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The Method of Alternating Projections

  • Frank Deutsch
Chapter
Part of the CMS Books in Mathematics / Ouvrages de mathématiques de la SMC book series (CMSBM)

Abstract

In this chapter we will describe a theoretically powerful method for computing best approximations from a closed convex set K that is the intersection of a finite number of closed convex sets, K = ∩<Stack><Subscript>1</Subscript><Superscript>r</Superscript></Stack> K i . This method is an iterative algorithm that reduces the problem to finding best approximations from the individual sets K i . The efficacy of the method thus depends on whether the given set K can be represented as the intersection of a finite number of sets K i from which it is “easy” to compute best approximations. This will be the case, for example, when the K i are either half-spaces, hyperplanes, finite-dimensional subspaces, or certain cones. Several applications will be made to a variety of problems including solving linear equations, solving linear inequalities, computing the best isotone and best convex regression functions, and solving the general shape-preserving interpolation problem.

Keywords

Hilbert Space Product Space Closed Subspace Convex Banach Space General Banach Space 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag New York, Inc. 2001

Authors and Affiliations

  • Frank Deutsch
    • 1
  1. 1.Department of MathematicsPennsylvania State UniversityUniversity ParkUSA

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