一般赋范空间中不动点问题的M-对偶性
M-duality in fixed-point problems for general normed spaces
- University of Pennsylvania（宾夕法尼亚大学）
- Seoul National University（首尔大学）
- UCLA（加州大学洛杉矶分校）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文在一般赋范空间中为Mann迭代建立M-对偶定理，并利用该对偶性提出新算法，其最坏情况残差界优于此前最优的Halpern迭代。
AI中文摘要：
通过减小残差界来求解不动点问题是不动点迭代中的核心方法之一，特别是Halpern迭代被已知能有效减小残差。近年来，在凸优化和不动点问题文献中发现了一种新型的对偶定理，该定理在互为对偶的算法的收敛行为中建立了某种对称性。然而，这些对偶定理依赖于Lyapunov式的证明结构，并且主要关注Hilbert空间。在本工作中，我们首先在一般赋范空间中，针对具有Lipschitz算子的Mann迭代建立了M-对偶定理，仅假设迭代系数满足温和条件。此外，利用此M-对偶性，我们推导出新的算法，其最坏情况残差界严格小于Halpern迭代的残差界，而Halpern迭代此前在一般赋范空间中被认为是性能最优的方法。
英文摘要：
Solving fixed-point problems by reducing residual bounds is one of the central approaches in fixed-point iteration, and, in particular, the Halpern iteration is known to effectively reduce the residual. Recently, a new type of duality theorem has been discovered in the convex optimization and fixed-point problem literature and establish a certain symmetry in the convergence behavior of algorithms that are dual to each other. However, these duality theorems rely on a Lyapunov-style proof structure and mostly focus on Hilbert spaces. In this work, we first establish an M-duality theorem for Mann iterations with Lipschitz operators in general normed spaces, assuming only mild conditions on the coefficients of the iterations. Moreover, leveraging this M-duality, we derive new algorithms that attain a strictly smaller worst-case residual bound than that of the Halpern iteration, which was previously known as the state-of-the-art method in general normed spaces.
