Anderson Acceleration for Nonsmooth Fixed Point Problems

In this talk, we consider the convergence behavior of Anderson acceleration algorithm for nonsmooth fixed point problems. First, we prove convergence of Anderson acceleration for a class of nonsmooth fixed-point problems for which the nonlinearities can be split into a smooth contractive part and a nonsmooth part which has a small Lipschitz constant. Second, we propose a smoothing approximation of the composite max function and show that the smoothing approximation for a class of contractive mappings is also a contraction mapping and has the same fixed point as the original problem, which confirms that the nonsmoothness can not affect the convergence properties of Anderson(m) algorithm when we use the proposed smoothing approximation instead of the original nonsmooth one. Third, we propose a novel Smoothing Anderson(m) algorithm by using a smoothing function of the original fixed point function and show its r-linear convergence. Furthermore, we prove the q-linear convergence of the Smoothing Anderson(1) algorithm. Finally, some numerical examples are given to demonstrate the efficiency of the proposed smoothing Anderson acceleration algorithms.

报告人简介：边伟，哈尔滨工业大学数学学院，教授、博士生导师。2004年和2009年于哈尔滨工业大学分别获得学士和博士学位，导师薛小平教授。2010-2012年访问香港理工大学，跟随陈小君教授从事博士后工作。主要研究领域为：最优化理论与算法。先后在Math. Program., Math. Oper. Res., SIAM J. Optim., SIAM J. Numer. Anal., SIAM J. Sci. Comput., SIAM J. Imaging Sci.等期刊发表多篇学术论文。先后获国家级青年人才称号和国家杰出青年基金项目。现任SCI期刊Journal of Optimization Theory and Application编委，中国运筹学会常务理事，黑龙江省数学会常务理事，中国运筹学会数学规划分会理事等。