An acceleration technique for methods for finding the nearest point in a polytope and computing the distance between two polytopes

TL;DR

This paper introduces an efficient acceleration method for computing Euclidean projections onto large convex polytopes by iteratively working with smaller subpolytopes, significantly reducing computation time especially for high-point polytopes.

Contribution

It proposes a novel acceleration technique applicable to any method for projection onto convex polytopes, improving efficiency when dealing with large point sets.

Findings

Significant reduction in computation time demonstrated through numerical experiments.

Efficiency gain increases with the number of points in the polytope.

Method extends to computing distances between convex polytopes.

Abstract

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.