Finding a best approximation pair of points for two polyhedra
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.
KeywordsBest approximation pair Convex polyhedra Alternating projections Half-spaces Cheney–Goldstein theorem Halpern–Lions–Wittmann–Bauschke algorithm
Mathematics Subject Classification65K05 90C20 90C25
We thank Yehuda Zur for Matlab programming work at the early stages of our research.
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