TY - JOUR

T1 - Finding a best approximation pair of points for two polyhedra

AU - Aharoni, Ron

AU - Censor, Yair

AU - Jiang, Zilin

N1 - Funding 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..
Publisher Copyright:
© 2018, Springer Science+Business Media, LLC, part of Springer Nature.

PY - 2018/11/1

Y1 - 2018/11/1

N2 - 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.

AB - 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.

KW - Alternating projections

KW - Best approximation pair

KW - Cheney–Goldstein theorem

KW - Convex polyhedra

KW - Half-spaces

KW - Halpern–Lions–Wittmann–Bauschke algorithm

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U2 - 10.1007/s10589-018-0021-3

DO - 10.1007/s10589-018-0021-3

M3 - Article

AN - SCOPUS:85049572052

SN - 0926-6003

VL - 71

SP - 509

EP - 523

JO - Computational Optimization and Applications

JF - Computational Optimization and Applications

IS - 2

ER -