Anderson Acceleration for Nonconvex ADMM Based on Douglas-Rachford Splitting

TL;DR

This paper introduces a novel approach that applies Anderson acceleration to the lower-dimensional fixed-point iteration of ADMM, based on Douglas-Rachford splitting, to improve convergence speed in nonconvex optimization problems in computer graphics.

Contribution

It reveals that ADMM can be accelerated by applying Anderson acceleration in the lower-dimensional space identified via Douglas-Rachford splitting, enhancing convergence in nonconvex settings.

Findings

Accelerates convergence of ADMM in nonconvex problems

Effective in geometry processing and physical simulation

Convergence analysis supports theoretical robustness

Abstract

The 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 problems including geometry processing and physical simulation.