重定位不动点迭代及其在变步长预解分裂中的应用
Relocated Fixed-Point Iterations with Applications to Variable Stepsize Resolvent Splitting
AI总结：
提出一种收敛框架，允许迭代算法中的非扩张算子族无公共不动点，通过重定位步骤实现变步长预解分裂，并应用于Douglas-Rachford算法。
摘要：
在这项工作中，我们开发了一个迭代算法的收敛框架，其更新可以由一个单参数族的非扩张算子描述。在该框架中，涉及主要算法算子的每一步之后都跟随第二步，该步将当前算子的不动点“重定位”到下一个算子。因此，我们的分析不要求非扩张算子族具有公共不动点，而这在文献中经常被假设。我们的分析使用了非扩张算子的半闭性原理的参数扩展。作为我们收敛结果的一个应用，我们开发了基于图的Douglas-Rachford算法的一个版本，用于寻找$N\geq 2$个极大单调算子之和的零点，该算法不要求预解参数在迭代过程中保持不变。
英文摘要：
In this work, we develop a convergence framework for iterative algorithms whose updates can be described by a one-parameter family of nonexpansive operators. Within the framework, each step involving one of the main algorithmic operators is followed by a second step which ''relocates'' fixed points of the current operator to the next. As a consequence, our analysis does not require the family of nonexpansive operators to have a common fixed point, as frequently assumed in the literature. Our analysis uses a parametric extension of the demiclosedness principle for nonexpansive operators. As an application of our convergence results, we develop a version of the graph-based extension of the Douglas--Rachford algorithm for finding a zero of the sum of $N\geq 2$ maximally monotone operators, which does not require the resolvent parameter to be constant across iterations.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
