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A Semidefinite Programming Approach to the Quadratic Knapsack Problem

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Listed:
  • C. Helmberg

    (Konrad Zuse Zentrum für Informationstechnik Berlin)

  • F. Rendl

    (Technische Universität Graz, Institut für Mathematik)

  • R. Weismantel

    (Universität Magdeburg, Institut für Mathematiche Optimierung)

Abstract

In order to gain insight into the quality of semidefinite relaxations of constrained quadratic 0/1 programming problems we study the quadratic knapsack problem. We investigate several basic semidefinite relaxations of this problem and compare their strength in theory and in practice. Various possibilities to improve these basic relaxations by cutting planes are discussed. The cutting planes either arise from quadratic representations of linear inequalities or from linear inequalities in the quadratic model. In particular, a large family of combinatorial cuts is introduced for the linear formulation of the knapsack problem in quadratic space. Computational results on a small class of practical problems illustrate the quality of these relaxations and cutting planes.

Suggested Citation

  • C. Helmberg & F. Rendl & R. Weismantel, 2000. "A Semidefinite Programming Approach to the Quadratic Knapsack Problem," Journal of Combinatorial Optimization, Springer, vol. 4(2), pages 197-215, June.
    • Handle: RePEc:spr:jcomop:v:4:y:2000:i:2:d:10.1023_a:1009898604624
    DOI: 10.1023/A:1009898604624
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    References listed on IDEAS

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    3. NESTEROV, Yu., 1998. "Semidefinite relaxation and nonconvex quadratic optimization," LIDAM Reprints CORE 1362, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    4. D. J. Laughhunn, 1970. "Quadratic Binary Programming with Application to Capital-Budgeting Problems," Operations Research, INFORMS, vol. 18(3), pages 454-461, June.
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    Cited by:

    1. Christoph Buchheim & Emiliano Traversi, 2018. "Quadratic Combinatorial Optimization Using Separable Underestimators," INFORMS Journal on Computing, INFORMS, vol. 30(3), pages 424-437, August.
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    3. Xiaojin Zheng & Xiaoling Sun & Duan Li & Yong Xia, 2010. "Duality Gap Estimation of Linear Equality Constrained Binary Quadratic Programming," Mathematics of Operations Research, INFORMS, vol. 35(4), pages 864-880, November.
    4. X. Zheng & X. Sun & D. Li & Y. Xu, 2012. "On reduction of duality gap in quadratic knapsack problems," Journal of Global Optimization, Springer, vol. 54(2), pages 325-339, October.
    5. Ming Huang & Yue Lu & Li Ping Pang & Zun Quan Xia, 2017. "A space decomposition scheme for maximum eigenvalue functions and its applications," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 85(3), pages 453-490, June.
    6. Britta Schulze & Michael Stiglmayr & Luís Paquete & Carlos M. Fonseca & David Willems & Stefan Ruzika, 2020. "On the rectangular knapsack problem: approximation of a specific quadratic knapsack problem," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 92(1), pages 107-132, August.
    7. Alexei Gaivoronski & Abdel Lisser & Rafael Lopez & Hu Xu, 2011. "Knapsack problem with probability constraints," Journal of Global Optimization, Springer, vol. 49(3), pages 397-413, March.
    8. Gicquel, C. & Lisser, A. & Minoux, M., 2014. "An evaluation of semidefinite programming based approaches for discrete lot-sizing problems," European Journal of Operational Research, Elsevier, vol. 237(2), pages 498-507.
    9. Sven Mallach, 2021. "Inductive linearization for binary quadratic programs with linear constraints," 4OR, Springer, vol. 19(4), pages 549-570, December.
    10. Ming Huang & Li-Ping Pang & Zun-Quan Xia, 2014. "The space decomposition theory for a class of eigenvalue optimizations," Computational Optimization and Applications, Springer, vol. 58(2), pages 423-454, June.
    11. Schauer, Joachim, 2016. "Asymptotic behavior of the quadratic knapsack problem," European Journal of Operational Research, Elsevier, vol. 255(2), pages 357-363.
    12. Alexandre d'Aspremont & Noureddine El Karoui, 2013. "Weak Recovery Conditions from Graph Partitioning Bounds and Order Statistics," Mathematics of Operations Research, INFORMS, vol. 38(2), pages 228-247, May.
    13. Timonina-Farkas, Anna & Katsifou, Argyro & Seifert, Ralf W., 2020. "Product assortment and space allocation strategies to attract loyal and non-loyal customers," European Journal of Operational Research, Elsevier, vol. 285(3), pages 1058-1076.
    14. Lv, Jian & Pang, Li-Ping & Wang, Jin-He, 2015. "Special backtracking proximal bundle method for nonconvex maximum eigenvalue optimization," Applied Mathematics and Computation, Elsevier, vol. 265(C), pages 635-651.
    15. Michele Garraffa & Federico Della Croce & Fabio Salassa, 2017. "An exact semidefinite programming approach for the max-mean dispersion problem," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 71-93, July.

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