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.
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© 2001 Springer-Verlag New York, Inc.
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Deutsch, F. (2001). The Method of Alternating Projections. In: Best Approximation in Inner Product Spaces. CMS Books in Mathematics / Ouvrages de mathématiques de la SMC. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-9298-9_9
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DOI: https://doi.org/10.1007/978-1-4684-9298-9_9
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4419-2890-0
Online ISBN: 978-1-4684-9298-9
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