This paper considers decentralized consensus optimization problems wheredifferent summands of a global objective function are available at nodes of anetwork that can communicate with neighbors only. The proximal method ofmultipliers is considered as a powerful tool that relies on proximal primaldescent and dual ascent updates on a suitably defined augmented Lagrangian. Thestructure of the augmented Lagrangian makes this problem non-decomposable,which precludes distributed implementations. This problem is regularlyaddressed by the use of the alternating direction method of multipliers. Theexact second order method (ESOM) is introduced here as an alternative thatrelies on: (i) The use of a separable quadratic approximation of the augmentedLagrangian. (ii) A truncated Taylor's series to estimate the solution of thefirst order condition imposed on the minimization of the quadraticapproximation of the augmented Lagrangian. The sequences of primal and dualvariables generated by ESOM are shown to converge linearly to their optimalarguments when the aggregate cost function is strongly convex and its gradientsare Lipschitz continuous. Numerical results demonstrate advantages of ESOMrelative to decentralized alternatives in solving least squares and logisticregression problems.
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