Problem-Driven Scenario Reduction and Scenario Approximation for Robust Optimization

TL;DR

This paper introduces a new method for reducing the size of uncertainty sets in robust optimization by considering feasible solution structures, leading to more accurate and practical robust models.

Contribution

It presents a novel theoretical framework and models for scenario reduction that incorporate solution feasibility, improving over previous methods.

Findings

Achieves better uncertainty sets than previous methods.

Provides theoretical guarantees for approximation quality.

Demonstrates improved computational results in experiments.

Abstract

In robust optimization, we would like to find a solution that is immunized against all scenarios that are modeled in an uncertainty set. Which scenarios to include in such a set is therefore of central importance for the tractability of the robust model and practical usefulness of the resulting solution. We consider problems with a discrete uncertainty set affecting only the objective function. Our aim is reduce the size of the uncertainty set, while staying as true as possible to the original robust problem, measured by an approximation guarantee. Previous reduction approaches ignored the structure of the set of feasible solutions in this process. We show how to achieve better uncertainty sets by taking into account what solutions are possible, providing a theoretical framework and models to this end. In computational experiments, we note that our new framework achieves better uncertainty sets than previous methods or a simple K-means approach.