改进非光滑优化的收敛速度，通过识别活跃流形实现快速稳定迭代。
Anderson Acceleration in Nonsmooth Problems: Local Convergence via Active Manifold Identification
- 基于活跃流形识别机制，设计非光滑问题的加速算法。
- 在驻点处实现局部线性收敛，理论证明收敛更快。
- 适合需要高效求解非光滑优化的科研与工程人员使用。
Anderson加速是一种提升定点迭代效率的有效技术；然而，在非光滑情形下分析其收敛性面临重大挑战。本文研究一类具有活跃流形识别特性的非光滑优化算法，包括近端点法、近端梯度法、近端线性法、近端坐标下降法、Douglas-Rachford分裂（或交替方向乘子法）以及迭代加权ℓ₁方法等。假设优化问题在驻点处存在活跃流形，我们建立了Anderson加速算法的局部R-线性收敛速率。大量数值实验进一步验证了所提方法的鲁棒性能。
原文摘要 · Abstract (English)
Anderson acceleration is an effective technique for enhancing the efficiency of fixed-point iterations; however, analyzing its convergence in nonsmooth settings presents significant challenges. In this paper, we investigate a class of nonsmooth optimization algorithms characterized by the active manifold identification property. This class includes a diverse array of methods such as the proximal point method, proximal gradient method, proximal linear method, proximal coordinate descent method, Douglas-Rachford splitting (or the alternating direction method of multipliers), and the iteratively reweighted $\ell_1$ method, among others. Under the assumption that the optimization problem possesses an active manifold at a stationary point, we establish a local R-linear convergence rate for the Anderson-accelerated algorithm. Our extensive numerical experiments further highlight the robust performance of the proposed Anderson-accelerated methods.
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