  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.
