Solving variational inequalities with stochastic mirror-prox algorithm

In this paper we consider iterative methods for stochastic variational (s. v. i. ) with monotone operators. Our basic assumption is that the possesses both smooth and nonsmooth components. Further, only noisy of the problem data are available. We develop a novel Stochastic-Prox (SMP) algorithm for solving s. v. i. and show that with the stepsize strategy it attains the optimal rates of convergence with to the problem parameters. We apply the SMP algorithm to Stochastic minimization and describe particular applications to Stochastic Feasability problem and Eigenvalue minimization.