On the centralization of the circumcentered-reflection method
On the centralization of the circumcentered-reflection method
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DOI：
10.1007/s10107-023-01978-w
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发表时间：
2021-11
期刊：
影响因子：
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通讯作者：
R. Behling;Yunier Bello-Cruz;A. Iusem;L. Santos
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作者：
R. Behling;Yunier Bello-Cruz;A. Iusem;L. Santos
本文给出了第一个不使用乘积空间变换的求两个闭凸集交点的外心迭代格式。我们介绍了一个所谓的集中式版本的环绕反射法（CRM）。CRM是为了加速经典的投影算法而开发的，它成功地跟踪了有限个仿射集的公共点。在一般凸集的情况下，如果不使用Pierra的积空间重构，CRM可能会出现分歧。在这项工作中，我们证明了存在一个容易到达的区域组成的，我们称之为集中点，纯外接圆步骤具有收敛性能。由此产生的算法被称为集中式CRM（cCRM）。除了具有全局收敛性之外，cCRM在误差界条件下线性收敛，并且如果两个目标集使得它们的交集具有非空内部并且它们的边界是局部可微流形，则超线性收敛。我们还进行了数值实验，取得了成功的结果。
This paper is devoted to deriving the first circumcenter iteration scheme that does not employ a product space reformulation for finding a point in the intersection of two closed convex sets. We introduce a so-called centralized version of the circumcentered-reflection method (CRM). Developed with the aim of accelerating classical projection algorithms, CRM is successful for tracking a common point of a finite number of affine sets. In the case of general convex sets, CRM was shown to possibly diverge if Pierra’s product space reformulation is not used. In this work, we prove that there exists an easily reachable region consisting of what we refer to as centralized points, where pure circumcenter steps possess properties yielding convergence. The resulting algorithm is called centralized CRM (cCRM). In addition to having global convergence, cCRM converges linearly under an error bound condition, and superlinearly if the two target sets are so that their intersection have nonempty interior and their boundaries are locally differentiable manifolds. We also run numerical experiments with successful results.