Abstract
We consider the specially structured (pure) integer Quadratic Multi-Knapsack Problem (QMKP) tackled in the paper “Exact solution methods to solve large scale integer quadratic knapsack problems” by D. Quadri, E. Soutif and P. Tolla (2009), recently appeared on this journal, where the problem is solved by transforming it into an equivalent 0–1 linearized Multi-Knapsack Problem (MKP). We show that, by taking advantage of the structure of the transformed (MKP), it is possible to derive an effective variable fixing procedure leading to an improved branch-and-bound approach. This procedure reduces dramatically the resulting linear problem size inducing an impressive improvement in the performances of the related branch and bound approach when compared to the results of the approach proposed by D. Quadri, E. Soutif and P. Tolla.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Bretthauer K, Shetty B (2002) The nonlinear knapsack problem—algorithms and applications. Eur J Oper Res 138(3):459–472
Chu PC, Beasley JE (1998) A genetic algorithm for the multidimensional knapsack problem. J Heuristics 4:63–86
Djerdjour M, Mathur K, Salkin H (1988) A Surrogate-based algorithm for the general quadratic multidimensional knapsack. Oper Res Lett 7(5):253–257
Faaland B (1974) An integer programming algorithm for portfolio selection. Manag Sci 20(10):1376–1384
Glover F (1975) Improved linear integer programming formulations of nonlinear integer problems. Manag Sci 22(4):455–460
Korner F (1985) Integer quadratic programming. Eur J Oper Res 19(2):268–273
Korner F (1990) On the numerical realization of the exact penalty method for quadratic programming algorithms. Eur J Oper Res 46(3):404–408
Li D, Sun XL (2006) Nonlinear integer programming. Springer, Berlin
Li D, Wang J, Sun XL (2007) Computing exact solution to nonlinear integer programming: Convergent Lagrangian and objective level cut method. J Glob Optim 39(1):127–154
Mathur K, Salkin H (1983) A branch and bound algorithm for a class of nonlinear knapsack problems. Oper Res Lett 2(4):155–160
Quadri D, Soutif E, Tolla P (2009) Exact solution methods to solve large scale integer quadratic knapsack problems. J Comb Optim 17:157–167
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Della Croce, F., Quadri, D. Improving an exact approach for solving separable integer quadratic knapsack problems. J Comb Optim 23, 21–28 (2012). https://doi.org/10.1007/s10878-010-9337-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10878-010-9337-3