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In this paper, we propose a new decomposition approach named the proximalprimal dual algorithm (Prox-PDA) for smooth nonconvex linearly constrainedoptimization problems. The proposed approach is primal-dual based, where theprimal step minimizes certain approximation of the augmented Lagrangian of theproblem, and the dual step performs an approximate dual ascent. Theapproximation used in the primal step is able to decompose the variable blocks,making it possible to obtain simple subproblems by leveraging the problemstructures. Theoretically, we show that whenever the penalty parameter in theaugmented Lagrangian is larger than a given threshold, the Prox-PDA convergesto the set of stationary solutions, globally and in a sublinear manner (i.e.,certain measure of stationarity decreases in the rate of $\mathcal{O}(1/r)$,where $r$ is the iteration counter). Interestingly, when applying a variant ofthe Prox-PDA to the problem of distributed nonconvex optimization (over aconnected undirected graph), the resulting algorithm coincides with the popularEXTRA algorithm [Shi et al 2014], which is only known to work in convex cases.Our analysis implies that EXTRA and its variants converge globally sublinearlyto stationary solutions of certain nonconvex distributed optimization problem.There are many possible extensions of the Prox-PDA, and we present oneparticular extension to certain nonconvex distributed matrix factorizationproblem.

Decomposing Linearly Constrained Nonconvex Problems by a Proximal Primal Dual Approach: Algorithms, Convergence, and Applications