最佳逼近元组：Cheney-Goldstein算法及结果到多集合情形的推广
The best approximation tuple: an extension of the Cheney-Goldstein algorithm and results to the multiple sets case
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中文总结 AI 辅助
本文推广Cheney-Goldstein算法至多集合情形，提出最佳逼近元组问题，通过区分枢轴与卫星集合克服理论障碍，并证明在严格凸紧致卫星集合下的收敛性。
中文摘要 AI 辅助
在本文中，我们将Cheney和Goldstein于1959年发表的著名论文中关于最佳逼近对（BAP）问题的算法及若干结果向两个独立方向进行推广。其一是考虑多于两个集合的情形。其二是能够将每个集合处理为有限族集合的交集。我们将由此产生的问题称为“最佳逼近元组（BAT）问题”。导致这些推广的基本观察是，识别并处理一个集合（“枢轴集合”）使其不同于其余集合（“卫星集合”），而不是寻求作为目标泛函极小元的循环。这使我们能够克服与一般泛函的循环和极小元相关的某个理论障碍。在欧几里得情形下，当卫星集合为严格凸且紧致时，我们证明了该算法收敛到问题的唯一解。由于缺乏Fejér单调性，我们的收敛性分析并非标准，而是基于正交投影在非扩张性定义中等式和不等式方面的几乎不为人知的属性。
英文摘要
We extend the algorithm and several results published in the celebrated 1959 paper of Cheney and Goldstein about the best approximation pair (BAP) problem in two separate directions. One is, for the first time, the ability to consider more than two sets, a task which has been in the mind of researchers for many years without much success in fulfilling it because of a certain theoretical obstacle related to cycles and minimizers of general functionals. The other direction is the ability to handle each set as an intersection of a finite family of sets. We call the resulting problem the "Best Approximation Tuple (BAT) problem". The fundamental observation that leads to these extensions is to recognize and handle one set (the "pivot set") as different from the remaining sets (the "satellite sets") instead of seeking cycles as the minimizers of a target functional. We prove the convergence of the algorithm to the unique solution of the problem in the Euclidean case with strictly convex and compact satellite sets. Because of the lack of Fejér monotonicity, our convergence analysis is not standard, and is based on almost unknown properties of orthogonal projections regarding equality and inequality in the definition of nonexpansiveness.
发表机构
- University of Haifa（海法大学）
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