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Valid Inequalities for Mixed Integer Bilevel Linear Optimization Problems

Sahar Tahernejad, Ted K. Ralphs

TL;DR

The paper advances the solution of mixed integer bilevel linear optimization problems by extending valid inequalities and cutting-plane methods from MILPs to the bilevel setting, and by detailing the open-source solver MibS 1.2. It presents a cohesive theory of convexification, separating solutions, and constructing diverse cuts (disjunctive, Chvátal, intersection, and Benders) tailored to MIBLPs, including specialized cuts for interdiction and hypercube-like regions. Empirically, it demonstrates that integrating stronger cuts, especially improving direction and interdiction-based cuts, substantially improves root bounds and reduces search trees, though cut efficacy depends heavily on problem structure and branching strategy. The work clarifies the interplay between cut generation, oracle calls, and branching, and provides a practical, tunable framework with extensive experimental results across multiple benchmark data sets, highlighting both achievements and remaining theoretical and computational challenges. Overall, the study offers a structured, extensible approach to leveraging MILP-cutting theory for MIBLPs and suggests avenues for future improvements in theory, algorithms, and solver design. The results indicate significant potential for improved performance in MIBLP solvers through dynamic, structure-aware cut generation and integrated branching strategies. Future work is poised to deepen the theoretical foundations and expand the arsenal of strong, scalable cuts for diverse bilevel applications.

Abstract

Despite the success of branch-and-cut methods for solving mixed integer bilevel linear optimization problems (MIBLPs) in practice, there are still gaps in both the theory and practice surrounding these methods. In the first part of this paper, we lay out a basic theory of valid inequalities and cutting-plane methods for MIBLPs that parallels the existing theory for mixed integer linear optimization problems (MILPs). We provide a general scheme for classifying valid inequalities and illustrate how the known classes of valid inequalities fit into this categorization, as well as generalizing several existing classes. In the second part of the paper, we assess the computational effectiveness of these valid inequalities and discuss the myriad challenges that arise in integrating methods of dynamically generating inequalities valid for MIBLPs into a branch-and-cut algorithms originally designed for solving MILPs. Although branch-and-cut methods for solving for MIBLPs are in principle straightforward generalizations of those used for MILP, there are subtle but important differences and there remain many unanswered questions regarding how to suitably modify control mechanisms and other algorithmic details in order to ensure performance in the MIBLP setting. We demonstrate that performance of version 1.2 of the open-source solver MibS was substantially improved over that of version 1.1 through a variety of improvements to the previous implementation.

Valid Inequalities for Mixed Integer Bilevel Linear Optimization Problems

TL;DR

The paper advances the solution of mixed integer bilevel linear optimization problems by extending valid inequalities and cutting-plane methods from MILPs to the bilevel setting, and by detailing the open-source solver MibS 1.2. It presents a cohesive theory of convexification, separating solutions, and constructing diverse cuts (disjunctive, Chvátal, intersection, and Benders) tailored to MIBLPs, including specialized cuts for interdiction and hypercube-like regions. Empirically, it demonstrates that integrating stronger cuts, especially improving direction and interdiction-based cuts, substantially improves root bounds and reduces search trees, though cut efficacy depends heavily on problem structure and branching strategy. The work clarifies the interplay between cut generation, oracle calls, and branching, and provides a practical, tunable framework with extensive experimental results across multiple benchmark data sets, highlighting both achievements and remaining theoretical and computational challenges. Overall, the study offers a structured, extensible approach to leveraging MILP-cutting theory for MIBLPs and suggests avenues for future improvements in theory, algorithms, and solver design. The results indicate significant potential for improved performance in MIBLP solvers through dynamic, structure-aware cut generation and integrated branching strategies. Future work is poised to deepen the theoretical foundations and expand the arsenal of strong, scalable cuts for diverse bilevel applications.

Abstract

Despite the success of branch-and-cut methods for solving mixed integer bilevel linear optimization problems (MIBLPs) in practice, there are still gaps in both the theory and practice surrounding these methods. In the first part of this paper, we lay out a basic theory of valid inequalities and cutting-plane methods for MIBLPs that parallels the existing theory for mixed integer linear optimization problems (MILPs). We provide a general scheme for classifying valid inequalities and illustrate how the known classes of valid inequalities fit into this categorization, as well as generalizing several existing classes. In the second part of the paper, we assess the computational effectiveness of these valid inequalities and discuss the myriad challenges that arise in integrating methods of dynamically generating inequalities valid for MIBLPs into a branch-and-cut algorithms originally designed for solving MILPs. Although branch-and-cut methods for solving for MIBLPs are in principle straightforward generalizations of those used for MILP, there are subtle but important differences and there remain many unanswered questions regarding how to suitably modify control mechanisms and other algorithmic details in order to ensure performance in the MIBLP setting. We demonstrate that performance of version 1.2 of the open-source solver MibS was substantially improved over that of version 1.1 through a variety of improvements to the previous implementation.

Paper Structure

This paper contains 77 sections, 13 theorems, 75 equations, 18 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Mixed Integer Bilevel Optimization
  3. Formulation
  4. Bilevel Feasible Region and Certificates of Infeasibility
  5. Assumptions
  6. Theory of Valid Inequalities for MIBLPs
  7. Convexification
  8. Cutting-plane Method
  9. Improving Valid Inequalities
  10. Methods for Construction
  11. Disjunctive Procedure
  12. Generalized Chvátal Procedure
  13. Intersection Inequalities
  14. Benders Cuts
  15. Valid Inequalities for MIBLPs
  16. ...and 62 more sections

Key Result

Proposition 1

For the vectors $x^1$ and $x^2\in\mathbb{R}_+^{n_1}$ with $x_{L}^1=x_{L}^2\in\mathbb{Z}^{L}$, we have where $x_{L}^1$ and $x_{L}^2$ represent the subvector of $x^1$ and $x^2$, respectively, corresponding to the linking variables.

Figures (18)

  • Figure 1: The feasible region and optimal solution of the the example from MooBarMixed90.
  • Figure 2: The polyhedral reformulation of the example from MooBarMixed90.
  • Figure 3: Types of infeasible points in the example from MooBarMixed90.
  • Figure 4: The generated integer no-good cut for the example shown in Figure \ref{['fig:mooreexample']}
  • Figure 5: The generated ISIC for the example shown in Figure \ref{['fig:mooreexample']}
  • ...and 13 more figures

Theorems & Definitions (33)

  • Definition 1: Certificate of Infeasibility
  • Proposition 1
  • Corollary 1
  • Proposition 2
  • proof
  • Theorem 1
  • proof
  • Theorem 2
  • proof
  • Definition 2
  • ...and 23 more