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We consider a multi agent optimization problem where a set of agentscollectively solves a global optimization problem with the objective functiongiven by the sum of locally known convex functions. We focus on the case wheninformation exchange among agents takes place over a directed network andpropose a distributed subgradient algorithm in which each agent performs localprocessing based on information obtained from his incoming neighbors. Ouralgorithm uses weight balancing to overcome the asymmetries caused by thedirected communication network, i.e., agents scale their outgoing informationwith dynamically updated weights that converge to balancing weights of thegraph. We show that both the objective function values and the consensusviolation, at the ergodic average of the estimates generated by the algorithm,converge with rate $O(\frac{\log T}{\sqrt{T}})$, where $T$ is the number ofiterations. A special case of our algorithm provides a new distributed methodto compute average consensus over directed graphs.

Graph Balancing for Distributed Subgradient Methods over Directed Graphs