  This paper is concerned with a lesser-studied problem in the context of
model-based, uncertainty quantification (UQ), that of
optimization/design/control under uncertainty. The solution of such problems is
hindered not only by the usual difficulties encountered in UQ tasks (e.g. the
high computational cost of each forward simulation, the large number of random
variables) but also by the need to solve a nonlinear optimization problem
involving large numbers of design variables and potentially constraints. We
propose a framework that is suitable for a large class of such problems and is
based on the idea of recasting them as probabilistic inference tasks. To that
end, we propose a Variational Bayesian (VB) formulation and an iterative
VB-Expectation-Maximization scheme that is also capable of identifying a
low-dimensional set of directions in the design space, along which, the
objective exhibits the largest sensitivity.
  We demonstrate the validity of the proposed approach in the context of two
numerical examples involving $\mathcal{O}(10^3)$ random and design variables.
In all cases considered the cost of the computations in terms of calls to the
forward model was of the order $\mathcal{O}(10^2)$. The accuracy of the
approximations provided is assessed by appropriate information-theoretic
metrics.
