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Although originally designed and analyzed for convex problems, the alternating direction method of multipliers (ADMM) and its close relatives, Douglas–Rachford splitting (DRS) and Peaceman–Rachford splitting (PRS), have been observed to perform remarkably well when applied to certain classes of structured nonconvex optimization problems. However, partial global convergence results in the nonconvex setting have only recently emerged. In this paper we show how the Douglas–Rachford envelope, introduced in 2014, can be employed to unify and considerably simplify the theory for devising global convergence guarantees for ADMM, DRS, and PRS applied to nonconvex problems under less restrictive conditions, larger prox-stepsizes, and overrelaxation parameters than previously known. In fact, our bounds are tight whenever the overrelaxation parameter ranges in (0, 2]. The analysis of ADMM uses a universal primal equivalence with DRS that generalizes the known duality of the algorithms. [ABSTRACT FROM AUTHOR]

The alternating direction method of multipliers (ADMM) is widely used to solve large-scale linearly constrained optimization problems, convex or nonconvex, in many engineering fields. However there is a general lack of theoretical understanding of the algorithm when the objective function is nonconvex. In this paper we analyze the convergence of the ADMM for solving certain nonconvex consensus and sharing problems. We show that the classical ADMM converges to the set of stationary solutions, provided that the penalty parameter in the augmented Lagrangian is chosen to be sufficiently large. For the sharing problems, we show that the ADMM is convergent regardless of the number of variable blocks. Our analysis does not impose any assumptions on the iterates generated by the algorithm and is broadly applicable to many ADMM variants involving proximal update rules and various flexible block selection rules. [ABSTRACT FROM AUTHOR]