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A Hybrid Approach of Bundle and Benders Applied Large Mixed Linear Integer Problem

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  • Placido Rogerio Pinheiro
  • Paulo Roberto Oliveira

Abstract

Consider a large mixed integer linear problem where structure of the constraint matrix is sparse, with independent blocks, and coupling constraints and variables. There is one of the groups of constraints to make difficult the application of Benders scheme decomposition. In this work, we propose the following algorithm; a Lagrangian relaxation is made on the mentioned set of constraints; we presented a process heuristic for the calculation of the multiplier through the resolution of the dual problem, structured starting from the bundle methods. According to the methodology proposed, for each iteration of the algorithm, we propose Benders decomposition where quotas are provided for the value function and ε‐subgradient.

Suggested Citation

  • Placido Rogerio Pinheiro & Paulo Roberto Oliveira, 2013. "A Hybrid Approach of Bundle and Benders Applied Large Mixed Linear Integer Problem," Journal of Applied Mathematics, John Wiley & Sons, vol. 2013(1).
    • Handle: RePEc:wly:jnljam:v:2013:y:2013:i:1:n:678783
    DOI: 10.1155/2013/678783
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    References listed on IDEAS

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    1. Krishan Rana & R. G. Vickson, 1988. "A Model and Solution Algorithm for Optimal Routing of a Time-Chartered Containership," Transportation Science, INFORMS, vol. 22(2), pages 83-95, May.
    2. Holmberg, Kaj & Jornsten, Kurt O., 1984. "Cross decomposition applied to the stochastic transportation problem," European Journal of Operational Research, Elsevier, vol. 17(3), pages 361-368, September.
    3. Marshall L. Fisher, 1981. "The Lagrangian Relaxation Method for Solving Integer Programming Problems," Management Science, INFORMS, vol. 27(1), pages 1-18, January.
    4. Holmberg, Kaj & Ling, Jonas, 1997. "A Lagrangean heuristic for the facility location problem with staircase costs," European Journal of Operational Research, Elsevier, vol. 97(1), pages 63-74, February.
    5. Cote, Gilles & Laughton, Michael A., 1984. "Large-scale mixed integer programming: Benders-type heuristics," European Journal of Operational Research, Elsevier, vol. 16(3), pages 327-333, June.
    6. George B. Dantzig & Philip Wolfe, 1960. "Decomposition Principle for Linear Programs," Operations Research, INFORMS, vol. 8(1), pages 101-111, February.
    7. 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).
    8. Michael Held & Richard M. Karp, 1970. "The Traveling-Salesman Problem and Minimum Spanning Trees," Operations Research, INFORMS, vol. 18(6), pages 1138-1162, December.
    9. Aardal, Karen & Larsson, Torbjorn, 1990. "A benders decomposition based heuristic for the hierarchical production planning problem," European Journal of Operational Research, Elsevier, vol. 45(1), pages 4-14, March.
    10. Kaj Holmberg, 1994. "Cross Decomposition Applied to Integer Programming Problems: Duality Gaps and Convexification in Parts," Operations Research, INFORMS, vol. 42(4), pages 657-668, August.
    11. Holmberg, Kaj, 1994. "On using approximations of the Benders master problem," European Journal of Operational Research, Elsevier, vol. 77(1), pages 111-125, August.
    12. Dale McDaniel & Mike Devine, 1977. "A Modified Benders' Partitioning Algorithm for Mixed Integer Programming," Management Science, INFORMS, vol. 24(3), pages 312-319, November.
    13. Tony J. Van Roy, 1986. "A Cross Decomposition Algorithm for Capacitated Facility Location," Operations Research, INFORMS, vol. 34(1), pages 145-163, February.
    14. 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.
    15. Holmberg, Kaj, 1997. "Mean value cross decomposition applied to integer programming problems," European Journal of Operational Research, Elsevier, vol. 97(1), pages 124-138, February.
    16. Holmberg, Kaj, 1994. "Solving the staircase cost facility location problem with decomposition and piecewise linearization," European Journal of Operational Research, Elsevier, vol. 75(1), pages 41-61, May.
    17. Aardal, Karen & Ari, Aysen, 1990. "On the resemblance between the Kornai-Liptak and cross decomposition techniques for block-angular linear programs," European Journal of Operational Research, Elsevier, vol. 46(3), pages 393-398, June.
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