Saddle-point dynamics for distributed convex optimization on general directed graphs

We show that the continuous-time saddle-point distributed convex optimization algorithm can be formulated as the trajectories of a distributed control systems, where the control input to the dynamics of each agent relies on an observer that estimates the average state. Using this observation and by incorporating a continuous-time version of the so-called push-sum algorithm, this paper relaxes the graph theoretic conditions under which the first component of the trajectories of this modified class of saddle-point dynamical systems for distributed optimization are asymptotically convergent to the set of optimizers. In particular, we prove that strong connectivity is sufficient under this modified dynamics, relaxing the known weight-balanced assumption. As a by product, we also show that the saddle-point distributed optimization dynamics can be extended to time-varying weight-balanced graphs which satisfy a persistency condition on the min-cut of the sequence of Laplacian matrices.