  Dual descent methods are commonly used to solve network optimization problems
because their implementation can be distributed through the network. However,
their convergence rates are typically very slow. This paper introduces a family
of dual descent algorithms that use approximate Newton directions to accelerate
the convergence rate of conventional dual descent. These approximate directions
can be computed using local information exchanges thereby retaining the
benefits of distributed implementations. The approximate Newton directions are
obtained through matrix splitting techniques and sparse Taylor approximations
of the inverse Hessian.We show that, similarly to conventional Newton methods,
the proposed algorithm exhibits superlinear convergence within a neighborhood
of the optimal value. Numerical analysis corroborates that convergence times
are between one to two orders of magnitude faster than existing distributed
optimization methods. A connection with recent developments that use consensus
iterations to compute approximate Newton directions is also presented.
