Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Problem-Driven Scenario Reduction and Scenario Approximation for Robust Optimization

Jamie Fairbrother, Marc Goerigk, Mohammad Khosravi

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.

Problem-Driven Scenario Reduction and Scenario Approximation for Robust Optimization

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.

Paper Structure

This paper contains 26 sections, 18 theorems, 44 equations, 16 figures, 1 table.

Table of Contents

  1. Introduction
  2. Scenario Reduction
  3. Setting
  4. Problem Definitions and Hardness
  5. Sufficiency Criteria
  6. Alternative Characterization of Sufficiency Criteria
  7. Finding Reduced Scenario Sets
  8. Example Problems
  9. The Selection Problem
  10. Relaxation for Other Combinatorial Problems
  11. Shortest Path Problem
  12. Scenario Approximation
  13. Definitions and Properties
  14. Optimization Models for Scenario Approximation
  15. Computational Experiments
  16. ...and 11 more sections

Key Result

Theorem 2.2.2

Let $X$ be a polyhedron given by an outer description, and let $\mathcal{U}=\{\pmb{c}^1,\ldots,\pmb{c}^N\}$ be a discrete set. Then the problem is NP-hard, if $k$ is part of the input.

Figures (16)

  • Figure 1: Example for dominated scenarios.
  • Figure 2: Illustration of different minimal sufficient sets for a two-dimension uncertainty set.
  • Figure 3: Reduction using for selection problem for dimensions $n=6,7,8,9$
  • Figure 4: Reduction using for selection problem for dimensions $n=15$, $25$ and $40$
  • Figure 5: Selection - performance of small instances
  • ...and 11 more figures

Theorems & Definitions (38)

  • Definition 2.2.1
  • Theorem 2.2.2: goerigk2022data
  • Theorem 2.2.3
  • proof
  • Definition 2.3.1
  • Theorem 2.3.2
  • proof
  • Theorem 2.3.3
  • proof
  • Corollary 2.3.4
  • ...and 28 more