寻找最大封闭凸集交集最大元素的投影方法

IF 4.4 3区 管理学 Q1 OPERATIONS RESEARCH & MANAGEMENT SCIENCE Annals of Operations Research Pub Date : 2024-07-19 DOI:10.1007/s10479-024-05980-z
Tomáš Dlask
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引用次数: 0

摘要

我们将重点放在寻找最大封闭凸集交集的最大元素问题上。为此,我们分析了著名的循环投影法,并证明如果对这种方法进行适当的初始化并应用于最大闭合凸集,就会收敛到它们交集的最大元素。此外,我们还提出了另一种称为递减投影的投影方法,在这种特殊情况下,它在理论和实践上都优于欧几里得投影。接下来,我们论证了几种已知算法,如最短路径的 Bellman-Ford 算法和 Floyd-Warshall 算法,或马尔可夫决策过程中值迭代的高斯-赛德尔变体,都可以解释为在某些最大闭合凸集上迭代应用递减投影的特例。最后,我们将递减投影(以及上述算法)与约束编程中的边界一致性联系起来。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

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Projection methods for finding the greatest element of the intersection of max-closed convex sets

We focus on the problem of finding the greatest element of the intersection of max-closed convex sets. For this purpose, we analyze the famous method of cyclic projections and show that, if this method is suitably initialized and applied to max-closed convex sets, it converges to the greatest element of their intersection. Moreover, we propose another projection method, called the decreasing projection, which turns out both theoretically and practically preferable to Euclidean projections in this particular case. Next, we argue that several known algorithms, such as Bellman-Ford and Floyd-Warshall algorithms for shortest paths or Gauss-Seidel variant of value iteration in Markov decision processes, can be interpreted as special cases of iteratively applying decreasing projections onto certain max-closed convex sets. Finally, we link decreasing projections (and thus also the aforementioned algorithms) to bounds consistency in constraint programming.

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来源期刊
Annals of Operations Research
Annals of Operations Research 管理科学-运筹学与管理科学
CiteScore
7.90
自引率
16.70%
发文量
596
审稿时长
8.4 months
期刊介绍: The Annals of Operations Research publishes peer-reviewed original articles dealing with key aspects of operations research, including theory, practice, and computation. The journal publishes full-length research articles, short notes, expositions and surveys, reports on computational studies, and case studies that present new and innovative practical applications. In addition to regular issues, the journal publishes periodic special volumes that focus on defined fields of operations research, ranging from the highly theoretical to the algorithmic and the applied. These volumes have one or more Guest Editors who are responsible for collecting the papers and overseeing the refereeing process.
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