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Annals of Operations Research

, Volume 272, Issue 1–2, pp 99–117 | Cite as

A note on linearized reformulations for a class of bilevel linear integer problems

  • M. Hosein Zare
  • Juan S. Borrero
  • Bo Zeng
  • Oleg A. Prokopyev
Advances in Theoretical and Applied Combinatorial Optimization

Abstract

We consider reformulations of a class of bilevel linear integer programs as equivalent linear mixed-integer programs (linear MIPs). The most common technique to reformulate such programs as a single-level problem is to replace the lower-level linear optimization problem by Karush–Kuhn–Tucker (KKT) optimality conditions. Employing the strong duality (SD) property of linear programs is an alternative method to perform such transformations. In this note, we describe two SD-based reformulations where the key idea is to exploit the binary expansion of upper-level integer variables. We compare the performance of an off-the-shelf MIP solver with the SD-based reformulations against the KKT-based one and show that the SD-based approaches can lead to orders of magnitude reduction in computational times for certain classes of instances.

Keywords

Bilevel optimization Bilevel integer programming Mixed integer linear programming 

Notes

Acknowledgements

The authors thank two anonymous referees and the Associate Editor for their constructive and helpful comments. This material is based upon work partially supported by the National Science Foundation [Grant CMMI-1634835]. The research of Oleg A. Prokopyev was in part performed while visiting the National Research University Higher School of Economics (Nizhny Novgorod), and was supported by the Laboratory of Algorithms and Technologies for Network Analysis (LATNA) and RSF grant 14-41-00039.

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Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  • M. Hosein Zare
    • 1
  • Juan S. Borrero
    • 2
  • Bo Zeng
    • 1
  • Oleg A. Prokopyev
    • 1
    • 3
  1. 1.Department of Industrial EngineeringUniversity of PittsburghPittsburghUSA
  2. 2.School of Industrial Engineering and ManagementOklahoma State UniversityStillwaterUSA
  3. 3.Laboratory of Algorithms and Technologies for Networks Analysis, National Research University Higher School of EconomicsNizhny NovgorodRussia

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