IDEAS home Printed from https://ideas.repec.org/a/spr/jcomop/v17y2009i2d10.1007_s10878-007-9105-1.html
   My bibliography  Save this article

Exact solution method to solve large scale integer quadratic multidimensional knapsack problems

Author

Listed:
  • D. Quadri

    (Université Paris-Dauphine)

  • E. Soutif

    (Conservatoire des Arts et Métiers)

  • P. Tolla

    (Université Paris-Dauphine)

Abstract

In this paper we develop a branch-and-bound algorithm for solving a particular integer quadratic multi-knapsack problem. The problem we study is defined as the maximization of a concave separable quadratic objective function over a convex set of linear constraints and bounded integer variables. Our exact solution method is based on the computation of an upper bound and also includes pre-procedure techniques in order to reduce the problem size before starting the branch-and-bound process. We lead a numerical comparison between our method and three other existing algorithms. The approach we propose outperforms other procedures for large-scaled instances (up to 2000 variables and constraints).

Suggested Citation

  • D. Quadri & E. Soutif & P. Tolla, 2009. "Exact solution method to solve large scale integer quadratic multidimensional knapsack problems," Journal of Combinatorial Optimization, Springer, vol. 17(2), pages 157-167, February.
    • Handle: RePEc:spr:jcomop:v:17:y:2009:i:2:d:10.1007_s10878-007-9105-1
    DOI: 10.1007/s10878-007-9105-1
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10878-007-9105-1
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10878-007-9105-1?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    1. W. David Pisinger & Anders Bo Rasmussen & Rune Sandvik, 2007. "Solution of Large Quadratic Knapsack Problems Through Aggressive Reduction," INFORMS Journal on Computing, INFORMS, vol. 19(2), pages 280-290, May.
    2. Duan Li & Xiaoling Sun, 2006. "Nonlinear Integer Programming," International Series in Operations Research and Management Science, Springer, number 978-0-387-32995-6, April.
    3. Mary W. Cooper, 1981. "A Survey of Methods for Pure Nonlinear Integer Programming," Management Science, INFORMS, vol. 27(3), pages 353-361, March.
    4. Bruce Faaland, 1974. "An Integer Programming Algorithm for Portfolio Selection," Management Science, INFORMS, vol. 20(10), pages 1376-1384, June.
    5. Fred Glover, 1975. "Improved Linear Integer Programming Formulations of Nonlinear Integer Problems," Management Science, INFORMS, vol. 22(4), pages 455-460, December.
    6. Fred Glover, 1975. "Surrogate Constraint Duality in Mathematical Programming," Operations Research, INFORMS, vol. 23(3), pages 434-451, June.
    7. Bretthauer, Kurt M. & Shetty, Bala, 2002. "The nonlinear knapsack problem - algorithms and applications," European Journal of Operational Research, Elsevier, vol. 138(3), pages 459-472, May.
    8. Billionnet, Alain & Soutif, Eric, 2004. "An exact method based on Lagrangian decomposition for the 0-1 quadratic knapsack problem," European Journal of Operational Research, Elsevier, vol. 157(3), pages 565-575, September.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. SYAFITRI, Utami & SARTONO, Bagus & GOOS, Peter, 2015. "D- and I-optimal design of mixture experiments in the presence of ingredient availability constraints," Working Papers 2015003, University of Antwerp, Faculty of Business and Economics.
    2. Kuen-Fang Jea & Jen-Ya Wang & Chih-Wei Hsu, 2019. "Two-Agent Advertisement Scheduling on Physical Books to Maximize the Total Profit," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 36(03), pages 1-24, June.
    3. Shuchen Wang & Suixiang Gao & Wenguo Yang, 2023. "Pilot pattern design scheme with branch and bound in PSA-OFDM system," Journal of Combinatorial Optimization, Springer, vol. 45(4), pages 1-16, May.
    4. X. J. Zheng & X. L. Sun & D. Li, 2010. "Separable Relaxation for Nonconvex Quadratic Integer Programming: Integer Diagonalization Approach," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 463-489, August.
    5. Federico Della Croce & Dominique Quadri, 2012. "Improving an exact approach for solving separable integer quadratic knapsack problems," Journal of Combinatorial Optimization, Springer, vol. 23(1), pages 21-28, January.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Federico Della Croce & Dominique Quadri, 2012. "Improving an exact approach for solving separable integer quadratic knapsack problems," Journal of Combinatorial Optimization, Springer, vol. 23(1), pages 21-28, January.
    2. David Bergman, 2019. "An Exact Algorithm for the Quadratic Multiknapsack Problem with an Application to Event Seating," INFORMS Journal on Computing, INFORMS, vol. 31(3), pages 477-492, July.
    3. Renaud Chicoisne, 2023. "Computational aspects of column generation for nonlinear and conic optimization: classical and linearized schemes," Computational Optimization and Applications, Springer, vol. 84(3), pages 789-831, April.
    4. Fabio Furini & Emiliano Traversi, 2019. "Theoretical and computational study of several linearisation techniques for binary quadratic problems," Annals of Operations Research, Springer, vol. 279(1), pages 387-411, August.
    5. Gabriel Lopez Zenarosa & Oleg A. Prokopyev & Eduardo L. Pasiliao, 2021. "On exact solution approaches for bilevel quadratic 0–1 knapsack problem," Annals of Operations Research, Springer, vol. 298(1), pages 555-572, March.
    6. Alidaee, Bahram, 2014. "Zero duality gap in surrogate constraint optimization: A concise review of models," European Journal of Operational Research, Elsevier, vol. 232(2), pages 241-248.
    7. Bretthauer, Kurt M. & Ross, Anthony & Shetty, Bala, 1999. "Nonlinear integer programming for optimal allocation in stratified sampling," European Journal of Operational Research, Elsevier, vol. 116(3), pages 667-680, August.
    8. 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.
    9. Juan S. Borrero & Colin Gillen & Oleg A. Prokopyev, 2017. "Fractional 0–1 programming: applications and algorithms," Journal of Global Optimization, Springer, vol. 69(1), pages 255-282, September.
    10. X. J. Zheng & X. L. Sun & D. Li, 2010. "Separable Relaxation for Nonconvex Quadratic Integer Programming: Integer Diagonalization Approach," Journal of Optimization Theory and Applications, Springer, vol. 146(2), pages 463-489, August.
    11. Richard J. Forrester & Warren P. Adams & Paul T. Hadavas, 2010. "Concise RLT forms of binary programs: A computational study of the quadratic knapsack problem," Naval Research Logistics (NRL), John Wiley & Sons, vol. 57(1), pages 1-12, February.
    12. Bueno, L.F. & Haeser, G. & Kolossoski, O., 2024. "On the paper “Augmented Lagrangian algorithms for solving the continuous nonlinear resource allocation problem”," European Journal of Operational Research, Elsevier, vol. 313(3), pages 1217-1222.
    13. Dori Hulst & Dick Hertog & Wim Nuijten, 2017. "Robust shift generation in workforce planning," Computational Management Science, Springer, vol. 14(1), pages 115-134, January.
    14. Yokoyama, Ryohei & Kitano, Hiroyuki & Wakui, Tetsuya, 2017. "Optimal operation of heat supply systems with piping network," Energy, Elsevier, vol. 137(C), pages 888-897.
    15. Tian, Xueyu & You, Fengqi, 2019. "Carbon-neutral hybrid energy systems with deep water source cooling, biomass heating, and geothermal heat and power," Applied Energy, Elsevier, vol. 250(C), pages 413-432.
    16. Longinidis, Pantelis & Georgiadis, Michael C., 2014. "Integration of sale and leaseback in the optimal design of supply chain networks," Omega, Elsevier, vol. 47(C), pages 73-89.
    17. Rostami, Borzou & Chassein, André & Hopf, Michael & Frey, Davide & Buchheim, Christoph & Malucelli, Federico & Goerigk, Marc, 2018. "The quadratic shortest path problem: complexity, approximability, and solution methods," European Journal of Operational Research, Elsevier, vol. 268(2), pages 473-485.
    18. John N. Hooker, 2002. "Logic, Optimization, and Constraint Programming," INFORMS Journal on Computing, INFORMS, vol. 14(4), pages 295-321, November.
    19. Unai Aldasoro & María Merino & Gloria Pérez, 2019. "Time consistent expected mean-variance in multistage stochastic quadratic optimization: a model and a matheuristic," Annals of Operations Research, Springer, vol. 280(1), pages 151-187, September.
    20. Janssens, Jochen & Talarico, Luca & Sörensen, Kenneth, 2016. "A hybridised variable neighbourhood tabu search heuristic to increase security in a utility network," Reliability Engineering and System Safety, Elsevier, vol. 145(C), pages 221-230.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:jcomop:v:17:y:2009:i:2:d:10.1007_s10878-007-9105-1. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.