分布式Optimization中对偶分解算法的收敛性分析：异步性和不精确性
Convergence Analysis of Dual Decomposition Algorithm in Distributed Optimization: Asynchrony and Inexactness
摘要：
对偶分解广泛应用于多智能体系统的分布式优化中。在实际应用中，由于通信的不完善，如时延和丢包等，对偶分解算法需要允许异步实现。此外，计算误差也存在时，个别代理解决自己的子问题。本文分析了分布式优化中对偶分解算法在通信不精确和子问题求解不精确同时存在的情况下的收敛性。我们发现，不精确性和不精确性之间的相互作用减慢了从$\mathcal{O} ( 1 / k )$到$\mathcal{O} ( 1 / \sqrt{k} )$的收敛速度。具体地说，当步长为常数时，目标函数的值收敛到最优值的邻域，而解收敛到精确最优解的邻域。此外，在$\mathcal{O} ( 1 / \sqrt{k} )$中约束的违反减少。我们的结果推广和统一了现有的只考虑模糊性或不精确性的结果。最后通过数值模拟验证了理论分析的正确性。
英文摘要：
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.
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