# Mixed Uncertainty Sets for Robust Combinatorial Optimization

## TL;DR

This paper introduces a novel robust optimization approach using multiple weighted uncertainty sets, demonstrating improved out-of-sample performance over traditional single-set methods through theoretical analysis and real-world computational experiments.

## Contribution

It extends classic robust optimization by modeling multiple uncertainty sets with belief weights, providing a flexible framework and demonstrating its effectiveness with empirical results.

## Key findings

- Multiple uncertainty sets can be effectively combined with weights.
- The approach improves out-of-sample robustness in shortest path problems.
- Modeling complexity remains comparable to classic methods.

## Abstract

In robust optimization, the uncertainty set is used to model all possible outcomes of uncertain parameters. In the classic setting, one assumes that this set is provided by the decision maker based on the data available to her. Only recently it has been recognized that the process of building useful uncertainty sets is in itself a challenging task that requires mathematical support. In this paper, we propose an approach to go beyond the classic setting, by assuming multiple uncertainty sets to be prepared, each with a weight showing the degree of belief that the set is a "true" model of uncertainty. We consider theoretical aspects of this approach and show that it is as easy to model as the classic setting. In an extensive computational study using a shortest path problem based on real-world data, we auto-tune uncertainty sets to the available data, and show that with regard to out-sample performance, the combination of multiple sets can give better results than each set on its own.