperative games over networks subject to local convex constraints have been developed, e.g., in [10], under strongly convex quadratic costs and time-invariant communication graphs; in [11] and [12], with differentiable cost functions with Lipschitz continuous gradients, strictly convex cost functions, and undirected, possibly time-varying, communication graphs; and in [7], where the communication is ruled by a possibly time-varying digraph and the cost functions are assumed to be convex. In some recent works, researchers have developed algorithms to solve games over networks subject to asynchronous updates of the agents: among others, in [13] and [14], the game is subject to affine coupling constraints, and the cost functions are differentiable, while the communication graph is assumed to be undirected.…”

“…To the best of the authors' knowledge, the main works that extensively focus on multiagent network games with proximal dynamics are [2], [7], where the authors consider local convex costs and quadratic proximal terms, time-invariant, and time-varying communication graphs, but subject to some technical restrictions.…”