On the Linear Convergence of Distributed Optimization over Directed Graphs

This paper develops a fast distributed algorithm, termed 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~directed graph. Existing algorithms solve the problem restricted to directed graphs with convergence rates of O(ln k/√k) for general convex objective functions and O(ln k/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(τk) for 0<τ<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.