Two generalizations of Dykstra’s cyclic projections algorithm

Dykstra’s cyclic projections algorithm allows one to compute best approximations to any pointx in a Hilbert space from the intersectionC = ⋂ _l ^r C _i of a finite number of closed convex setsC _i , by reducing it to a sequence of best approximation problems from theindividual setsC _i . Here we present two generalizations of this algorithm. First we allow the number of setsC _i to beinfinite rather than finite; secondly, we allow arandom, rather than cyclic, ordering of the setsC _i . This author was supported by NSF Grant DMS-9303705.