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Surrogate Gradient Algorithm for Lagrangian Relaxation

Author

Listed:
  • X. Zhao

    (University of Connecticut)

  • P. B. Luh

    (University of Connecticut)

  • J. Wang

    (University of Connecticut)

Abstract

The subgradient method is used frequently to optimize dual functions in Lagrangian relaxation for separable integer programming problems. In the method, all subproblems must be solved optimally to obtain a subgradient direction. In this paper, the surrogate subgradient method is developed, where a proper direction can be obtained without solving optimally all the subproblems. In fact, only an approximate optimization of one subproblem is needed to get a proper surrogate subgradient direction, and the directions are smooth for problems of large size. The convergence of the algorithm is proved. Compared with methods that take effort to find better directions, this method can obtain good directions with much less effort and provides a new approach that is especially powerful for problems of very large size.

Suggested Citation

  • X. Zhao & P. B. Luh & J. Wang, 1999. "Surrogate Gradient Algorithm for Lagrangian Relaxation," Journal of Optimization Theory and Applications, Springer, vol. 100(3), pages 699-712, March.
    • Handle: RePEc:spr:joptap:v:100:y:1999:i:3:d:10.1023_a:1022646725208
    DOI: 10.1023/A:1022646725208
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    References listed on IDEAS

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    1. Sarin, Sanjiv & Karwan, Mark H. & Rardin, Ronald L., 1988. "Surrogate duality in a branch-and-bound procedure for integer programming," European Journal of Operational Research, Elsevier, vol. 33(3), pages 326-333, February.
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    5. L Tang & H Xuan, 2006. "Lagrangian relaxation algorithms for real-time hybrid flowshop scheduling with finite intermediate buffers," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 57(3), pages 316-324, March.
    6. Mikhail A. Bragin & Peter B. Luh & Joseph H. Yan & Nanpeng Yu & Gary A. Stern, 2015. "Convergence of the Surrogate Lagrangian Relaxation Method," Journal of Optimization Theory and Applications, Springer, vol. 164(1), pages 173-201, January.
    7. Larsson, Torbjorn & Patriksson, Michael & Stromberg, Ann-Brith, 2003. "On the convergence of conditional [var epsilon]-subgradient methods for convex programs and convex-concave saddle-point problems," European Journal of Operational Research, Elsevier, vol. 151(3), pages 461-473, December.
    8. G. Rius-Sorolla & J. Maheut & Jairo R. Coronado-Hernandez & J. P. Garcia-Sabater, 2020. "Lagrangian relaxation of the generic materials and operations planning model," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 28(1), pages 105-123, March.
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    10. X.H. Guan & Q.Z. Zhai & F. Lai, 2002. "New Lagrangian Relaxation Based Algorithm for Resource Scheduling with Homogeneous Subproblems," Journal of Optimization Theory and Applications, Springer, vol. 113(1), pages 65-82, April.
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    13. X. Zhao & P.B. Luh, 2002. "New Bundle Methods for Solving Lagrangian Relaxation Dual Problems," Journal of Optimization Theory and Applications, Springer, vol. 113(2), pages 373-397, May.
    14. Lixin Tang & Gongshu Wang & Jiyin Liu & Jingyi Liu, 2011. "A combination of Lagrangian relaxation and column generation for order batching in steelmaking and continuous‐casting production," Naval Research Logistics (NRL), John Wiley & Sons, vol. 58(4), pages 370-388, June.
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    17. Chen, Haoxun & Luh, Peter B., 2003. "An alternative framework to Lagrangian relaxation approach for job shop scheduling," European Journal of Operational Research, Elsevier, vol. 149(3), pages 499-512, September.
    18. Azadian, Farshid & Murat, Alper & Chinnam, Ratna Babu, 2015. "Integrated production and logistics planning: Contract manufacturing and choice of air/surface transportation," European Journal of Operational Research, Elsevier, vol. 247(1), pages 113-123.
    19. Yu-Wei An & Hong-Sen Yan, 2016. "Lagrangean relaxation approach to joint optimization for production planning and scheduling of synchronous assembly lines," International Journal of Production Research, Taylor & Francis Journals, vol. 54(22), pages 6718-6735, November.
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