IEEE Transactions on Automatic Control, Vol.65, No.7, 3098-3103, 2020
A Modified Saddle-Point Dynamics for Distributed Convex Optimization on General Directed Graphs
In this technical note, we show that the continuous-time saddle-point distributed convex optimization dynamics can be cast as a distributed control system, where each agent implements a control input using an estimate of the average state, generated through an observer. Using this, and by incorporating a continuous-time version of the so-called push-sum algorithm, we relax the graph-theoretic conditions under which the first component of the trajectories of this modified class of saddle-point dynamical systems is 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 minimum-cut, of the sequence of Laplacian matrices.