Douglas--Rachford Splitting and ADMM for Nonconvex Optimization: Tight Convergence Results

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.