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An efficient global algorithm for indefinite separable quadratic knapsack problems with box constraints

Author

Listed:
  • Shaoze Li

    (University of Chinese Academy of Sciences)

  • Zhibin Deng

    (University of Chinese Academy of Sciences)

  • Cheng Lu

    (North China Electric Power University)

  • Junhao Wu

    (North China Electric Power University)

  • Jinyu Dai

    (Beijing University of Posts and Telecommunications)

  • Qiao Wang

    (University of Chinese Academy of Sciences)

Abstract

The indefinite separable quadratic knapsack problem (ISQKP) with box constraints is known to be NP-hard. In this paper, we propose a new branch-and-bound algorithm based on a convex envelope relaxation that can be efficiently solved by exploiting its special dual structure. Benefiting from a new branching strategy, the complexity of the proposed algorithm is quadratic in terms of the number of variables when the number of negative eigenvalues in the objective function of ISQKP is fixed. We then improve the proposed algorithm for the case that ISQKP has symmetric structures. The improvement is achieved by constructing tight convex relaxations based on the aggregate functions. Numerical experiments on large-size instances show that the proposed algorithm is much faster than Gurobi and CPLEX. It turns out that the proposed algorithm can solve the instances of size up to three million in less than twenty seconds on average and its improved version is still very efficient for problems with symmetric structures.

Suggested Citation

  • Shaoze Li & Zhibin Deng & Cheng Lu & Junhao Wu & Jinyu Dai & Qiao Wang, 2023. "An efficient global algorithm for indefinite separable quadratic knapsack problems with box constraints," Computational Optimization and Applications, Springer, vol. 86(1), pages 241-273, September.
    • Handle: RePEc:spr:coopap:v:86:y:2023:i:1:d:10.1007_s10589-023-00488-x
    DOI: 10.1007/s10589-023-00488-x
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    References listed on IDEAS

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    1. Edirisinghe, Chanaka & Jeong, Jaehwan, 2019. "Indefinite multi-constrained separable quadratic optimization: Large-scale efficient solution," European Journal of Operational Research, Elsevier, vol. 278(1), pages 49-63.
    2. Leo Liberti & James Ostrowski, 2014. "Stabilizer-based symmetry breaking constraints for mathematical programs," Journal of Global Optimization, Springer, vol. 60(2), pages 183-194, October.
    3. Alberto Marchi, 2022. "On a primal-dual Newton proximal method for convex quadratic programs," Computational Optimization and Applications, Springer, vol. 81(2), pages 369-395, March.
    4. G. Liuzzi & M. Locatelli & V. Piccialli & S. Rass, 2021. "Computing mixed strategies equilibria in presence of switching costs by the solution of nonconvex QP problems," Computational Optimization and Applications, Springer, vol. 79(3), pages 561-599, July.
    5. Jing Zhou & Shu-Cherng Fang & Wenxun Xing, 2017. "Conic approximation to quadratic optimization with linear complementarity constraints," Computational Optimization and Applications, Springer, vol. 66(1), pages 97-122, January.
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