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A Computational Study for Solving Decision-Dependent Robust Problems as Bilevel Optimization Problems

Henri Lefebvre, Martin Schmidt, Simon Stevens, Johannes Thürauf

TL;DR

This paper is the first one that provides an extensive computational study for solving strictly robust optimization problems with decision-dependent uncertainty sets as equivalent bilevel optimization problems and indicates that the bilevel approach performs better in terms of computation times.

Abstract

Both bilevel and robust optimization are established fields of mathematical optimization and operations research. However, only until recently, the similarities in their mathematical structure has neither been studied theoretically nor exploited computationally. Based on the recent results by Goerigk et al. (2025), this paper is the first one that provides an extensive computational study for solving strictly robust optimization problems with decision-dependent uncertainty sets as equivalent bilevel optimization problems. If the uncertainty set can be dualized, the respective bilevel techniques to obtain a single-level reformulation are very similar compared with the classic dualization techniques used in robust optimization but lead to larger single-level problems to be solved. Our numerical study shows that this usually leads to larger computation times. For the more challenging case of decision-dependent uncertainty sets represented by mixed-integer linear models, one cannot apply classic dualization techniques from robust optimization. Thus, we compare the presented bilevel approach with an established method from the literature, which is based on quantified mixed-integer linear programs. Our numerical results indicate that, for the problem class of decision-dependent robust optimization problems with mixed-integer linear uncertainty sets, the bilevel approach performs better in terms of computation times.

A Computational Study for Solving Decision-Dependent Robust Problems as Bilevel Optimization Problems

TL;DR

This paper is the first one that provides an extensive computational study for solving strictly robust optimization problems with decision-dependent uncertainty sets as equivalent bilevel optimization problems and indicates that the bilevel approach performs better in terms of computation times.

Abstract

Both bilevel and robust optimization are established fields of mathematical optimization and operations research. However, only until recently, the similarities in their mathematical structure has neither been studied theoretically nor exploited computationally. Based on the recent results by Goerigk et al. (2025), this paper is the first one that provides an extensive computational study for solving strictly robust optimization problems with decision-dependent uncertainty sets as equivalent bilevel optimization problems. If the uncertainty set can be dualized, the respective bilevel techniques to obtain a single-level reformulation are very similar compared with the classic dualization techniques used in robust optimization but lead to larger single-level problems to be solved. Our numerical study shows that this usually leads to larger computation times. For the more challenging case of decision-dependent uncertainty sets represented by mixed-integer linear models, one cannot apply classic dualization techniques from robust optimization. Thus, we compare the presented bilevel approach with an established method from the literature, which is based on quantified mixed-integer linear programs. Our numerical results indicate that, for the problem class of decision-dependent robust optimization problems with mixed-integer linear uncertainty sets, the bilevel approach performs better in terms of computation times.

Paper Structure

This paper contains 47 sections, 1 theorem, 78 equations, 13 figures.

Table of Contents

  1. Introduction
  2. Reformulations of a General DDRO Problem
  3. Classic Robust Reformulation
  4. Bilevel Reformulation
  5. Strong-Duality-Based Single-Level Reformulation
  6. KKT-Based Single-Level Reformulation
  7. Uncertainty Sets
  8. Decision-Dependent Budgeted Uncertainty
  9. Decision-Dependent Knapsack Uncertainty
  10. Applications
  11. Shortest-Path Problem
  12. Budgeted Uncertainty
  13. Knapsack Uncertainty
  14. Knapsack Problem
  15. Budgeted Uncertainty
  16. ...and 32 more sections

Key Result

Lemma 1

For every bilevel-feasible point $(x, r^u, u, r^l)$ of Problem eq:app-a:both-levels, the point $(x,u,r^l)$ is bilevel-feasible for Problem eq:app-a:lower-level with the same objective function value. Moreover, for every bilevel-feasible point $(x,u,r^l)$ of Problem eq:app-a:lower-level, the point $(

Figures (13)

  • Figure 1: Heatmap of the computation times depending on the knapsacks capacity for the knapsack problem with continuous knapsack uncertainty set.
  • Figure 2: ECDF of computation time for the shortest-path problem (left), the knapsack problem (middle), and the portfolio selection problem (right) with continuous budgeted uncertainty sets. The numbers in the legend indicate the number of instances solved within the time limit of 2 hours out of 100 instances. Solid blue: bilevel approach. Dotted orange: robust approach.
  • Figure 3: ECDF of BnB nodes for the shortest-path problem (left), the knapsack problem (middle), and the portfolio selection problem (right) with continuous budgeted uncertainty sets. The numbers in the legend indicate the number of instances solved within the time limit of 2 hours out of 100 instances. Solid blue: bilevel approach. Dotted orange: robust approach.
  • Figure 4: ECDF of computation time for the shortest-path problem (left), the knapsack problem (middle), and the portfolio selection problem (right) with continuous knapsack uncertainty sets. The numbers in the legend indicate the number of instances solved within the time limit of 2 hours out of 100 instances. Solid blue: bilevel approach. Dotted orange: robust approach.
  • Figure 5: ECDF of BnB nodes for the shortest-path problem (left), the knapsack problem (middle), and the portfolio selection problem (right) with continuous knapsack uncertainty sets. The numbers in the legend indicate the number of instances solved within the time limit of 2 hours out of 100 instances. Solid blue: bilevel approach. Dotted orange: robust approach.
  • ...and 8 more figures

Theorems & Definitions (8)

  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Lemma 1
  • proof
  • Remark 5
  • Remark 6