  We investigate an optimization problem governed by an elliptic partial
differential equation with uncertain parameters. We introduce a robust
optimization framework that accounts for uncertain model parameters. The
resulting non-linear optimization problem has a bi-level structure due to the
min-max formulation. To approximate the worst-case in the optimization problem
we propose linear and quadratic approximations. However, this approach still
turns out to be very expensive, therefore we propose an adaptive model order
reduction technique which avoids long offline stages and provides a certified
reduced order surrogate model for the parametrized PDE which is then utilized
in the numerical optimization. Numerical results are presented to validate the
presented approach.
