Distributed Subgradient Projection Algorithm over Directed Graphs
  We propose a distributed algorithm, termed the Directed-Distributed Projected Subgradient (D-DPS), to solve a constrained optimization problem over a multi-agent network, where the goal of agents is to collectively minimize the sum of locally known convex functions. Each agent in the network owns only its local objective function, constrained to a commonly known convex set. We focus on the circumstance when communications between agents are described by a directed network. The D-DPS augments an additional variable for each agent, to overcome the asymmetry caused by the directed communication network. The convergence analysis shows that D-DPS converges at a rate of $O(\frac{\ln k}{\sqrt{k}})$, where k is the number of iterations. 
Chenguang Xi, Usman A. Khan
math.OC