Variable-Sized Uncertainty and Inverse Problems in Robust Optimization

TL;DR

This paper explores robust optimization with variable-sized uncertainty sets, proposing methods to construct minimal robust solution sets and analyzing inverse problems where nominal solutions lose robustness, supported by theoretical bounds and experimental data.

Contribution

It introduces a novel approach to handle variable-sized uncertainty in robust optimization and formulates inverse problems with mixed-integer linear programming.

Findings

Constructed minimal robust solution sets for variable uncertainty

Provided bounds on the size of these solution sets

Developed MILP formulations for inverse robust optimization

Abstract

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.