  In classic robust optimization, it is assumed that a set of possible
parameter realizations, the uncertainty set, is modeled in a previous step and
part of the input. As recent work has shown, finding the most suitable
uncertainty set is in itself already a difficult task. We consider robust
problems where the uncertainty set is not completely defined. Only the shape is
known, but not its size. Such a setting is known as variable-sized uncertainty.
  In this work we present an approach how to find a single robust solution,
that performs well on average over all possible uncertainty set sizes. We
demonstrate that this approach can be solved efficiently for min-max robust
optimization, but is more involved in the case of min-max regret, where
positive and negative complexity results for the selection problem, the minimum
spanning tree problem, and the shortest path problem are provided. We introduce
an iterative solution procedure, and evaluate its performance in an
experimental comparison.
