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Dual decomposition is widely utilized in distributed optimization ofmulti-agent systems. In practice, the dual decomposition algorithm is desiredto admit an asynchronous implementation due to imperfect communication, such astime delay and packet drop. In addition, computational errors also exist whenindividual agents solve their own subproblems. In this paper, we analyze theconvergence of the dual decomposition algorithm in distributed optimizationwhen both the asynchrony in communication and the inexactness in solvingsubproblems exist. We find that the interaction between asynchrony andinexactness slows down the convergence rate from $\mathcal{O} ( 1 / k )$ to$\mathcal{O} ( 1 / \sqrt{k} )$. Specifically, with a constant step size, thevalue 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 onlyconsider either asynchrony or inexactness. Finally, numerical simulationsvalidate the theoretical results.

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