  We propose new Markov Chain Monte Carlo algorithms to sample probability
distributions on submanifolds, which generalize previous methods by allowing
the use of set-valued maps in the proposal step of the MCMC algorithms. The
motivation for this generalization is that the numerical solvers used to
project proposed moves to the submanifold of interest may find several
solutions. We show that the new algorithms indeed sample the target probability
measure correctly, thanks to some carefully enforced reversibility property. We
demonstrate the interest of the new MCMC algorithms on illustrative numerical
examples.
