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Sequence Independent Lifting in Mixed Integer Programming

Author

Listed:
  • Zonghao Gu

    (Georgia Institute of Technology)

  • George L. Nemhauser

    (Georgia Institute of Technology)

  • Martin W.P. Savelsbergh

    (Georgia Institute of Technology)

Abstract

We investigate lifting, i.e., the process of taking a valid inequality for a polyhedron and extending it to a valid inequality in a higher dimensional space. Lifting is usually applied sequentially, that is, variables in a set are lifted one after the other. This may be computationally unattractive since it involves the solution of an optimization problem to compute a lifting coefficient for each variable. To relieve this computational burden, we study sequence independent lifting, which only involves the solution of one optimization problem. We show that if a certain lifting function is superadditive, then the lifting coefficients are independent of the lifting sequence. We introduce the idea of valid superadditive lifting functions to obtain good aproximations to maximum lifting. We apply these results to strengthen Balas' lifting theorem for cover inequalities and to produce lifted flow cover inequalities for a single node flow problem.

Suggested Citation

  • Zonghao Gu & George L. Nemhauser & Martin W.P. Savelsbergh, 2000. "Sequence Independent Lifting in Mixed Integer Programming," Journal of Combinatorial Optimization, Springer, vol. 4(1), pages 109-129, March.
    • Handle: RePEc:spr:jcomop:v:4:y:2000:i:1:d:10.1023_a:1009841107478
    DOI: 10.1023/A:1009841107478
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    References listed on IDEAS

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    Cited by:

    1. Absi, Nabil & Kedad-Sidhoum, Safia, 2008. "The multi-item capacitated lot-sizing problem with setup times and shortage costs," European Journal of Operational Research, Elsevier, vol. 185(3), pages 1351-1374, March.
    2. Kaparis, Konstantinos & Letchford, Adam N., 2008. "Local and global lifted cover inequalities for the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 186(1), pages 91-103, April.
    3. Gamvros, Ioannis & Raghavan, S., 2012. "Multi-period traffic routing in satellite networks," European Journal of Operational Research, Elsevier, vol. 219(3), pages 738-750.
    4. Ahmet B. Keha & Ismael R. de Farias & George L. Nemhauser, 2006. "A Branch-and-Cut Algorithm Without Binary Variables for Nonconvex Piecewise Linear Optimization," Operations Research, INFORMS, vol. 54(5), pages 847-858, October.
    5. Xiaoyi Gu & Santanu S. Dey & Jean-Philippe P. Richard, 2024. "Solving Sparse Separable Bilinear Programs Using Lifted Bilinear Cover Inequalities," INFORMS Journal on Computing, INFORMS, vol. 36(3), pages 884-899, May.
    6. Agostinho Agra & Cristina Requejo & Eulália Santos, 2016. "Implicit cover inequalities," Journal of Combinatorial Optimization, Springer, vol. 31(3), pages 1111-1129, April.
    7. Alper Atamtürk & Muhong Zhang, 2008. "The Flow Set with Partial Order," Mathematics of Operations Research, INFORMS, vol. 33(3), pages 730-746, August.
    8. Atamturk, Alper & Munoz, Juan Carlos, 2002. "A Study of the Lot-Sizing Polytope," University of California Transportation Center, Working Papers qt6zz2g0z4, University of California Transportation Center.
    9. Christopher Hojny & Tristan Gally & Oliver Habeck & Hendrik Lüthen & Frederic Matter & Marc E. Pfetsch & Andreas Schmitt, 2020. "Knapsack polytopes: a survey," Annals of Operations Research, Springer, vol. 292(1), pages 469-517, September.
    10. Lisa A. Miller & Yanjun Li & Jean‐Philippe P. Richard, 2008. "New inequalities for finite and infinite group problems from approximate lifting," Naval Research Logistics (NRL), John Wiley & Sons, vol. 55(2), pages 172-191, March.
    11. Yongjia Song & James R. Luedtke & Simge Küçükyavuz, 2014. "Chance-Constrained Binary Packing Problems," INFORMS Journal on Computing, INFORMS, vol. 26(4), pages 735-747, November.
    12. Alper Atamtürk, 2004. "Sequence Independent Lifting for Mixed-Integer Programming," Operations Research, INFORMS, vol. 52(3), pages 487-490, June.
    13. Yang, Zhen & Chu, Feng & Chen, Haoxun, 2012. "A cut-and-solve based algorithm for the single-source capacitated facility location problem," European Journal of Operational Research, Elsevier, vol. 221(3), pages 521-532.
    14. Miller, Andrew J. & Nemhauser, George L. & Savelsbergh, Martin W. P., 2000. "On the capacitated lot-sizing and continuous 0-1 knapsack polyhedra," European Journal of Operational Research, Elsevier, vol. 125(2), pages 298-315, September.
    15. Santanu S. Dey & Jean-Philippe Richard, 2009. "Linear-Programming-Based Lifting and Its Application to Primal Cutting-Plane Algorithms," INFORMS Journal on Computing, INFORMS, vol. 21(1), pages 137-150, February.
    16. Yang, Zhen & Chen, Haoxun & Chu, Feng & Wang, Nengmin, 2019. "An effective hybrid approach to the two-stage capacitated facility location problem," European Journal of Operational Research, Elsevier, vol. 275(2), pages 467-480.
    17. Amar K. Narisetty & Jean-Philippe P. Richard & George L. Nemhauser, 2011. "Lifted Tableaux Inequalities for 0--1 Mixed-Integer Programs: A Computational Study," INFORMS Journal on Computing, INFORMS, vol. 23(3), pages 416-424, August.
    18. Yongjia Song & Siqian Shen, 2016. "Risk-Averse Shortest Path Interdiction," INFORMS Journal on Computing, INFORMS, vol. 28(3), pages 527-539, August.
    19. Detienne, Boris, 2014. "A mixed integer linear programming approach to minimize the number of late jobs with and without machine availability constraints," European Journal of Operational Research, Elsevier, vol. 235(3), pages 540-552.
    20. Shanshan Wang & Jinlin Li & Sanjay Mehrotra, 2021. "Chance-Constrained Multiple Bin Packing Problem with an Application to Operating Room Planning," INFORMS Journal on Computing, INFORMS, vol. 33(4), pages 1661-1677, October.
    21. Yogesh Agarwal & Yash Aneja, 2012. "Fixed-Charge Transportation Problem: Facets of the Projection Polyhedron," Operations Research, INFORMS, vol. 60(3), pages 638-654, June.
    22. Dimitri J. Papageorgiou & Alejandro Toriello & George L. Nemhauser & Martin W. P. Savelsbergh, 2012. "Fixed-Charge Transportation with Product Blending," Transportation Science, INFORMS, vol. 46(2), pages 281-295, May.
    23. Michele Conforti & Gérard Cornuéjols & Giacomo Zambelli, 2011. "A Geometric Perspective on Lifting," Operations Research, INFORMS, vol. 59(3), pages 569-577, June.
    24. Atamturk, Alper & Munoz, Juan Carlos, 2002. "A Study of the Lot-Sizing Polytope," University of California Transportation Center, Working Papers qt9gx170tx, University of California Transportation Center.

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    integer programming; lifting;

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