  We analyze the Douglas-Rachford splitting method for weakly convex
optimization problems, by the token of the Douglas-Rachford envelope, a merit
function akin to the Moreau envelope. First, we use epi-convergence techniques
to show that this artifact approximates the original objective function via
epigraphs. Secondly, we present how global convergence and local linear
convergence rates for Douglas-Rachford can be obtained using such envelope,
under mild regularity assumptions. The keystone of the convergence analysis is
the fact that the Douglas-Rachford envelope evaluated at the generated iterates
satisfies a sufficient descent inequality, a feature that allows us to use
arguments usually employed for descent methods. We report the results of
numerical experiments that use weakly convex penalty functions, which are
comparable with the known behavior of the method in the convex case.
