Optimal Scenario Reduction for One- and Two-Stage Robust Optimization

TL;DR

This paper introduces scenario reduction methods for robust optimization that preserve solution quality, enabling more efficient solving of large-scale problems with minimal loss in objective value.

Contribution

It formulates scenario reduction as a problem dependent only on the uncertainty set, providing guarantees on solution performance and outperforming standard clustering methods.

Findings

Reduced uncertainty sets maintain close objective values to original sets.

Proposed methods outperform K-means clustering in preserving solution quality.

Experimental results demonstrate improved robustness and computational efficiency.

Abstract

Robust optimization typically follows a worst-case perspective, where a single scenario may determine the objective value of a given solution. Accordingly, it is a challenging task to reduce the size of an uncertainty set without changing the resulting objective value too much. On the other hand, robust optimization problems with many scenarios tend to be hard to solve, in particular for two-stage problems. Hence, a reduced uncertainty set may be central to find solutions in reasonable time. We propose scenario reduction methods that give guarantees on the performance of the resulting robust solution. Scenario reduction problems for one- and two-stage robust optimization are framed as optimization problems that only depend on the uncertainty set and not on the underlying decision making problem. Experimental results indicate that objective values for the reduced uncertainty sets are closely correlated to original objective values, resulting in better solutions than when using general-purpose clustering methods such as K-means.