The Method of Alternating Projections
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.
KeywordsHilbert Space Product Space Closed Subspace Convex Banach Space General Banach Space
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