The pseudomonotone stochastic variational inequality problem: Analytical statements and stochastic extragradient schemes

Variational inequality problems find wide applicability in modeling a range of optimization and equilibrium problems. We consider the stochastic generalization of such a problem wherein the mapping is pseudomonotone and make two sets of contributions in this paper. First, we provide sufficiency conditions for the solvability of such problems that do not require evaluating the expectation. Second, we consider an extragradient variant of stochastic approximation for the solution of such problems and under suitable conditions, show that this scheme produces iterates that converge in an almost-sure sense.