Variable-sized uncertainty and inverse problems in robust optimization

- We consider robust combinatorial optimization problems with parametrized uncertainty.
- For min-max robustness, we develop methods to find a set of robust solutions.
- This set contains an optimal robust solution for each possible uncertainty size.
- For min-max regret robustness we consider the inverse robust problem.
- The aim is to find the largest uncertainty such that a fixed solution stays optimal.

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.