  In this paper, we propose two new solution schemes to solve the stochastic
strongly monotone variational inequality problems: the stochastic extra-point
solution scheme and the stochastic extra-momentum solution scheme. The first
one is a general scheme based on updating the iterative sequence and an
auxiliary extra-point sequence. In the case of deterministic VI model, this
approach includes several state-of-the-art first-order methods as its special
cases. The second scheme combines two momentum-based directions: the so-called
heavy-ball direction and the optimism direction, where only one projection per
iteration is required in its updating process. We show that, if the variance of
the stochastic oracle is appropriately controlled, then both schemes can be
made to achieve optimal iteration complexity of
$\mathcal{O}\left(\kappa\ln\left(\frac{1}{\epsilon}\right)\right)$ to reach an
$\epsilon$-solution for a strongly monotone VI problem with condition number
$\kappa$. We show that these methods can be readily incorporated in a
zeroth-order approach to solve stochastic minimax saddle-point problems, where
only noisy and biased samples of the objective can be obtained, with a total
sample complexity of
$\mathcal{O}\left(\frac{\kappa^2}{\epsilon}\ln\left(\frac{1}{\epsilon}\right)\right)$
