The rate of convergence of dykstra's cyclic projections algorithm: The polyhedral case

Suppose K is the intersection of a finite number of closed half-spaces in a Hilbert space X. Starting with any point xεX, it is shown that the sequence of iterates {x n } generated by Dykstra's cyclic projections algorithm satisfies the inequality for all n, where P K (x) is the nearest point i...

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Bibliographic Details
Published in:Numerical functional analysis and optimization Vol. 15; no. 5-6; pp. 537 - 565
Main Authors: Deutsch, Frank, Hundal, Hein
Format: Journal Article
Language:English
Published: Marcel Dekker, Inc 01.01.1994
Subjects:
alternating projections
best approximations from polyhedra
convex regression
cyclic projections
Hildreth's algorithm
isotone regression
iterative projections
linear inequalities
linear programming
Primary 41A65
quadratic programming
rate of convergence of Dykstra's algorithm
Secondary 47N10
Secondary 49M30
subject classification
ISSN:0163-0563, 1532-2467
Online Access:Get full text
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Summary:Suppose K is the intersection of a finite number of closed half-spaces in a Hilbert space X. Starting with any point xεX, it is shown that the sequence of iterates {x n } generated by Dykstra's cyclic projections algorithm satisfies the inequality for all n, where P K (x) is the nearest point in K to x;, ρ is a constant, and 0 ≤c<1. In the case when K is the intersection of just two closed half-spaces, a stronger result is established: the sequence of iterates is either finite or satisfies for all n, where c is the cosine of the angle between the two functionals which define the half-spaces. Moreover, the constant c is the best possible. Applications are made to isotone and convex regression, and linear and quadratic programming.
ISSN:0163-0563
1532-2467
DOI:10.1080/01630569408816580