Finding best approximation pairs relative to two closed convex sets in Hilbert spaces
Finding best approximation pairs relative to two closed convex sets in Hilbert spaces
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DOI：
10.1016/j.jat.2004.02.006
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发表时间：
2004-04-01
影响因子：
0.9
通讯作者：
Luke, DR
中科院分区：
文献类型：
--
作者：
Bauschke, HH;Combettes, PL;Luke, DR
我们考虑找到最佳近似对的问题，即，在希尔伯特空间中，两个点达到两个闭凸集之间的最小距离。当集合相交时，所考虑的方法，称为AAR的平均交替反射，是一个特殊的例子，由于狮子和Mercier的算法找到一个零的总和两个最大的单调算子。我们系统地研究了AAR的渐近行为在一般情况下，当集合不一定相交，并表明该方法产生最佳逼近对，只要它们存在。许多集处理的产品空间，在这种情况下，AAR方法示出符合Spingarn的方法的部分逆的一个特殊情况。(C)2004年爱思唯尔公司All rights reserved.
We consider the problem of finding a best approximation pair, i.e., two points which achieve the minimum distance between two closed convex sets in a Hilbert space. When the sets intersect, the method under consideration, termed AAR for averaged alternating reflections, is a special instance of an algorithm due to Lions and Mercier for finding a zero of the sum of two maximal monotone operators. We investigate systematically the asymptotic behavior of AAR in the general case when the sets do not necessarily intersect and show that the method produces best approximation pairs provided they exist. Finitely many sets are handled in a product space, in which case the AAR method is shown to coincide with a special case of Spingarn's method of partial inverses. (C) 2004 Elsevier Inc. All rights reserved.