Accelerating ADMM for Efficient Simulation and Optimization

TL;DR

This paper introduces an Anderson acceleration method to speed up ADMM, improving convergence in both convex and certain non-convex optimization problems common in computer graphics applications.

Contribution

It extends ADMM acceleration using Anderson acceleration, including convergence analysis for specific non-convex problems in graphics.

Findings

Significant reduction in convergence time for various graphics optimization tasks

Theoretical proof of ADMM convergence on certain non-convex problems

Enhanced efficiency in physical simulation and geometry processing

Abstract

The alternating direction method of multipliers (ADMM) is a popular approach for solving optimization problems that are potentially non-smooth and with hard constraints. It has been applied to various computer graphics applications, including physical simulation, geometry processing, and image processing. However, ADMM can take a long time to converge to a solution of high accuracy. Moreover, many computer graphics tasks involve non-convex optimization, and there is often no convergence guarantee for ADMM on such problems since it was originally designed for convex optimization. In this paper, we propose a method to speed up ADMM using Anderson acceleration, an established technique for accelerating fixed-point iterations. We show that in the general case, ADMM is a fixed-point iteration of the second primal variable and the dual variable, and Anderson acceleration can be directly applied. Additionally, when the problem has a separable target function and satisfies certain conditions, ADMM becomes a fixed-point iteration of only one variable, which further reduces the computational overhead of Anderson acceleration. Moreover, we analyze a particular non-convex problem structure that is common in computer graphics, and prove the convergence of ADMM on such problems under mild assumptions. We apply our acceleration technique on a variety of optimization problems in computer graphics, with notable improvement on their convergence speed.