  Robust optimization has been established as a leading methodology to approach
decision problems under uncertainty. To derive a robust optimization model, a
central ingredient is to identify a suitable model for uncertainty, which is
called the uncertainty set. An ongoing challenge in the recent literature is to
derive uncertainty sets from given historical data that result in solutions
that are robust regarding future scenarios. In this paper we use an
unsupervised deep learning method to learn and extract hidden structures from
data, leading to non-convex uncertainty sets and better robust solutions. We
prove that most of the classical uncertainty classes are special cases of our
derived sets and that optimizing over them is strongly NP-hard. Nevertheless,
we show that the trained neural networks can be integrated into a robust
optimization model by formulating the adversarial problem as a convex quadratic
mixed-integer program. This allows us to derive robust solutions through an
iterative scenario generation process. In our computational experiments, we
compare this approach to a similar approach using kernel-based support vector
clustering. We find that uncertainty sets derived by the unsupervised deep
learning method find a better description of data and lead to robust solutions
that outperform the comparison method both with respect to objective value and
feasibility.
