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This paper studies the continuous-time distributed optimization of a sum ofconvex functions over directed graphs. Contrary to what is known in theconsensus literature, where the same dynamics works for both undirected anddirected scenarios, we show that the consensus-based dynamics that solves thecontinuous-time distributed optimization problem for undirected graphs fails toconverge when transcribed to the directed setting. This study sets the basisfor the design of an alternative distributed dynamics which we show isguaranteed to converge, on any strongly connected weight-balanced digraph, tothe set of minimizers of a sum of convex differentiable functions with globallyLipschitz gradients. Our technical approach combines notions of invariance andcocoercivity with the positive definiteness properties of graph matrices toestablish the results.

Distributed continuous-time convex optimization on weight-balanced digraphs