On the Linear Convergence of Distributed Optimization over Directed Graphs

This paper develops a fast distributed algorithm, termed \emph{DEXTRA}, to solve the optimization problem when~$n$ agents reach agreement and collaboratively minimize the sum of their local objective functions over the network, where the communication between the agents is described by a~\emph{directed} graph. Existing algorithms solve the problem restricted to directed graphs with convergence rates of $O(\ln k\mathrm{/}\sqrt{k})$ for general convex objective functions and $O(\ln k\mathrm{/}k)$ when the objective functions are strongly-convex, where~$k$ is the number of iterations. We show that, with the appropriate step-size, DEXTRA converges at a linear rate $O({\tau }^k)$ for $0<\tau <1$, given that the objective functions are restricted strongly-convex. The implementation of DEXTRA requires each agent to know its local out-degree. Simulation examples further illustrate our findings.