Advertisement

Journal of Combinatorial Optimization

, Volume 23, Issue 1, pp 21–28 | Cite as

Improving an exact approach for solving separable integer quadratic knapsack problems

  • Federico Della CroceEmail author
  • Dominique Quadri
Article
  • 140 Downloads

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.

Keywords

Integer quadratic knapsack problem Variable fixing Linearization 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Bretthauer K, Shetty B (2002) The nonlinear knapsack problem—algorithms and applications. Eur J Oper Res 138(3):459–472 CrossRefzbMATHMathSciNetGoogle Scholar
  2. Chu PC, Beasley JE (1998) A genetic algorithm for the multidimensional knapsack problem. J Heuristics 4:63–86 CrossRefzbMATHGoogle Scholar
  3. Djerdjour M, Mathur K, Salkin H (1988) A Surrogate-based algorithm for the general quadratic multidimensional knapsack. Oper Res Lett 7(5):253–257 CrossRefzbMATHMathSciNetGoogle Scholar
  4. Faaland B (1974) An integer programming algorithm for portfolio selection. Manag Sci 20(10):1376–1384 CrossRefzbMATHMathSciNetGoogle Scholar
  5. Glover F (1975) Improved linear integer programming formulations of nonlinear integer problems. Manag Sci 22(4):455–460 CrossRefMathSciNetGoogle Scholar
  6. Korner F (1985) Integer quadratic programming. Eur J Oper Res 19(2):268–273 CrossRefMathSciNetGoogle Scholar
  7. Korner F (1990) On the numerical realization of the exact penalty method for quadratic programming algorithms. Eur J Oper Res 46(3):404–408 CrossRefMathSciNetGoogle Scholar
  8. Li D, Sun XL (2006) Nonlinear integer programming. Springer, Berlin zbMATHGoogle Scholar
  9. 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 CrossRefzbMATHMathSciNetGoogle Scholar
  10. Mathur K, Salkin H (1983) A branch and bound algorithm for a class of nonlinear knapsack problems. Oper Res Lett 2(4):155–160 CrossRefzbMATHMathSciNetGoogle Scholar
  11. Quadri D, Soutif E, Tolla P (2009) Exact solution methods to solve large scale integer quadratic knapsack problems. J Comb Optim 17:157–167 CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.DAIPolitecnico di TorinoTorinoItaly
  2. 2.LIAUniversité d’AvignonAvignonFrance

Personalised recommendations