Gradient-Consensus Method for Distributed Optimization in Directed Multi-Agent Networks

In this article, a distributed optimization problem for minimizing a sum, Sigma(n)(i=1) f(i), of convex objective functions, f(i), on directed graph topologies is addressed. Here each function f(i) is a function of n variables, private to agent i which defines the agent's objective. These f(i)'s are assumed to be Lipschitz-differentiable convex functions. For solving this optimization problem, we develop a novel distributed algorithm, which we term as the gradient-consensus method. The gradient-consensus scheme uses a finite-time terminated consensus protocol called rho-consensus, which allows each local estimate to be rho-close to each other at every iteration. The parameter rho is a fixed constant independent of the network size and topology. It is shown that the estimate of the optimal solution at any local agent i converges geometrically to the optimal solution within an O(rho) neighborhood, where rho can be chosen to be arbitrarily small.