  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.
