  This paper is concerned with some new projection methods for solving
variational inequality problems with monotone and Lipschitz-continuous mapping
in Hilbert space. First, we propose the projected reflected gradient algorithm
with a constant stepsize. It is similar to the projected gradient method,
namely, the method requires only one projection onto the feasible set and only
one value of the mapping per iteration. This distinguishes our method from most
other projection-type methods for variational inequalities with monotone
mapping. Also we prove that it has R-linear rate of convergence under the
strong monotonicity assumption. The usual drawback of algorithms with constant
stepsize is the requirement to know the Lipschitz constant of the mapping. To
avoid this, we modify our first algorithm so that the algorithm needs at most
two projections per iteration. In fact, our computational experience shows that
such cases with two projections are very rare. This scheme, at least
theoretically, seems to be very effective. All methods are shown to be globally
convergent to a solution of the variational inequality. Preliminary results
from numerical experiments are quite promising.
