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Robust Bilevel Optimization for Near-Optimal Lower-Level Solutions

Mathieu Besançon, Miguel F. Anjos, Luce Brotcorne

TL;DR

The paper introduces near-optimal robustness (NRB) for bilevel optimization, defining a near-optimal set $\,\mathcal{Z}(x;\delta)$ to protect the upper level from lower-level deviations and unifying optimistic, pessimistic, and bounded-rationality models. For convex lower levels, NRB admits a closed-form single-level reformulation via dual adversarial certificates; in the linear case, an extended MILP formulation with disjunctive constraints over dual-vertex solutions is developed. Efficient solution strategies are proposed, including lazy subproblem expansion and a single-vertex heuristic, complemented by valid inequalities to tighten relaxations. Computational experiments show that the extended formulation outperforms naive bilinear approaches, that lazy and heuristic methods deliver near-optimal robust solutions quickly, and that NRB effectively guards upper-level feasibility under bounded follower suboptimality with favorable scalability and practical impact.

Abstract

Bilevel optimization problems embed the optimality of a subproblem as a constraint of another optimization problem. We introduce the concept of near-optimality robustness for bilevel optimization, protecting the upper-level solution feasibility from limited deviations from the optimal solution at the lower level. General properties and necessary conditions for the existence of solutions are derived for near-optimal robust versions of general bilevel optimization problems. A duality-based solution method is defined when the lower level is convex, leveraging the methodology from the robust and bilevel literature. Numerical results assess the efficiency of exact and heuristic methods and the impact of valid inequalities on the solution time.

Robust Bilevel Optimization for Near-Optimal Lower-Level Solutions

TL;DR

The paper introduces near-optimal robustness (NRB) for bilevel optimization, defining a near-optimal set to protect the upper level from lower-level deviations and unifying optimistic, pessimistic, and bounded-rationality models. For convex lower levels, NRB admits a closed-form single-level reformulation via dual adversarial certificates; in the linear case, an extended MILP formulation with disjunctive constraints over dual-vertex solutions is developed. Efficient solution strategies are proposed, including lazy subproblem expansion and a single-vertex heuristic, complemented by valid inequalities to tighten relaxations. Computational experiments show that the extended formulation outperforms naive bilinear approaches, that lazy and heuristic methods deliver near-optimal robust solutions quickly, and that NRB effectively guards upper-level feasibility under bounded follower suboptimality with favorable scalability and practical impact.

Abstract

Bilevel optimization problems embed the optimality of a subproblem as a constraint of another optimization problem. We introduce the concept of near-optimality robustness for bilevel optimization, protecting the upper-level solution feasibility from limited deviations from the optimal solution at the lower level. General properties and necessary conditions for the existence of solutions are derived for near-optimal robust versions of general bilevel optimization problems. A duality-based solution method is defined when the lower level is convex, leveraging the methodology from the robust and bilevel literature. Numerical results assess the efficiency of exact and heuristic methods and the impact of valid inequalities on the solution time.

Paper Structure

This paper contains 18 sections, 7 theorems, 68 equations, 6 figures, 2 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Near-optimal set and near-optimal robust bilevel problem
  3. Near-optimal robust bilevel problems with a convex lower level
  4. Linear near-optimal robust bilevel problem
  5. Extended formulation
  6. Bounded example
  7. Valid inequalities
  8. Solution algorithm
  9. Lazy subproblem expansion
  10. Single vertex heuristic
  11. Computational experiments
  12. Instance sets
  13. Computational setup
  14. Bilinear and extended formulation
  15. Robustness of optimistic solutions and influence of $\delta$
  16. ...and 3 more sections

Key Result

proposition \@thmcounterproposition

For a given pair $(x,\delta)$, any of the following properties is sufficient for $\mathcal{Z}(x;\delta)$ to be a bounded set:

Figures (6)

  • Figure 1: Linear bilevel problem
  • Figure 2: Linear bilevel problem with a near-optimality robustness constraint
  • Figure 3: Representation of the bilevel problem.
  • Figure 4: Near-optimal robustness constraints.
  • Figure 5: Comparing the different solution methods on the two instance sets.
  • ...and 1 more figures

Theorems & Definitions (17)

  • proposition \@thmcounterproposition
  • proof
  • definition \@thmcounterdefinition: Radius of near-optimal feasibility
  • proposition \@thmcounterproposition
  • proof
  • proposition \@thmcounterproposition
  • proof
  • proposition \@thmcounterproposition
  • proof
  • corollary \@thmcountercorollary
  • ...and 7 more