  The monotone Variational Inequality (VI) is a general model with important
applications in various engineering and scientific domains. In numerous
instances, the VI problems are accompanied by function constraints that can be
data-driven, making the usual projection operator challenging to compute. This
paper presents novel first-order methods for the function-constrained
Variational Inequality (FCVI) problem in smooth or nonsmooth settings with
possibly stochastic operators and constraints. We introduce the AdOpEx method,
which employs an operator extrapolation on the KKT operator of the FCVI in a
smooth deterministic setting. Since this operator is not uniformly Lipschitz
continuous in the Lagrange multipliers, we employ an adaptive two-timescale
algorithm leading to bounded multipliers and achieving the optimal $O(1/T)$
convergence rate. For the nonsmooth and stochastic VIs, we introduce design
changes to the AdOpEx method and propose a novel P-OpEx method that takes
partial extrapolation. It converges at the rate of $O(1/\sqrt{T})$ when both
the operator and constraints are stochastic or nonsmooth. This method has
suboptimal dependence on the noise and Lipschitz constants of function
constraints. We propose a constraint extrapolation approach leading to the
OpConEx method that improves this dependence by an order of magnitude. All our
algorithms easily extend to saddle point problems with function constraints
that couple the primal and dual variables while maintaining the same complexity
results. To the best of our knowledge, all our complexity results are new in
the literature
