## 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.

Original language | English |
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Pages (from-to) | 509-523 |

Number of pages | 15 |

Journal | Computational Optimization and Applications |

Volume | 71 |

Issue number | 2 |

DOIs | |

State | Published - 1 Nov 2018 |

### Bibliographical note

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

## Keywords

- Alternating projections
- Best approximation pair
- Cheney–Goldstein theorem
- Convex polyhedra
- Half-spaces
- Halpern–Lions–Wittmann–Bauschke algorithm

## ASJC Scopus subject areas

- Control and Optimization
- Computational Mathematics
- Applied Mathematics