  We consider minimization of a sum of convex objective functions where the
components of the objective are available at different nodes of a network and
nodes are allowed to only communicate with their neighbors. The use of
distributed subgradient or gradient methods is widespread but they often suffer
from slow convergence since they rely on first order information, which leads
to a large number of local communications between nodes in the network. In this
paper we propose the Network Newton (NN) method as a distributed algorithm that
incorporates second order information via distributed evaluation of
approximations to Newton steps. We also introduce adaptive (A)NN in order to
establish exact convergence. Numerical analyses show significant improvement in
both convergence time and number of communications for NN relative to existing
(first order) alternatives.
