Two generalizations of Dykstra's cyclic projections algorithm

Dykstra's cyclic projections algorithm allows one to compute best approximations to any point x in a Hilbert space from the intersection C = ∩^r1 Ci of a finite number of closed convex sets Ci, by reducing it to a sequence of best approximation problems from the individual sets Ci. Here we present two generalizations of this algorithm. First we allow the number of sets Ci to be infinite rather than finite; secondly, we allow a random, rather than cyclic, ordering of the sets Ci.