  We consider distributed optimization problems where networked nodes
cooperatively minimize the sum of their locally known convex costs. A popular
class of methods to solve these problems are the distributed gradient methods,
which are attractive due to their inexpensive iterations, but have a drawback
of slow convergence rates. This motivates the incorporation of second-order
information in the distributed methods, but this task is challenging: although
the Hessians which arise in the algorithm design respect the sparsity of the
network, their inverses are dense, hence rendering distributed implementations
difficult. We overcome this challenge and propose a class of distributed
Newton-like methods, which we refer to as Distributed Quasi Newton (DQN). The
DQN family approximates the Hessian inverse by: 1) splitting the Hessian into
its diagonal and off-diagonal part, 2) inverting the diagonal part, and 3)
approximating the inverse of the off-diagonal part through a weighted linear
function. The approximation is parameterized by the tuning variables which
correspond to different splittings of the Hessian and by different weightings
of the off-diagonal Hessian part. Specific choices of the tuning variables give
rise to different variants of the proposed general DQN method -- dubbed DQN-0,
DQN-1 and DQN-2 -- which mutually trade-off communication and computational
costs for convergence. Simulations demonstrate the effectiveness of the
proposed DQN methods.
