Noise-to-state exponentially stable distributed convex optimization on weight-balanced digraphs

This paper studies the robustness under additive persistent noise of a class of continuous-time distributed algorithms for convex optimization. A group of agents, each with its own private objective function and communicating over a weight-balanced digraph, seeks to determine the global decision vector that minimizes the sum of all the functions. Under mild conditions on the local objective functions, we establish that the distributed algorithm is noise-to-state exponentially stable in second moment with respect to the optimal solution. Our technical approach combines notions and tools from graph theory, stochastic differential equations, and Lyapunov stability analysis. Simulations illustrate our results.