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Surrogate projection methods for finding fixed points of firmly nonexpansive mappings. (English) Zbl 0905.47044
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 H. H. Bauschke and J. M. Borwein [SIAM Rev. 38, No. 3, 367-426 (1996; Zbl 0865.47039)] on the convergence of projection methods.
MSC:
47H09Mappings defined by “shrinking” properties
47J25Iterative procedures (nonlinear operator equations)
90C25Convex programming
47H10Fixed point theorems for nonlinear operators on topological linear spaces
65J15Equations with nonlinear operators (numerical methods)