交替同时 Halpern-Lions-Wittmann-Bauschke 算法：寻找两个不相交凸集交集的最佳逼近对
The alternating simultaneous Halpern-Lions-Wittmann-Bauschke algorithm for finding the best approximation pair for two disjoint intersections of convex sets
AI总结：
针对两个不相交凸集交集的最佳逼近对问题，提出基于投影的交替同时 Halpern-Lions-Wittmann-Bauschke 迭代算法，免除了对交集本身的投影，并证明了其在特定条件下的收敛性，推广了已有研究。
摘要：
给定两个非空且不相交的闭凸子集交集，我们寻找相对于它们的最佳逼近对，即在每个交集中各取一个点，使两点间的距离达到这两个不相交交集之间的最小距离。我们提出了一种基于向生成这些交集的子集进行投影的迭代过程。该过程受 Halpern-Lions-Wittmann-Bauschke 算法及 Cheney 和 Goldstein 的经典交替过程启发，其优势在于无需直接对交集本身进行投影，因为这一任务可能相当繁重。我们证明了在特定条件下，两个交错子序列会收敛至最佳逼近对。这些条件在空间为欧几里得空间且生成交集的子集是紧致且严格凸时成立。我们的结果推广了 Aharoni、Censor 和 Jiang 的研究，后者仅考虑了有限维多面体的情况。
英文摘要：
Given two nonempty and disjoint intersections of closed and convex subsets, we look for a best approximation pair relative to them, i.e., a pair of points, one in each intersection, attaining the minimum distance between the disjoint intersections. We propose an iterative process based on projections onto the subsets which generate the intersections. The process is inspired by the Halpern-Lions-Wittmann-Bauschke algorithm and the classical alternating process of Cheney and Goldstein, and its advantage is that there is no need to project onto the intersections themselves, a task which can be rather demanding. We prove that under certain conditions the two interlaced subsequences converge to a best approximation pair. These conditions hold, in particular, when the space is Euclidean and the subsets which generate the intersections are compact and strictly convex. Our result extends the one of Aharoni, Censor and Jiang ["Finding a best approximation pair of points for two polyhedra'', Computational Optimization and Applications 71 (2018), 509--523] who considered the case of finite-dimensional polyhedra.
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