Anderson Acceleration for Nonconvex ADMM Based on Douglas‐Rachford Splitting

Wenqing Ouyang,Yue Peng,Yuxin Yao,Juyong Zhang,Bailin Deng

doi:10.1111/cgf.14081

Abstract

AbstractThe alternating direction multiplier method (ADMM) is widely used in computer graphics for solving optimization problems that can be nonsmooth and nonconvex. It converges quickly to an approximate solution, but can take a long time to converge to a solution of high‐accuracy. Previously, Anderson acceleration has been applied to ADMM, by treating it as a fixed‐point iteration for the concatenation of the dual variables and a subset of the primal variables. In this paper, we note that the equivalence between ADMM and Douglas‐Rachford splitting reveals that ADMM is in fact a fixed‐point iteration in a lower‐dimensional space. By applying Anderson acceleration to such lower‐dimensional fixed‐point iteration, we obtain a more effective approach for accelerating ADMM. We analyze the convergence of the proposed acceleration method on nonconvex problems, and verify its effectiveness on a variety of computer graphics including geometry processing and physical simulation.

Highlights

The alternating direction multiplier method (ADMM) is widely used in computer graphics for solving optimization problems that can be nonsmooth and nonconvex
We evaluate our method on a variety of nonconvex ADMM solvers used in computer graphics and other domains
We focus on ADMM for the following optimization problem with a separable target function: min f (x) + g(z) s.t

Summary

Introduction

The alternating direction multiplier method (ADMM) is widely used in computer graphics for solving optimization problems that can be nonsmooth and nonconvex. The basic idea is to introduce auxiliary variables to replace some of the original variables in the target function, while enforcing consistency between the original variables and the auxiliary variables with a soft or hard constraint This often allows to problem to be solved via alternating update of the variables, which reduces to simple sub-problems that can be solved efficiently. One example of such proximal splitting methods is the local-global solvers commonly used for geometry processing and physical simulation [SA07, LZX∗08, BDS∗12, LBOK13, BML∗14]. This article may be used for non-commercial purposes in accordance with the Wiley Self-Archiving Policy

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