**Keywords:**stochastic approximation | randomized algorithms | stochastic variational inequalities | incremental methods | extragradient method | variance-reduction | weak-sharpness | Tykhonov regularization

Stochastic approximation methods are well established for optimization problems.

The appeal of these methods is due largely to their ability to cope efficiently and robustly

with inexact information about the underlying optimization problem. This

thesis proposes stochastic approximation methods for the solution of stochastic

variational inequalities, paying attention to asymptotic convergence (stability),

convergence rate, oracle complexity, knowledge of problem parameters, data availability

and distributed solution. In chapter 3, we propose a method that combines

stochastic approximation with incremental constraint projections, meaning that, at

each iteration, the random operator is sampled and a component of the intersection

defining the feasible set is chosen at random. Our method allows the distributed

solution of Cartesian stochastic variational inequalities with partial coordination

between users of a network. Such sequential scheme is well suited for applications

involving large data sets, online optimization and distributed learning. We analyse

this method for the class of weak-sharp monotone operators (without regularization)

and for the class of plain monotone operators with regularization. In chapter

4, we propose a stochastic extragradient method for pseudo-monotone operators

with a novel iterative variance reduction procedure. We present convergence and

complexity analysis relaxing previous assumptions used for stochastic approximation

and accelerating the convergence rate while maintaining a near-optimal oracle

complexity. Our extragradient method is also suitable for the distributed case. In

chapter 5, we propose two stochastic extragradient methods with linear search

with the same set of assumptions as in chapter 4, except that we do not require

the knowledge of the Lipschitz constant or Lipschitz continuity.