一种用于求多面体中最近点及计算两个多面体之间距离的方法的加速技术
An acceleration technique for methods for finding the nearest point in a polytope and computing the distance between two polytopes
AI总结：
针对高维多点凸多面体投影问题，本文提出通过逐步移动小子多面体来加速任意投影算法，并扩展到两凸多面体距离计算，实验显示显著且随点数增长的加速效果。
摘要：
我们提出了一种简单而高效的加速技术，适用于任意计算点到凸多面体的欧几里得投影的方法；该凸多面体定义为有限数量点的凸包，且适用于多面体中点的数量远大于空间维数的情形。该技术的做法是：将任意给定方法应用于原始多面体的一个“小”子多面体，并逐步移动该子多面体，直到给定点到该子多面体的投影与其到原始多面体的投影重合。数值实验结果证明了所提出加速技术的高效率。特别是，结果表明，计算时间的减少随多面体中点数的增加而增大，并且对于某些方法与该点数成正比。在论文的第二部分，我们还讨论了将所提出的加速技术直接扩展到任意计算两个凸多面体之间距离的方法的情形，这两个凸多面体定义为有限点集的凸包。
英文摘要：
We present a simple and efficient acceleration technique for an arbitrary method for computing the Euclidean projection of a point onto a convex polytope, defined as the convex hull of a finite number of points, in the case when the number of points in the polytope is much greater than the dimension of the space. The technique consists in applying any given method to a "small" subpolytope of the original polytope and gradually shifting it, till the projection of the given point onto the subpolytope coincides with its projection onto the original polytope. The results of numerical experiments demonstrate the high efficiency of the proposed acceleration technique. In particular, they show that the reduction of computation time increases with an increase of the number of points in the polytope and is proportional to this number for some methods. In the second part of the paper, we also discuss a straightforward extension of the proposed acceleration technique to the case of arbitrary methods for computing the distance between two convex polytopes, defined as the convex hulls of finite sets of points.
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