In this work, we introduce ADAPD, $textbf{A}$ $textbf{D}$ecentr$textbf{A}$lized $textbf{P}$rimal-$textbf{D}$ual algorithmic framework for solving non-convex and smooth consensus optimization problems over a network of distributed agents. ADAPD makes use of an inexact ADMM-type update. During each iteration, each agent first inexactly solves a local strongly convex subproblem and then performs a neighbor communication while updating a set of dual variables. Two variations to ADAPD are presented. The variants allow agents to balance the communication and computation workload while they collaboratively solve the consensus optimization problem. The optimal convergence rate for non-convex and smooth consensus optimization problems is established; namely, ADAPD achieves $varepsilon$-stationarity in $mathcal{O}(varepsilon^{-1})$ iterations. Numerical experiments demonstrate the superiority of ADAPD over several existing decentralized methods.
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