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.

翻译：给定两个非空且不相交的闭凸子集的交集，我们寻找相对于它们的最佳逼近对，即分别位于两个交集中且达到不相交交集间最小距离的一对点。我们提出一种基于生成这些交集的子集投影的迭代过程。该过程受Halpern-Lions-Wittmann-Bauschke算法以及Cheney和Goldstein的经典交替过程启发，其优势在于无需直接投影到交集本身——这一任务通常相当困难。我们证明，在特定条件下，两个交织子序列收敛到最佳逼近对。这些条件在欧几里得空间且生成交集的子集为紧致严格凸集时成立。我们的结果推广了Aharoni、Censor和Jiang的结论（见《寻找两个多面体的最佳逼近点对》，Computational Optimization and Applications 71 (2018), 509--523），他们仅考虑了有限维多面体的情形。