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On a class of bilevel linear mixed-integer programs in adversarial settings

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Abstract

We consider a class of bilevel linear mixed-integer programs (BMIPs), where the follower’s optimization problem is a linear program. A typical assumption in the literature for BMIPs is that the follower responds to the leader optimally, i.e., the lower-level problem is solved to optimality for a given leader’s decision. However, this assumption may be violated in adversarial settings, where the follower may be willing to give up a portion of his/her optimal objective function value, and thus select a suboptimal solution, in order to inflict more damage to the leader. To handle such adversarial settings we consider a modeling approach referred to as \(\alpha \)-pessimistic BMIPs. The proposed method naturally encompasses as its special classes pessimistic BMIPs and max–min (or min–max) problems. Furthermore, we extend this new modeling approach by considering strong-weak bilevel programs, where the leader is not certain if the follower is collaborative or adversarial, and thus attempts to make a decision by taking into account both cases via a convex combination of the corresponding objective function values. We study basic properties of the proposed models and provide numerical examples with a class of the defender–attacker problems to illustrate the derived results. We also consider some related computational complexity issues, in particular, with respect to optimistic and pessimistic bilevel linear programs.

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Notes

  1. In the remainder of the paper we use “her” and “his” whenever we refer to the leader and the follower, respectively.

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Acknowledgements

This material is based upon work partially supported by the National Science Foundation [Grants CMMI-1400009 and CMMI-1634835], DoD DURIP Grant FA2386-12-1-3032, the Air Force Research Laboratory (AFRL) Mathematical Modeling and Optimization Institute and the Air Force Office of Scientific Research (AFOSR). The authors thank two anonymous referees and the Associate Editor for their constructive and helpful comments.

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Authors and Affiliations

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

    M. Hosein Zare & Oleg A. Prokopyev

  2. Edward P. Fitts Department of Industrial and Systems Engineering, North Carolina State University, Raleigh, NC, USA

    Osman Y. Özaltın

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  1. M. Hosein Zare
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  2. Osman Y. Özaltın
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  3. Oleg A. Prokopyev
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Correspondence to Oleg A. Prokopyev.

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Zare, M.H., Özaltın, O.Y. & Prokopyev, O.A. On a class of bilevel linear mixed-integer programs in adversarial settings. J Glob Optim 71, 91–113 (2018). https://doi.org/10.1007/s10898-017-0549-2

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  • DOI: https://doi.org/10.1007/s10898-017-0549-2

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