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Finding a best approximation pair of points for two polyhedra

Ron Aharoni(raharoni***at***gmail.com)
Yair Censor(yair***at***math.haifa.ac.il)
Zilin Jiang(libragold***at***gmail.com)

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.

Keywords: Best approximation pair, convex polyhedra, alternating projections, half-spaces, Cheney–Goldstein theorem, Halpern–Lions–Wittmann–Bauschke algorithm

Category 1: Convex and Nonsmooth Optimization (Convex Optimization )

Category 2: Combinatorial Optimization (Polyhedra )

Citation: Technical Report, July 30, 2017.

Download: [PDF]

Entry Submitted: 08/02/2017
Entry Accepted: 08/02/2017
Entry Last Modified: 08/02/2017

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