  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 $\min_{{\boldsymbol x}}\big\{F({\boldsymbol
x})=\sum_{i\in V}f_i({\boldsymbol x})\big\}$ 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. Our numerical
demonstrations also indicate that the proposed divide-and-conquer algorithm has
superior performance than popular decentralized optimization methods do for the
least squares problem with/without $\ell^1$ penalty.
