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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Veröffentlicht in:Numerical functional analysis and optimization Jg. 15; H. 5-6; S. 537 - 565
Hauptverfasser: Deutsch, Frank, Hundal, Hein
Format: Journal Article
Sprache:Englisch
Veröffentlicht: Marcel Dekker, Inc 01.01.1994
Schlagworte:
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
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Beschreibung
Zusammenfassung: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