  In robust optimization, the general aim is to find a solution that performs
well over a set of possible parameter outcomes, the so-called uncertainty set.
In this paper, we assume that the uncertainty size is not fixed, and instead
aim at finding a set of robust solutions that covers all possible uncertainty
set outcomes. We refer to these problems as robust optimization with
variable-sized uncertainty. We discuss how to construct smallest possible sets
of min-max robust solutions and give bounds on their size.
  A special case of this perspective is to analyze for which uncertainty sets a
nominal solution ceases to be a robust solution, which amounts to an inverse
robust optimization problem. We consider this problem with a min-max regret
objective and present mixed-integer linear programming formulations that can be
applied to construct suitable uncertainty sets.
  Results on both variable-sized uncertainty and inverse problems are further
supported with experimental data.
