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Solving the generalized knapsack problem with variable coefficients

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  • Kaj Holmberg
  • Kurt Jörnsten

Abstract

In this article we present methods based on Lagrangian duality and decomposition techniques for the generalized knapsack problem with variable coefficients. The Lagrangian dual is solved with subgradient optimization or interval bisection. We also describe a heuristic that yields primal feasible solutions. Combining the Lagrangian relaxation with a primal (Benders) subproblem yields the subproblem phase in cross decomposition. By using averages in this procedure, we get the new mean‐value cross‐decomposition method. Finally, we describe how to insert this into a globally convergent generalized Benders decomposition framework, in the case that there is a duality gap. Encouraging computational results for the optimal generating unit commitment problem are presented. © 1996 John Wiley & Sons, Inc.

Suggested Citation

  • Kaj Holmberg & Kurt Jörnsten, 1996. "Solving the generalized knapsack problem with variable coefficients," Naval Research Logistics (NRL), John Wiley & Sons, vol. 43(5), pages 673-689, August.
    • Handle: RePEc:wly:navres:v:43:y:1996:i:5:p:673-689
    DOI: 10.1002/(SICI)1520-6750(199608)43:53.0.CO;2-2
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    References listed on IDEAS

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    1. Holmberg, Kaj, 1992. "Linear mean value cross decomposition: A generalization of the Kornai-Liptak method," European Journal of Operational Research, Elsevier, vol. 62(1), pages 55-73, October.
    2. Robert Wilson, 1966. "Programming Variable Factors," Management Science, INFORMS, vol. 13(1), pages 144-151, September.
    3. T. L. Magnanti & R. T. Wong, 1981. "Accelerating Benders Decomposition: Algorithmic Enhancement and Model Selection Criteria," Operations Research, INFORMS, vol. 29(3), pages 464-484, June.
    4. VAN ROY, Tony J., 1983. "Cross decomposition for mixed integer programming," LIDAM Reprints CORE 496, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. Bazaraa, Mokhtar S. & Goode, Jamie J., 1979. "A survey of various tactics for generating Lagrangian multipliers in the context of Lagrangian duality," European Journal of Operational Research, Elsevier, vol. 3(4), pages 322-338, July.
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