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In this work, we study two first-order primal-dual based algorithms, theGradient Primal-Dual Algorithm (GPDA) and the Gradient Alternating DirectionMethod of Multipliers (GADMM), for solving a class of linearly constrainednon-convex optimization problems. We show that with random initialization ofthe primal and dual variables, both algorithms are able to compute second-orderstationary solutions (ss2) with probability one. This is the first resultshowing that primal-dual algorithm is capable of finding ss2 when only usingfirst-order information, it also extends the existing results for first-order,but primal-only algorithms.  An important implication of our result is that it also gives rise to thefirst global convergence result to the ss2, for two classes of unconstraineddistributed non-convex learning problems over multi-agent networks.

Gradient Primal-Dual Algorithm Converges to Second-Order Stationary Solutions for Nonconvex Distributed Optimization