  An algorithm is proposed to solve robust control problems constrained by
partial differential equations with uncertain coefficients, based on the
so-called MG/OPT framework. The levels in this MG/OPT hierarchy correspond to
discretization levels of the PDE, as usual. For stochastic problems, the
relevant quantities (such as the gradient) contain expected value operators on
each of these levels. They are estimated using a multilevel Monte Carlo method,
the specifics of which depend on the MG/OPT level. Each of the optimization
levels then contains multiple underlying multilevel Monte Carlo levels. The
MG/OPT hierarchy allows the algorithm to exploit the structure inherent in the
PDE, speeding up the convergence to the optimum. In contrast, the multilevel
Monte Carlo hierarchy exists to exploit structure present in the stochastic
dimensions of the problem. A statement about the rate of convergence of the
algorithm is proven, and some additional properties are discussed. The
performance of the algorithm is numerically investigated for three test cases.
A reduction in the number of samples required on expensive levels and therefore
in computational time can be observed.
