  We consider a multi agent optimization problem where a set of agents
collectively solves a global optimization problem with the objective function
given by the sum of locally known convex functions. We focus on the case when
information exchange among agents takes place over a directed network and
propose a distributed subgradient algorithm in which each agent performs local
processing based on information obtained from his incoming neighbors. Our
algorithm uses weight balancing to overcome the asymmetries caused by the
directed communication network, i.e., agents scale their outgoing information
with dynamically updated weights that converge to balancing weights of the
graph. We show that both the objective function values and the consensus
violation, 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 of
iterations. A special case of our algorithm provides a new distributed method
to compute average consensus over directed graphs.
