  In this work, we consider two-stage quadratic optimization problems under
ellipsoidal uncertainty. In the first stage, one needs to decide upon the
values of a subset of optimization variables (control variables). In the second
stage, the uncertainty is revealed and the rest of the optimization variables
(state variables) are set up as a solution to a known system of possibly
non-linear equations. This type of problem occurs, for instance, in
optimization for dynamical systems, such as electric power systems as well as
gas and water networks. We propose a convergent iterative algorithm to build a
sequence of approximately robustly feasible solutions with an improving
objective value. At each iteration, the algorithm optimizes over a subset of
the feasible set and uses affine approximations of the second-stage equations
while preserving the non-linearity of other constraints. We implement our
approach and demonstrate its performance on Matpower instances. This paper
focuses on quadratic problems, but the approach is suitable for more general
setups.
