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

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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.

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Data Availability statement

The data that support the findings of this study are available from the corresponding author upon request.

Notes

  1. Note that all the constraints in (SP) are linear, the optimal solution of (SP) must satisfy the KKT condition, without further assumption on constraint qualifications.

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Funding

Lu’s research has been supported by National Natural Science Foundation of China Grant No. 12171151. Deng’s research has been supported by the National Natural Science Foundation of China Grant No. T2293774, by the Fundamental Research Funds for the Central Universities E2ET0808X2, and by the grant from MOE Social Science Laboratory of Digital Economic Forecast and Policy Simulation at UCAS. Wang’s research has been supported by China Postdoctoral Research Foundation No. 2022M713102 and the Fundamental Research Funds for the Central Universities No. E2E49801.

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Authors and Affiliations

  1. School of Economics and Management, University of Chinese Academy of Sciences, Beijing, 100190, China

    Shaoze Li & Zhibin Deng

  2. School of Economics and Management, North China Electric Power University, Beijing, 102206, China

    Cheng Lu & Junhao Wu

  3. School of Science, Beijing University of Posts and Telecommunications, Beijing, 100876, China

    Jinyu Dai

  4. Institute of Science City Industry, University of Chinese Academy of Sciences, Beijing, 100190, China

    Qiao Wang

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  1. Shaoze Li
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  2. Zhibin Deng
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  3. Cheng Lu
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  4. Junhao Wu
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  5. Jinyu Dai
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  6. Qiao Wang
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Correspondence to Zhibin Deng or Qiao Wang.

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Li, S., Deng, Z., Lu, C. et al. An efficient global algorithm for indefinite separable quadratic knapsack problems with box constraints. Comput Optim Appl 86, 241–273 (2023). https://doi.org/10.1007/s10589-023-00488-x

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