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Zbl 0905.47044
Kiwiel, Krzysztof C.; Lopuch, Bozena
Surrogate projection methods for finding fixed points of firmly nonexpansive mappings.
(English)
[J] SIAM J. Optim. 7, No.4, 1084-1102 (1997). ISSN 1052-6234; ISSN 1095-7189/e

Summary: We present methods for finding common fixed points of finitely many firmly nonexpansive mappings on a Hilbert space. At every iteration, an approximation to each mapping generates a halfspace containing its set of fixed points. The next iterate is found by projecting the current iterate on a surrogate halfspace formed by taking a convex combination of the halfspace inequalities. This acceleration technique extends one for convex feasibility problems (CFPs), since projection operators onto closed convex sets are firmly nonexpansive. The resulting methods are block iterative and, hence, lend themselves to parallel implementation. We extend to accelerated methods some recent results of {\it H. H. Bauschke} and {\it J. M. Borwein} [SIAM Rev. 38, No. 3, 367-426 (1996; Zbl 0865.47039)] on the convergence of projection methods.
MSC 2000:
*47H09 Mappings defined by "shrinking" properties
47J25 Methods for solving nonlinear operator equations (general)
90C25 Convex programming
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
65J15 Equations with nonlinear operators (numerical methods)

Keywords: firmly nonexpansive mappings; successive projections; relaxation methods; convex feasibility problems; surrogate inequalities; accelerated methods; convergence of projection methods

Citations: Zbl 0865.47039

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