  An efficient proximal-gradient-based method, called proximal extrapolated
gradient method, is designed for solving monotone variational inequality in
Hilbert space. The proposed method extends the acceptable range of parameters
to obtain larger step sizes. The step size is predicted based a local
information of the operator and corrected by linesearch procedures to satisfy a
very weak condition, which is even weaker than the boundedness of sequence
generated and always holds when the operator is the gradient of a convex
function. We establish its convergence and ergodic convergence rate in theory
under the larger range of parameters. Furthermore, we improve numerical
efficiency by employing the proposed method with non-monotonic step size, and
obtain the upper bound of the parameter relating to step size by an extremely
simple example. Related numerical experiments illustrate the improvements in
efficiency from the larger step size.
