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Mixed integer bilevel optimization with a k-optimal follower: a hierarchy of bounds

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Abstract

We consider mixed integer bilevel linear optimization problems in which the decision variables of the lower-level (follower’s) problem are all binary. We propose a general modeling and solution framework motivated by the practical reality that in a Stackelberg game, the follower does not always solve their optimization problem to optimality. They may instead implement a locally optimal solution with respect to a given upper-level decision. Such scenarios may occur when the follower’s computational capabilities are limited, or when the follower is not completely rational. Our framework relaxes the typical assumption of perfect rationality that underlies the standard modeling framework by defining a hierarchy of increasingly stringent assumptions about the behavior of the follower. Namely, at level k of this hierarchy, it is assumed that the follower produces a k-optimal solution. Associated with this hierarchy is a hierarchy of upper and lower bounds that are in fact valid for the classical case in which complete rationality of the follower is assumed. Two mixed integer linear programming (MILP) formulations are derived for the resulting optimization problems. Extensive computational results are provided to demonstrate the effectiveness of the proposed MILP formulations and the quality of the bounds produced. The latter are shown to dominate the standard approach based on a single-level relaxation at a reasonable computational cost. Finally, we also explore a class of bilevel problems for which 2-optimal lower-level solutions imply global optimality, and hence we can solve these bilevel problems exactly using the developed MILP formulations.

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Data availability statement

The data used during this study are publicly available with doi 10.5281/zenodo.6972912 at https://github.com/xyshi25/BilevelBounds

Code availability statement

The code used during this study are publicly available with doi 10.5281/zenodo.6972912 at https://github.com/xyshi25/BilevelBounds

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Acknowledgements

The authors would like to thank the Review Team for their helpful and constructive comments.

Funding

This research is based upon work supported by the Office of Naval Research [Grant N00014-19-1-2330].

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Authors and Affiliations

  1. Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA, USA

    Xueyu Shi & Oleg A. Prokopyev

  2. Department of Industrial and System Engineering, Lehigh University, Bethlehem, PA, USA

    Ted K. Ralphs

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Correspondence to Ted K. Ralphs.

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Shi, X., Prokopyev, O.A. & Ralphs, T.K. Mixed integer bilevel optimization with a k-optimal follower: a hierarchy of bounds. Math. Prog. Comp. 15, 1–51 (2023). https://doi.org/10.1007/s12532-022-00227-z

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