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Exact solution approaches for a class of bilevel fractional programs

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Abstract

We consider a class of bilevel programming problems (BPPs) where the leader’s decision variables are all binary, the follower’s decision variables are all continuous, and a fractional objective function appears in the follower’s problem. This class of problems generalizes standard bilevel linear mixed-integer programs with a linear program (LP) in the lower level. One motivating application example for this generalization arises in a network interdiction context, where the follower (i.e., the evader) instead of minimizing his/her shortest path, optimizes some fractional objective function, e.g., a cost-to-time ratio. By applying Charnes-Cooper transformation, we first reformulate the original BPP as an equivalent BPP with a fractional objective in the upper level, but an LP in the lower level. Using a combination of the LP strong-duality property and linearization techniques, we show how to address the resulting reformulation via a parametric approach that solves a sequence of linear mixed-integer programs. The latter can be handled by off-the-shelf solvers, which implies that our overall solution scheme is easy to implement. Finally, we perform a brief computational study to illustrate the performance of the proposed approaches.

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Notes

  1. The generated data sets are available from the corresponding author on reasonable request (as required by the journal data policy).

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Acknowledgements

This research is supported by the National Science Foundation [Grant CMMI-1634835], the Office of Naval Research [Grant N00014-19-1-2330] and the Central Research Development Fund (CRDF) at the University of Pittsburgh. The research of O.A. Prokopyev was also supported by the U.S. Air Force Summer Faculty Fellowship. The authors would like to thank the Associate Editor and two anonymous referees for their constructive and helpful comments.

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  1. Department of Industrial Engineering, University of Pittsburgh, Pittsburgh, PA, 15261, USA

    Jing Yang, Xueyu Shi & Oleg A. Prokopyev

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  1. Jing Yang
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  3. Oleg A. Prokopyev
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Correspondence to Oleg A. Prokopyev.

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Yang, J., Shi, X. & Prokopyev, O.A. Exact solution approaches for a class of bilevel fractional programs. Optim Lett 17, 191–210 (2023). https://doi.org/10.1007/s11590-022-01869-7

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