  Convex feasibility problems require to find a point in the intersection of a
finite family of convex sets. We propose to solve such problems by performing
set-enlargements and applying a new kind of projection operators called valiant
projectors. A valiant projector onto a convex set implements a special
relaxation strategy, proposed by Goffin in 1971, that dictates the move toward
the projection according to the distance from the set. Contrary to past
realizations of this strategy, our valiant projection operator implements the
strategy in a continuous fashion. We study properties of valiant projectors and
prove convergence of our new valiant projections method. These results include
as a special case and extend the 1985 automatic relaxation method of Censor.
