  In this paper we propose new algorithms for solving a class of structured
monotone variational inequality (VI) problems over compact feasible sets. By
identifying the gradient components existing in the operator of VI, we show
that it is possible to skip computations of the gradients from time to time,
while still maintaining the optimal iteration complexity for solving these VI
problems. Specifically, for deterministic VI problems involving the sum of the
gradient of a smooth convex function $\nabla G$ and a monotone operator $H$, we
propose a new algorithm, called the mirror-prox sliding method, which is able
to compute an $\varepsilon$-approximate weak solution with at most
$O((L/\varepsilon)^{1/2})$ evaluations of $\nabla G$ and
$O((L/\varepsilon)^{1/2}+M/\varepsilon)$ evaluations of $H$, where $L$ and $M$
are Lipschitz constants of $\nabla G$ and $H$, respectively. Moreover, for the
case when the operator $H$ can only be accessed through its stochastic
estimators, we propose a stochastic mirror-prox sliding method that can compute
a stochastic $\varepsilon$-approximate weak solution with at most
$O((L/\varepsilon)^{1/2})$ evaluations of $\nabla G$ and
$O((L/\varepsilon)^{1/2}+M/\varepsilon + \sigma^2/\varepsilon^2)$ samples of
$H$, where $\sigma$ is the variance of the stochastic samples of $H$.
