  We introduce the decentralized Broyden-Fletcher-Goldfarb-Shanno (D-BFGS)
method as a variation of the BFGS quasi-Newton method for solving decentralized
optimization problems. The D-BFGS method is of interest in problems that are
not well conditioned, making first order decentralized methods ineffective, and
in which second order information is not readily available, making second order
decentralized methods impossible. D-BFGS is a fully distributed algorithm in
which nodes approximate curvature information of themselves and their neighbors
through the satisfaction of a secant condition. We additionally provide a
formulation of the algorithm in asynchronous settings. Convergence of D-BFGS is
established formally in both the synchronous and asynchronous settings and
strong performance advantages relative to first order methods are shown
numerically.
