Accelerating ADMM for efficient simulation and optimization

Yue Peng,Juyong Zhang,Bailin Deng,Wenqing Ouyang

doi:10.1145/3355089.3356491

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.

Highlights

Many tasks in computer graphics involve solving optimization problems
When the problem structure satisfies some mild conditions, one of these two variables can be determined from the other one; in this case alternating direction method of multipliers (ADMM) becomes a fixed-point iteration of only one variable with less computational overhead, and we can accept an accelerated iterate based on a more simple condition. We apply this method to a variety of ADMM solvers for computer graphics problems, and observe a notable improvement in their convergence rates
We show that ADMM can be interpreted as a fixed-point iteration of the second primal variable and the dual variable in the general case, and of only one of them when the problem has a separable target function and satisfies certain conditions

Summary

Introduction

A geometry processing task may compute the vertex positions of a deformed mesh by minimizing its deformation energy [Sorkine and Alexa 2007], whereas a physical simulation task may optimize the node positions of a system to enforce physics laws that govern its behavior [Martin et al 2011; Schumacher et al 2012]. Such tasks are often formulated as unconstrained optimization, where the target function penalizes the violation of certain conditions so that they are satisfied as much as possible by the solution.

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