  We consider a risk-averse optimal control problem governed by an elliptic
variational inequality (VI) subject to random inputs. By deriving KKT-type
optimality conditions for a penalised and smoothed problem and studying
convergence of the stationary points with respect to the penalisation
parameter, we obtain two forms of stationarity conditions. The lack of
regularity with respect to the uncertain parameters and complexities induced by
the presence of the risk measure give rise to new challenges unique to the
stochastic setting. We also propose a path-following stochastic approximation
algorithm using variance reduction techniques and demonstrate the algorithm on
a modified benchmark problem.
