  In this paper we study variational inequalities (VI) defined by the
conditional value-at-risk (CVaR) of uncertain functions. We introduce
stochastic approximation schemes that employ an empirical estimate of the CVaR
at each iteration to solve these VIs. We investigate convergence of these
algorithms under various assumptions on the monotonicity of the VI and accuracy
of the CVaR estimate. Our first algorithm is shown to converge to the exact
solution of the VI when the estimation error of the CVaR becomes progressively
smaller along any execution of the algorithm. When the estimation error is
nonvanishing, we provide two algorithms that provably converge to a
neighborhood of the solution of the VI. For these schemes, under strong
monotonicity, we provide an explicit relationship between sample size,
estimation error, and the size of the neighborhood to which convergence is
achieved. A simulation example illustrates our theoretical findings.
