Abstract
We consider the knapsack problem in which the objective function is uncertain, and given by a finite set of possible realizations. The resulting robust optimization problem is a max–min problem that follows the pessimistic view of optimizing the worst-case behavior. Several branch-and-bound algorithms have been proposed in the recent literature. In this short note, we show that by using a simple upper bound that is tailored to balance out the drawbacks of the current best approach based on surrogate relaxation, computation times improve by up to an order of magnitude. Additionally, one can make use of any upper bound for the classic knapsack problem as an upper bound for the robust problem.
This is a preview of subscription content, log in via an institution to check access.
References
Aissi, H., Bazgan, C., & Vanderpooten, D. (2009). Minmax and minmax regret versions of combinatorial optimization problems: A survey. European Journal of Operational Research, 197(2), 427–438. doi:10.1016/j.ejor.2008.09.012.
Ben-Tal, A., & Nemirovski, A. (1998). Robust convex optimization. Mathematics of Operations Research, 23(4), 769–805.
Ben-Tal, A., & Nemirovski, A. (1999). Robust solutions of uncertain linear programs. Operations Research Letters, 25, 1–13.
Ben-Tal, A., Ghaoui, L. E., & Nemirovski, A. (2009). Robust optimization. Princeton and Oxford: Princeton University Press.
Büsing, C., Koster, A., & Kutschka, M. (2011). Recoverable robust knapsacks: The discrete scenario case. Optimization Letters, 5(3), 379–392.
Dantzig, G. B. (1957). Discrete-variable extremum problems. Operations Research, 5(2), 266–288.
Iida, H. (1999). A note on the max–min 0–1 knapsack problem. Journal of Combinatorial Optimization, 3(1), 89–94. doi:10.1023/A:1009821323279.
Kellerer, H., Pferschy, U., & Pisinger, D. (2004). Knapsack problems. New York: Springer.
Kouvelis, P., & Yu, G. (1997). Robust discrete optimization and Its applications. Boston: Kluwer Academic Publishers.
Martello, S., & Toth, P. (1977). An upper bound for the zero–one knapsack problem and a branch and bound algorithm. European Journal of Operational Research, 1(3), 169–175.
Martello, S., & Toth, P. (1988). A new algorithm for the 0–1 knapsack problem. Management Science, 34(5), 633–644.
Martello, S., Pisinger, D., & Toth, P. (1999). Dynamic programming and strong bounds for the 0–1 knapsack problem. Management Science, 45(3), 414–424.
Monaci, M., Pferschy, U., & Serafini, P. (2013). Exact solution of the robust knapsack problem. Computers & Operations Research, 40(11), 2625–2631.
Soyster, A. (1973). Convex programming with set-inclusive constraints and applications to inexact linear programming. Operations Research, 21, 1154–1157.
Taniguchi, F., Yamada, T., & Kataoka, S. (2008). Heuristic and exact algorithms for the max-min optimization of the multi-scenario knapsack problem. Computers & Operations Research, 35(6), 2034–2048. doi:10.1016/j.cor.2006.10.002.
You, B., Yokoya, D., & Yamada, T. (2012). An improved reduction method for the robust optimization of the assignment problem. Asia-Pacific Journal of Operational Research, 29(5), 1250032.
Acknowledgments
Effort sponsored by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-13-1-3066. The U.S Government is authorized to reproduce and distribute reprints for Governmental purpose notwithstanding any copyright notation thereon.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Goerigk, M. A note on upper bounds to the robust knapsack problem with discrete scenarios. Ann Oper Res 223, 461–469 (2014). https://doi.org/10.1007/s10479-014-1618-2
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-014-1618-2