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New Bundle Methods for Solving Lagrangian Relaxation Dual Problems

Author

Listed:
  • X. Zhao

    (I2 Technologies)

  • P.B. Luh

    (University of Connecticut)

Abstract

Bundle methods have been used frequently to solve nonsmooth optimization problems. In these methods, subgradient directions from past iterations are accumulated in a bundle, and a trial direction is obtained by performing quadratic programming based on the information contained in the bundle. A line search is then performed along the trial direction, generating a serious step if the function value is improved by ∈ or a null step otherwise. Bundle methods have been used to maximize the nonsmooth dual function in Lagrangian relaxation for integer optimization problems, where the subgradients are obtained by minimizing the performance index of the relaxed problem. This paper improves bundle methods by making good use of near-minimum solutions that are obtained while solving the relaxed problem. The bundle information is thus enriched, leading to better search directions and less number of null steps. Furthermore, a simplified bundle method is developed, where a fuzzy rule is used to combine linearly directions from near-minimum solutions, replacing quadratic programming and line search. When the simplified bundle method is specialized to an important class of problems where the relaxed problem can be solved by using dynamic programming, fuzzy dynamic programming is developed to obtain efficiently near-optimal solutions and their weights for the linear combination. This method is then applied to job shop scheduling problems, leading to better performance than previously reported in the literature.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joptap:v:113:y:2002:i:2:d:10.1023_a:1014839227049
    DOI: 10.1023/A:1014839227049
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    References listed on IDEAS

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    1. Niklas Kohl & Oli B. G. Madsen, 1997. "An Optimization Algorithm for the Vehicle Routing Problem with Time Windows Based on Lagrangian Relaxation," Operations Research, INFORMS, vol. 45(3), pages 395-406, June.
    2. 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.
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    Cited by:

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    3. 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.
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    5. Antonio Frangioni, 2005. "About Lagrangian Methods in Integer Optimization," Annals of Operations Research, Springer, vol. 139(1), pages 163-193, October.
    6. Alireza Hosseini & S. M. Hosseini, 2013. "A New Steepest Descent Differential Inclusion-Based Method for Solving General Nonsmooth Convex Optimization Problems," Journal of Optimization Theory and Applications, Springer, vol. 159(3), pages 698-720, December.

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