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On exact solution approaches for bilevel quadratic 0–1 knapsack problem

  • S.I.: CoDIT2017-Combinatorial Optimization
  • Published:
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Abstract

We consider the bilevel quadratic knapsack problem (BQKP) that model settings where a leader appropriates a budget for a follower, who solves a quadratic 0–1 knapsack problem. BQKP generalizes the bilevel knapsack problem introduced by Dempe and Richter (Cent Eur J Oper Res 8(2):93–107, 2000), where the follower solves a linear 0–1 knapsack problem. We present an exact-solution approach for BQKP based on extensions of dynamic programs for QKP bounds and the branch-and-backtrack algorithm. We compare our approach against a two-phase method computed using an optimization solver in both phases: it first computes the follower’s value function for all feasible leader’s decisions, and then solves a single-level, value-function reformulation of BQKP as a mixed-integer quadratically constrained program. Our computational experiments show that our approach for solving BQKP outperforms the two-phase method computed by a commercial state-of-the-art optimization software package.

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Notes

  1. Rodrigues et al. (2012) presented valid inequalities using three types of upper planes: a variant of the first kind of upper planes of Gallo et al. (1980) and those of the third and fourth kind of Caprara et al. (1999).

  2. Beheshti et al. (2015) factored the diagonal entries of their leader’s Q-matrix into the linear component of their leader’s objective function.

  3. BARON 17.8.9 uses CPLEX 12.7.1 as the default linear and mixed-integer programming solver.

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Acknowledgements

The authors are grateful to Dr. Behdad Beheshti and three anonymous reviewers for their constructive comments on earlier versions of this note. The U.S. Government is authorized to reproduce and distribute reprints for Governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of AFRL/RW or the U.S. Government.

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Authors and Affiliations

  1. Systems Engineering and Engineering Management, UNC Charlotte, 9201 University City Blvd., Charlotte, NC, 28223, USA

    Gabriel Lopez Zenarosa

  2. Department of Industrial Engineering, University of Pittsburgh, 3700 O’Hara Street, Pittsburgh, PA, 15261, USA

    Oleg A. Prokopyev

  3. AFRL Munitions Directorate, Building 13, Eglin AFB, FL, 32542, USA

    Eduardo L. Pasiliao

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  1. Gabriel Lopez Zenarosa
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  2. Oleg A. Prokopyev
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  3. Eduardo L. Pasiliao
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Correspondence to Oleg A. Prokopyev.

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Contract Grant sponsor: U.S. Air Force Research Laboratory (AFRL) Mathematical Modeling and Optimization Institute.

Contract Grant sponsor: U.S. Air Force Office of Scientic Research (AFOSR).

Contract Grant sponsor: U.S. Air Force Summer Faculty Fellowship and by AFRL/RW.

Contract Grant sponsor: National Science Foundation, NSF CMMI 1634835.

Contract Grant sponsor: University of Pittbusrgh Central Research Development Funds (CRDF).

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Zenarosa, G.L., Prokopyev, O.A. & Pasiliao, E.L. On exact solution approaches for bilevel quadratic 0–1 knapsack problem. Ann Oper Res 298, 555–572 (2021). https://doi.org/10.1007/s10479-018-2970-4

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