  The paper concerns with novel first-order methods for monotone variational
inequalities. They use a very simple linesearch procedure that takes into
account a local information of the operator. Also the methods do not require
Lipschitz-continuity of the operator and the linesearch procedure uses only
values of the operator. Moreover, when operator is affine our linesearch
becomes very simple, namely, it needs only vector-vector multiplication. For
all our methods we establish the ergodic convergence rate. Although the
proposed methods are very general, sometimes they may show much better
performance even for optimization problems. The reason for this is that they
often can use larger stepsizes without additional expensive computation.
