  We introduce the primal-dual quasi-Newton (PD-QN) method as an approximated
second order method for solving decentralized optimization problems. The PD-QN
method performs quasi-Newton updates on both the primal and dual variables of
the consensus optimization problem to find the optimal point of the augmented
Lagrangian. By optimizing the augmented Lagrangian, the PD-QN method is able to
find the exact solution to the consensus problem with a linear rate of
convergence. We derive fully decentralized quasi-Newton updates that
approximate second order information to reduce the computational burden
relative to dual methods and to make the method more robust in ill-conditioned
problems relative to first order methods. The linear convergence rate of PD-QN
is established formally and strong performance advantages relative to existing
dual and primal-dual methods are shown numerically.
