  This paper considers consensus optimization problems where each node of a
network has access to a different summand of an aggregate cost function. Nodes
try to minimize the aggregate cost function, while they exchange information
only with their neighbors. We modify the dual decomposition method to
incorporate a curvature correction inspired by the
Broyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method. The resulting dual
D-BFGS method is a fully decentralized algorithm in which nodes approximate
curvature information of themselves and their neighbors through the
satisfaction of a secant condition. Dual D-BFGS is of interest in consensus
optimization problems that are not well conditioned, making first order
decentralized methods ineffective, and in which second order information is not
readily available, making decentralized second order methods infeasible.
Asynchronous implementation is discussed and convergence of D-BFGS is
established formally for both synchronous and asynchronous implementations.
Performance advantages relative to alternative decentralized algorithms are
shown numerically.
