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This paper considers consensus optimization problems where each node of anetwork has access to a different summand of an aggregate cost function. Nodestry to minimize the aggregate cost function, while they exchange informationonly with their neighbors. We modify the dual decomposition method toincorporate a curvature correction inspired by theBroyden-Fletcher-Goldfarb-Shanno (BFGS) quasi-Newton method. The resulting dualD-BFGS method is a fully decentralized algorithm in which nodes approximatecurvature information of themselves and their neighbors through thesatisfaction of a secant condition. Dual D-BFGS is of interest in consensusoptimization problems that are not well conditioned, making first orderdecentralized methods ineffective, and in which second order information is notreadily available, making decentralized second order methods infeasible.Asynchronous implementation is discussed and convergence of D-BFGS isestablished formally for both synchronous and asynchronous implementations.Performance advantages relative to alternative decentralized algorithms areshown numerically.

A Decentralized Quasi-Newton Method for Dual Formulations of Consensus Optimization