Anderson Acceleration for Nonsmooth Fixed Point Problems

TL;DR

This paper establishes new convergence results for Anderson acceleration applied to nonsmooth fixed point problems, demonstrating that smoothing approximations preserve convergence rates and improve efficiency in various applications.

Contribution

The paper introduces a smoothing approximation for the composite max function, proving it maintains contraction properties and does not affect Anderson acceleration's convergence rate.

Findings

Anderson(1) and EDIIS(1) are q-linear convergent with improved q-factors.

The smoothing approximation is a contraction mapping with the same fixed point.

Numerical experiments show the method's efficiency in constrained minimax, complementarity, and nonsmooth differential equations.

Abstract

We give new convergence results of Anderson acceleration for the composite $\max$ fixed point problem. We prove that Anderson(1) and EDIIS(1) are q-linear convergent with a smaller q-factor than existing q-factors. Moreover, we propose a smoothing approximation of the composite max function in the contractive fixed point problem. We show that the smoothing approximation is a contraction mapping with the same fixed point as the composite $\max$ fixed point problem. Our results rigorously confirm that the nonsmoothness does not affect the convergence rate of Anderson acceleration method when we use the proposed smoothing approximation for the composite $\max$ fixed point problem. Numerical results for constrained minimax problems, complementarity problems and nonsmooth differential equations are presented to show the efficiency and good performance of the proposed Anderson acceleration method with smoothing approximation.