Continuous-time distributed convex optimization on weight-balanced digraphs

Abstract

This paper studies the continuous-time distributed optimization of a sum of convex functions over directed graphs. Contrary to what is known in the consensus literature, where the same dynamics works for both undirected and directed scenarios, we show that the consensus-based dynamics that solves the continuous-time distributed optimization problem for undirected graphs fails to converge when transcribed to the directed setting. This study sets the basis for the design of an alternative distributed dynamics which we show is guaranteed to converge, on any strongly connected weightbalanced digraph, to the set of minimizers of a sum of convex differentiable functions with globally Lipschitz gradients. Our technical approach combines notions of invariance and cocoercivity with the positive definiteness properties of graph matrices to establish the results.

Distributed zero‐gradient‐sum algorithm for convex optimization with time‐varying communication delays and switching networks

“…However, the basis consensus‐based subgradient algorithms need to select the diminishing step sizes, which lead to the slow convergence rate. To overcome the drawbacks caused by the diminishing step sizes, some novel distributed optimization algorithms based on auxiliary‐variables method are developed . The main feature of these algorithms is that the step sizes are fixed (or nonincreasing) so as to ensure fast and exact convergence.…”

“…To overcome the drawbacks caused by the diminishing step sizes, some novel distributed optimization algorithms based on auxiliary-variables method are developed. [12][13][14][15] The main feature of these algorithms is that the step sizes are fixed (or nonincreasing) so as to ensure fast and exact convergence. However, the price is the increasing of the computation and communication burden.…”

Distributed continuous‐time algorithm for a general nonsmooth monotropic optimization problem

“…To date, although a wide spectrum of results have been reported for discrete-time networks with various scenarios in the literature, ranging from distributed optimization problems in the absence of constraints to those subject to constraints, [1][2][3][4][5] continuous-time algorithms have attracted an increasing interest in recent years mostly due to the fact that a lot of physical systems operate in a continuum domain, such as the current flow in smart grid. [6][7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23] For instance, distributed convex optimization problems have been studied in the work of Yang et al 17 subject to local feasible constraints, local inequality and equality constraints, where a proportional-integral continuous-time algorithm has been designed with output information exchange.…”

Distributed algorithm design for aggregative games of disturbed multiagent systems over weight‐balanced digraphs

“…Besides, the algorithm analysis over directed graphs is more difficult than that over undirected graphs. Moreover, Gharesifard and Cortés pointed out that a convergent algorithm over undirected graphs may be divergent for some digraphs. It is worthwhile mentioning that many distributed GNE seeking algorithms depend on the undirected graphs such as those in other works, while there are few results about aggregative games over directed graphs.…”