A divide-and-conquer algorithm for distributed optimization on networks

In this paper, we consider networks with topologies described by some connected undirected graph $G=(V,E)$ and with some agents (fusion centers) equipped with processing power and local peer-to-peer communication, and optimization problem ${\min }_x\left\{F\right(x)={\sum }_{i\in V}{f}_i(x\left)\right\}$ 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.