Convergence Analysis of Dual Decomposition Algorithm in Distributed Optimization: Asynchrony and Inexactness

TL;DR

This paper analyzes how asynchrony and inexactness in distributed dual decomposition algorithms affect convergence, showing a slowdown from $ ext{O}(1/k)$ to $ ext{O}(1/\sqrt{k})$, with theoretical and numerical validation.

Contribution

It provides a unified convergence analysis for dual decomposition algorithms considering both asynchrony and inexactness, extending prior work that addressed only one factor.

Findings

Convergence rate slows to $ ext{O}(1/\sqrt{k})$ due to combined effects.

Objective function converges to a neighborhood of the optimum with constant step size.

Constraint violations decrease at a rate of $ ext{O}(1/\sqrt{k})$.

Abstract

Dual decomposition is widely utilized in distributed optimization of multi-agent systems. In practice, the dual decomposition algorithm is desired to admit an asynchronous implementation due to imperfect communication, such as time delay and packet drop. In addition, computational errors also exist when individual agents solve their own subproblems. In this paper, we analyze the convergence of the dual decomposition algorithm in distributed optimization when both the asynchrony in communication and the inexactness in solving subproblems exist. We find that the interaction between asynchrony and inexactness slows down the convergence rate from $\mathcal{O} ( 1 / k )$ to $\mathcal{O} ( 1 / \sqrt{k} )$. Specifically, with a constant step size, the value of objective function converges to a neighborhood of the optimal value, and the solution converges to a neighborhood of the exact optimal solution. Moreover, the violation of the constraints diminishes in $\mathcal{O} ( 1 / \sqrt{k} )$. Our result generalizes and unifies the existing ones that only consider either asynchrony or inexactness. Finally, numerical simulations validate the theoretical results.