SURROGATE PROJECTION METHODS FOR FINDING FIXED-POINTS OF FIRMLY NONEXPANSIVE-MAPPINGS

We present methods for finding common fixed points of finitely many fi rmly 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 on e for convex feasibility problems (CFPs),since projection operators on to closed convex sets are firmly nonexpansive. The resulting methods a re block iterative and, hence, lend themselves to parallel implementat ion. We extend to accelerated methods some recent results of Bauschke and Borwein [SIAM Rev., 38 (1996), pp. 367-426] on the convergence of projection methods.