Understanding the Douglas–Rachford splitting method through the lenses of Moreau-type envelopes
分析了Douglas–Rachford分裂方法在弱凸优化问题中的表现，通过引入类似Moreau包络的Douglas–Rachford包络，证明了全局收敛和局部线性收敛速率，并用数值实验验证了效果。
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.