Finding a best approximation pair of points for two polyhedra

@article{Aharoni2018FindingAB,
  title={Finding a best approximation pair of points for two polyhedra},
  author={Ron Aharoni and Yair Censor and Zilin Jiang},
  journal={Computational Optimization and Applications},
  year={2018},
  volume={71},
  pages={509-523}
}
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… 

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References

SHOWING 1-10 OF 31 REFERENCES

Proximity maps for convex sets

The method of successive approximation is applied to the problem of obtaining points of minimum distance on two convex sets. Specifically, given a closed convex set K in Hilbert space, let P be the

Finding Best Approximation Pairs Relative to a Convex and Prox-Regular Set in a Hilbert Space

  • D. R. Luke
  • Mathematics, Computer Science
    SIAM J. Optim.
  • 2008
TLDR
This work studies the convergence of an iterative projection/reflection algorithm originally proposed for solving phase retrieval problems in optics, and investigates the asymptotic behavior of the RAAR algorithm for the general problem of finding points that achieve the minimum distance between two closed convex sets in a Hilbert space with empty intersection.

Computational acceleration of projection algorithms for the linear best approximation problem

Projection methods: an annotated bibliography of books and reviews

TLDR
This annotated bibliography includes books and review papers on, or related to, projection methods that the authors know about, use and like.

Algorithms and Convergence Results of Projection Methods for Inconsistent Feasibility Problems: A Review

TLDR
This paper brings under one roof and telegraphically review some recent works on inconsistent CFPs and investigates the behavior of algorithms that are designed to solve a consistent CFP when applied to inconsistent problems.

Best Approximation In Inner Product Spaces

I can still remember being surprised and delighted to discover, back in high school, that drawing the 'best' straight line to fit a set of data was not a matter of artistic skill, but a

Alternating Projection Methods

This book describes and analyzes all available alternating projection methods for solving the general problem of finding a point in the intersection of several given sets belonging to a Hilbert

Best approximation in inner product spaces

Inner Product Spaces.- Best Approximation.- Existence and Uniqueness of Best Approximations.- Characterization of Best Approximations.- The Metric Projection.- Bounded Linear Functionals and Best

Dykstra's Alternating Projection Algorithm for Two Sets

We analyze Dykstra?s algorithm for two arbitrary closed convex sets in a Hilbert space. Our technique also applies to von Neumann?s algorithm. Various convergence results follow. An example allows