In this work we focus on the problem of minimizing the sum of convex cost functions in a distributed fashion over a peer-to-peer network. In particular, we are interested in the case in which communications between nodes are prone to failures and the agents are not synchronized among themselves. We address the problem proposing a modified version of the relaxed ADMM, which corresponds to the Peaceman-Rachford splitting method applied to the dual. By exploiting results from operator theory, we are able to prove the almost sure convergence of the proposed algorithm under general assumptions on the distribution of communication loss and node activation events. By further assuming the cost functions to be strongly convex, we prove the linear convergence of the algorithm in mean to a neighborhood of the optimal solution, and provide an upper bound to the convergence rate. Finally, we present numerical results testing the proposed method in different scenarios.

中文翻译：

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通过松弛 ADMM 对有损网络进行异步分布式优化：稳定性和线性收敛

在这项工作中，我们专注于在点对点网络上以分布式方式最小化凸成本函数的总和的问题。特别是，我们对节点之间的通信容易出现故障并且代理之间不同步的情况感兴趣。我们解决了提出宽松 ADMM 的修改版本的问题，该版本对应于应用于对偶的 Peaceman-Rachford 分裂方法。通过利用算子理论的结果，我们能够证明所提出的算法在通信丢失和节点激活事件分布的一般假设下几乎肯定收敛。通过进一步假设成本函数是强凸的，我们证明了算法对最优解的邻域的线性收敛性，并提供收敛速度的上限。最后，我们展示了在不同场景下测试所提出方法的数值结果。