Improving the speed of convergence in the method of projections onto convex sets

A serious drawback of the method of projections onto convex sets to find a point in the intersection of a finite number of closed convex sets in an Euclidean space, is its often very slow convergence. This bad behaviour, sometimes called the "tunneling effect", seems a.o. to be connected with the monotone behaviour of the usual algorithms. We present a new algorithm that can interrupt at different steps this monotone behaviour; this can substantially improve the speed of convergence.