Compromise Solutions for Robust Combinatorial Optimization with Variable-Sized Uncertainty

TL;DR

This paper addresses robust combinatorial optimization under variable-sized uncertainty sets, proposing an efficient method to find solutions that perform well across all possible uncertainty sizes, with theoretical and experimental validation.

Contribution

It introduces an approach to identify single solutions that are robust over all uncertainty set sizes, including an iterative solution procedure and complexity analysis.

Findings

Efficient solution method for min-max robust optimization.

Complexity results for min-max regret problems.

Experimental evaluation demonstrating approach effectiveness.

Abstract

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.