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Neur2BiLO: Neural Bilevel Optimization

Justin Dumouchelle, Esther Julien, Jannis Kurtz, Elias B. Khalil

TL;DR

The proposed framework, Neur2BiLO, embeds a neural network approximation of the leader's or follower's value function, trained via supervised regression, into an easy-to-solve mixed-integer program that produces high-quality solutions extremely fast for four applications with linear and non-linear objectives and pure and mixed-integer variables.

Abstract

Bilevel optimization deals with nested problems in which a leader takes the first decision to minimize their objective function while accounting for a follower's best-response reaction. Constrained bilevel problems with integer variables are particularly notorious for their hardness. While exact solvers have been proposed for mixed-integer linear bilevel optimization, they tend to scale poorly with problem size and are hard to generalize to the non-linear case. On the other hand, problem-specific algorithms (exact and heuristic) are limited in scope. Under a data-driven setting in which similar instances of a bilevel problem are solved routinely, our proposed framework, Neur2BiLO, embeds a neural network approximation of the leader's or follower's value function, trained via supervised regression, into an easy-to-solve mixed-integer program. Neur2BiLO serves as a heuristic that produces high-quality solutions extremely fast for four applications with linear and non-linear objectives and pure and mixed-integer variables.

Neur2BiLO: Neural Bilevel Optimization

TL;DR

The proposed framework, Neur2BiLO, embeds a neural network approximation of the leader's or follower's value function, trained via supervised regression, into an easy-to-solve mixed-integer program that produces high-quality solutions extremely fast for four applications with linear and non-linear objectives and pure and mixed-integer variables.

Abstract

Bilevel optimization deals with nested problems in which a leader takes the first decision to minimize their objective function while accounting for a follower's best-response reaction. Constrained bilevel problems with integer variables are particularly notorious for their hardness. While exact solvers have been proposed for mixed-integer linear bilevel optimization, they tend to scale poorly with problem size and are hard to generalize to the non-linear case. On the other hand, problem-specific algorithms (exact and heuristic) are limited in scope. Under a data-driven setting in which similar instances of a bilevel problem are solved routinely, our proposed framework, Neur2BiLO, embeds a neural network approximation of the leader's or follower's value function, trained via supervised regression, into an easy-to-solve mixed-integer program. Neur2BiLO serves as a heuristic that produces high-quality solutions extremely fast for four applications with linear and non-linear objectives and pure and mixed-integer variables.
Paper Structure (54 sections, 3 theorems, 27 equations, 6 figures, 13 tables, 2 algorithms)

This paper contains 54 sections, 3 theorems, 27 equations, 6 figures, 13 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. A motivating application.
  3. Scope of this work.
  4. Background
  5. Assumptions.
  6. Value function reformulation.
  7. Methodology
  8. Neur2BiLO
  9. Upper-level approximation.
  10. Lower-level approximation.
  11. Bilevel feasibility.
  12. Upper- v.s. lower-level level approximation.
  13. Limitations.
  14. Model architecture
  15. Approximation guarantees
  16. ...and 39 more sections

Key Result

Theorem 3.1

If the leader and the follower have the same objective function and $\lambda>1$, Neur2BiLO returns a feasible solution $(\mathbf{x}^\star,\mathbf{y}^\star)$ for Problem eq:bilevel with objective value where opt is the optimal value of eq:bilevel and $\lambda$ the penalty term in eq:objective_with_slack .

Figures (6)

  • Figure 1: Box plot of relative errors for KIP with interdiction budget of $k=n/4$.
  • Figure 2: Box plot of relative errors for KIP with interdiction budget of $k=n/2$.
  • Figure 3: Box plot of relative errors for KIP with interdiction budget of $k=3n/4$.
  • Figure 4: Box plot of relative errors for CNP. B&C does not find any upper-level solutions for 2 of the 300 instances of size $|V| = 500$, so these are excluded from the plot.
  • Figure 5: Box plot of relative errors for DNDP with 10 edges. MKKT-{5,10,30} corresponds to MKKT run with each respective time limit.
  • ...and 1 more figures

Theorems & Definitions (6)

  • Theorem 3.1
  • Example C.1
  • Lemma D.2
  • proof
  • Theorem 1: \ref{['thm:approx_guarantee']}
  • proof