Finding a best approximation pair of points for two polyhedra

TL;DR

This paper introduces a new projection-based method to find the closest points between two disjoint convex polyhedra, improving computational feasibility over traditional approaches.

Contribution

It proposes a novel process using half-space projections and the Halpern--Lions--Wittmann--Bauschke algorithm for efficient approximation of the best pair of points.

Findings

The method converges to a best approximation pair.

It simplifies projections by using half-spaces instead of entire polyhedra.

The approach enhances computational efficiency for convex set approximations.

Abstract

Given two disjoint convex polyhedra, we look for a best approximation pair relative to them, i.e., a pair of points, one in each polyhedron, attaining the minimum distance between the sets. Cheney and Goldstein showed that alternating projections onto the two sets, starting from an arbitrary point, generate a sequence whose two interlaced subsequences converge to a best approximation pair. We propose a process based on projections onto the half-spaces defining the two polyhedra, which are more negotiable than projections on the polyhedra themselves. A central component in the proposed process is the Halpern--Lions--Wittmann--Bauschke algorithm for approaching the projection of a given point onto a convex set.