  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 (DRE), 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 over-relaxation
parameters than previously known. In fact, our bounds are tight whenever the
over-relaxation parameter ranges in $(0,2]$. The analysis of ADMM uses a
universal primal equivalence with DRS that generalizes the known duality of the
algorithms.
