Finding a best approximation pair of points for two polyhedra
This work addresses the computational bottleneck of projecting onto polyhedra for solving the best approximation pair problem, offering a more negotiable alternative.
The paper proposes a new iterative method for finding the minimum distance between two disjoint convex polyhedra, using projections onto their defining half-spaces instead of the polyhedra themselves, and proves convergence to a best approximation pair.
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.