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We analyze the Douglas-Rachford splitting method for weakly convexoptimization problems, by the token of the Douglas-Rachford envelope, a meritfunction akin to the Moreau envelope. First, we use epi-convergence techniquesto show that this artifact approximates the original objective function viaepigraphs. Secondly, we present how global convergence and local linearconvergence rates for Douglas-Rachford splitting can be obtained using suchenvelope, under mild regularity assumptions. The keystone of the convergenceanalysis is the fact that the Douglas-Rachford envelope satisfies a sufficientdescent inequality alongside the generated sequence, a feature that allows usto use arguments usually employed for descent methods. We report numericalexperiments that use weakly convex penalty functions, which are comparable withthe known behavior of the method in the convex case.

Understanding the Douglas-Rachford splitting method through the lenses of Moreau-type envelopes