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Towards Robust Interpretable Surrogates for Optimization

Marc Goerigk, Michael Hartisch, Sebastian Merten

TL;DR

This work presents suitable models based on different variants to model uncertainty, and solution methods to create decision trees as surrogates for the optimization process that are more robust to perturbations and still inherently interpretable.

Abstract

An important factor in the practical implementation of optimization models is the acceptance by the intended users. This is influenced among other factors by the interpretability of the solution process. Decision rules that meet this requirement can be generated using the framework for inherently interpretable optimization models. In practice, there is often uncertainty about the parameters of an optimization problem. An established way to deal with this challenge is the concept of robust optimization. The goal of our work is to combine both concepts: to create decision trees as surrogates for the optimization process that are more robust to perturbations and still inherently interpretable. For this purpose we present suitable models based on different variants to model uncertainty, and solution methods. Furthermore, the applicability of heuristic methods to perform this task is evaluated. Both approaches are compared with the existing framework for inherently interpretable optimization models.

Towards Robust Interpretable Surrogates for Optimization

TL;DR

This work presents suitable models based on different variants to model uncertainty, and solution methods to create decision trees as surrogates for the optimization process that are more robust to perturbations and still inherently interpretable.

Abstract

An important factor in the practical implementation of optimization models is the acceptance by the intended users. This is influenced among other factors by the interpretability of the solution process. Decision rules that meet this requirement can be generated using the framework for inherently interpretable optimization models. In practice, there is often uncertainty about the parameters of an optimization problem. An established way to deal with this challenge is the concept of robust optimization. The goal of our work is to combine both concepts: to create decision trees as surrogates for the optimization process that are more robust to perturbations and still inherently interpretable. For this purpose we present suitable models based on different variants to model uncertainty, and solution methods. Furthermore, the applicability of heuristic methods to perform this task is evaluated. Both approaches are compared with the existing framework for inherently interpretable optimization models.

Paper Structure

This paper contains 23 sections, 4 theorems, 14 equations, 7 figures, 5 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Motivation
  3. Motivating Example
  4. Related Work
  5. Contribution and Structure of the Paper
  6. Contribution of this paper.
  7. Course of the work.
  8. Robust Interpretable Surrogates
  9. Problem Formulation and Uncertainty Sets
  10. Iterative Solution Approach
  11. Heuristics
  12. Baseline Heuristic H$_1$: Single Solution
  13. Heuristic H$_\textnormal{tree}$: Fix Tree, Optimize Solution
  14. Heuristic H$_\textnormal{sol}$: Fix Solutions, Optimize Tree
  15. Heuristic H$_\textnormal{alt}$: Alternate H$_\textnormal{tree}$ and H$_\textnormal{sol}$
  16. ...and 8 more sections

Key Result

Theorem 4.1

Let $\mathcal{T}$ be the set of univariate trees of depth $D\in\mathbb{N}$ as described in Section subs:optApproach:uncertainty. Let $M=\max_{i\in[n]}\left\lbrace \max_{j\in[N]} c_{j,i}-\min_{j\in[N]}c_{j,i}\right\rbrace$. For a global budgeted uncertainty set with $\Gamma^{\text{glob}}> D\cdot N \c

Figures (7)

  • Figure 1: Example graph
  • Figure 2: Decision tree optimal for the nominal case
  • Figure 3: Decision tree optimal for the robust case
  • Figure 4: Scheme of the alternating heuristic for a global budget (H$_\textnormal{alt}^\textnormal{glob}$).
  • Figure 5: Plots experiment 1.
  • ...and 2 more figures

Theorems & Definitions (7)

  • Theorem 4.1
  • proof
  • Corollary 4.1
  • Corollary 4.2
  • Definition 4.1
  • Theorem 4.2
  • proof