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

In this article, a distributed optimization problem for minimizing a sum, sum {-{i = 1}-n{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 ρ-consensus, which allows each local estimate to be ρ-close to each other at every iteration. The parameter ρ 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(ρ) neighborhood, where ρ can be chosen to be arbitrarily small.