Distributed continuous-time convex optimization on weight-balanced digraphs

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 weight-balanced 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.

💡 Research Summary

The paper addresses the problem of continuously‑time distributed optimization of a global objective that is the sum of locally known convex functions, each held by an agent in a network. While the consensus‑based dynamics (\dot{x}= -\nabla f(x) - Lx) is known to converge for undirected (symmetric) graphs, the authors demonstrate that a straightforward transcription of this dynamics to directed graphs generally fails to converge, even when the directed graph is strongly connected. This observation motivates the design of a new dynamics that is provably convergent on any strongly connected weight‑balanced digraph.

The authors first formalize the setting: each agent (i) knows a differentiable convex function (f_i:\mathbb{R}^d\to\mathbb{R}) whose gradient is globally Lipschitz with constant (L). The global objective is (f(x)=\sum_{i=1}^N f_i(x)). The communication topology is described by a directed graph (\mathcal{G}) with adjacency matrix (A) and Laplacian (L = D - A). The graph is assumed to be weight‑balanced, i.e., for every node the sum of outgoing weights equals the sum of incoming weights, which guarantees that the vector of all ones lies in the nullspace of both (L) and its transpose (L^{\top}).

The key contribution is the introduction of the dynamics
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