In this paper, we consider networks with topologies described by some connected undirected graph $\mathcal{G}=(V,E)$ and with some agents (fusion centers) equipped with processing power and local peer-to-peer communication, and optimization problem ${\min }_{\mathit{x}}\{F(\mathit{x})={\sum }_{i\in V}{f}_i(\mathit{x})\}$ with local objective functions $f_i$ depending only on neighboring variables of the vertex $i\in V$. We introduce a divide-and-conquer algorithm to solve the above optimization problem in a distributed and decentralized manner. The proposed divide-and-conquer algorithm has exponential convergence, its computational cost is almost linear with respect to the size of the network, and it can be fully implemented at fusion centers of the network. In addition, our numerical demonstrations indicate that the proposed divide-and-conquer algorithm has superior performance than popular decentralized optimization methods in solving the least squares problem, both with and without the ${\ell }^1$ penalty, and exhibits great performance on networks equipped with asynchronous local peer-to-peer communication.

在本文中，我们考虑具有由某些连通无向图描述的拓扑的网络$\mathcal{G}=（V,乙）$并与一些配备处理能力和本地点对点通信的代理（融合中心）以及优化问题${\mathrm{分钟}}_{\mathit{X}}\{F（\mathit{X}）={\Sigma }_{我\epsilon V}{F}_{我}（\mathit{X}）\}$具有局部目标函数$F_{我}$仅取决于顶点的邻近变量$我\epsilon V$。我们引入分治算法以分布式、去中心化的方式解决上述优化问题。所提出的分而治之算法具有指数收敛性，其计算成本与网络规模几乎呈线性，并且可以在网络的融合中心完全实现。此外，我们的数值演示表明，所提出的分而治之算法在解决最小二乘问题时比流行的分散优化方法具有更优越的性能，无论有没有${\ell }^1$惩罚，并且在配备异步本地对等通信的网络上表现出出色的性能。