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.

中文翻译：

---

分布式优化中对偶分解算法的收敛性分析：异步与不精确

对偶分解被广泛用于多主体系统的分布式优化中。在实践中，由于不完善的通信（例如时间延迟和数据包丢失），需要双重分解算法来接受异步实现。此外，当单个代理解决自己的子问题时，也会存在计算错误。在本文中，我们分析了当通信中的异步性和解决子问题的不精确性同时存在时，对偶分解算法在分布式优化中的收敛性。我们发现异步和不精确性之间的相互作用使收敛速度从$ \ mathcal {O}（1 / k）$减至$ \ mathcal {O}（1 / \ sqrt {k}）$。具体来说，在步长恒定的情况下，目标函数的值收敛到最优值的附近，并且解收敛到精确最优解的附近。而且，违反约束的情况在$ \ mathcal {O}（1 / \ sqrt {k}）$中减小。我们的结果归纳并统一了仅考虑异步或不精确性的现有结果。最后，数值模拟验证了理论结果。