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

Abstract: 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 splitting can be obtained using such envelope, under mild regularity assumptions. The keystone of the convergence analysis is the fact that the Douglas-Rachford envelope satisfies a sufficient descent inequality alongside the generated sequence, a feature that allows us to use arguments usually employed for descent methods. We report numerical experiments that use weakly convex penalty functions, which are comparable with the known behavior of the method in the convex case.