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.
Bibliographical noteFunding Information:
Ron Aharoni: Supported in part by the United States–Israel Binational Science Foundation (BSF) Grant No. 2012031, the Israel Science Foundation (ISF) Grant No. 2023464 and the Discount Bank Chair at the Technion. Yair Censor: Supported in part by BSF Grant No. 2013003. Zilin Jiang: Supported in part by ISF Grant Nos. 1162/15, 936/16..
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- Alternating projections
- Best approximation pair
- Cheney–Goldstein theorem
- Convex polyhedra
- Halpern–Lions–Wittmann–Bauschke algorithm
ASJC Scopus subject areas
- Control and Optimization
- Computational Mathematics
- Applied Mathematics