  We present a combination technique based on mixed differences of both spatial
approximations and quadrature formulae for the stochastic variables to solve
efficiently a class of Optimal Control Problems (OCPs) constrained by random
partial differential equations. The method requires to solve the OCP for
several low-fidelity spatial grids and quadrature formulae for the objective
functional. All the computed solutions are then linearly combined to get a
final approximation which, under suitable regularity assumptions, preserves the
same accuracy of fine tensor product approximations, while drastically reducing
the computational cost. The combination technique involves only tensor product
quadrature formulae, thus the discretized OCPs preserve the (possible)
convexity of the continuous OCP. Hence, the combination technique avoids the
inconveniences of Multilevel Monte Carlo and/or sparse grids approaches, but
remains suitable for high dimensional problems. The manuscript presents an
a-priori procedure to choose the most important mixed differences and an
asymptotic complexity analysis, which states that the asymptotic complexity is
exclusively determined by the spatial solver. Numerical experiments validate
the results.
