A Prediction-Correction Primal-Dual Algorithm for Distributed Optimization

In this work, we propose a novel primal-dual algorithm for solving distributed optimization problems with consensus constraints over a network of agents. Our starting point is to form the Lagrangian of the constrained problem. The agents update their dual variables using dual gradient ascent (the dual step). By interpreting the dual variables as time-varying parameters, the agents track the minimizer of the resulting time-varying Lagrangian using a prediction-correction scheme (the primal step). Each iteration of the resulting algorithm requires two rounds of communication and two local Hessian inversions per agent. In particular, we establish exponential convergence of the resulting algorithm to a neighborhood of the optimal solution. Numerical experiments support the theoretical conclusions.