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Bilevel optimization based on iterative approximation of multiple mappings

Author

Listed:
  • Ankur Sinha

    (Indian Institute of Management)

  • Zhichao Lu

    (Michigan State University)

  • Kalyanmoy Deb

    (Michigan State University)

  • Pekka Malo

    (Aalto University School of Economics)

Abstract

A large number of application problems involve two levels of optimization, where one optimization task is nested inside the other. These problems are known as bilevel optimization problems and have been studied by both classical optimization community and evolutionary optimization community. Most of the solution procedures proposed until now are either computationally very expensive or applicable to only small classes of bilevel optimization problems adhering to mathematically simplifying assumptions. In this paper, we propose an evolutionary optimization method that tries to reduce the computational expense by iteratively approximating two important mappings in bilevel optimization; namely, the lower level rational reaction mapping and the lower level optimal value function mapping. The algorithm has been tested on a large number of test problems and comparisons have been performed with other algorithms. The results show the performance gain to be quite significant. To the best knowledge of the authors, a combined theory-based and population-based solution procedure utilizing mappings has not been suggested yet for bilevel problems.

Suggested Citation

  • Ankur Sinha & Zhichao Lu & Kalyanmoy Deb & Pekka Malo, 2020. "Bilevel optimization based on iterative approximation of multiple mappings," Journal of Heuristics, Springer, vol. 26(2), pages 151-185, April.
    • Handle: RePEc:spr:joheur:v:26:y:2020:i:2:d:10.1007_s10732-019-09426-9
    DOI: 10.1007/s10732-019-09426-9
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    1. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(6), pages 1157-1160, December.
    2. Miles G. Nicholls, 1995. "Aluminum Production Modeling—A Nonlinear Bilevel Programming Approach," Operations Research, INFORMS, vol. 43(2), pages 208-218, April.
    3. Martine Labbé & Patrice Marcotte & Gilles Savard, 1998. "A Bilevel Model of Taxation and Its Application to Optimal Highway Pricing," Management Science, INFORMS, vol. 44(12-Part-1), pages 1608-1622, December.
    4. Jerome Bracken & James T. McGill, 1974. "Defense Applications of Mathematical Programs with Optimization Problems in the Constraints," Operations Research, INFORMS, vol. 22(5), pages 1086-1096, October.
    5. Jerome Bracken & James T. McGill, 1973. "Mathematical Programs with Optimization Problems in the Constraints," Operations Research, INFORMS, vol. 21(1), pages 37-44, February.
    6. ,, 2001. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 17(5), pages 1025-1031, October.
    7. Hecheng Li, 2015. "A genetic algorithm using a finite search space for solving nonlinear/linear fractional bilevel programming problems," Annals of Operations Research, Springer, vol. 235(1), pages 543-558, December.
    8. J. A. Mirrlees, 1999. "The Theory of Moral Hazard and Unobservable Behaviour: Part I," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 66(1), pages 3-21.
    9. Xinmin Hu & Daniel Ralph, 2007. "Using EPECs to Model Bilevel Games in Restructured Electricity Markets with Locational Prices," Operations Research, INFORMS, vol. 55(5), pages 809-827, October.
    10. Bo An & Fernando Ordóñez & Milind Tambe & Eric Shieh & Rong Yang & Craig Baldwin & Joseph DiRenzo & Kathryn Moretti & Ben Maule & Garrett Meyer, 2013. "A Deployed Quantal Response-Based Patrol Planning System for the U.S. Coast Guard," Interfaces, INFORMS, vol. 43(5), pages 400-420, October.
    11. Sinha, Ankur & Malo, Pekka & Deb, Kalyanmoy, 2017. "Evolutionary algorithm for bilevel optimization using approximations of the lower level optimal solution mapping," European Journal of Operational Research, Elsevier, vol. 257(2), pages 395-411.
    12. Yuping Wang & Hong Li & Chuangyin Dang, 2011. "A New Evolutionary Algorithm for a Class of Nonlinear Bilevel Programming Problems and Its Global Convergence," INFORMS Journal on Computing, INFORMS, vol. 23(4), pages 618-629, November.
    13. Gerald Brown & Matthew Carlyle & Douglas Diehl & Jeffrey Kline & Kevin Wood, 2005. "A Two-Sided Optimization for Theater Ballistic Missile Defense," Operations Research, INFORMS, vol. 53(5), pages 745-763, October.
    14. Nathan Williams & P. K. Kannan & Shapour Azarm, 2011. "Retail Channel Structure Impact on Strategic Engineering Product Design," Management Science, INFORMS, vol. 57(5), pages 897-914, May.
    15. Luce Brotcorne & Martine Labbé & Patrice Marcotte & Gilles Savard, 2001. "A Bilevel Model for Toll Optimization on a Multicommodity Transportation Network," Transportation Science, INFORMS, vol. 35(4), pages 345-358, November.
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    Cited by:

    1. Jayaswal, Sachin & Sinha, Ankur, 2022. "Bilevel Optimization: Applications, Models and Solution Approaches," IIMA Working Papers WP 2022-05-02, Indian Institute of Management Ahmedabad, Research and Publication Department.
    2. Mejía-de-Dios, Jesús-Adolfo & Mezura-Montes, Efrén & Toledo-Hernández, Porfirio, 2022. "Pseudo-feasible solutions in evolutionary bilevel optimization: Test problems and performance assessment," Applied Mathematics and Computation, Elsevier, vol. 412(C).
    3. Rihab Said & Maha Elarbi & Slim Bechikh & Lamjed Ben Said, 2022. "Solving combinatorial bi-level optimization problems using multiple populations and migration schemes," Operational Research, Springer, vol. 22(3), pages 1697-1735, July.

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