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In this work, we consider two-stage quadratic optimization problems underellipsoidal uncertainty. In the first stage, one needs to decide upon thevalues of a subset of optimization variables (control variables). In the secondstage, the uncertainty is revealed and the rest of the optimization variables(state variables) are set up as a solution to a known system of possiblynon-linear equations. This type of problem occurs, for instance, inoptimization for dynamical systems, such as electric power systems as well asgas and water networks. We propose a convergent iterative algorithm to build asequence of approximately robustly feasible solutions with an improvingobjective value. At each iteration, the algorithm optimizes over a subset ofthe feasible set and uses affine approximations of the second-stage equationswhile preserving the non-linearity of other constraints. We implement ourapproach and demonstrate its performance on Matpower instances. This paperfocuses on quadratic problems, but the approach is suitable for more generalsetups.

Two-Stage Robust Quadratic Optimization with Equalities and its Application to Optimal Power Flow