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A Benders decomposition method for discretely-constrained mathematical programs with equilibrium constraints

Author

Listed:
  • S A Gabriel

    (University of Maryland)

  • Y Shim

    (University of Maryland)

  • A J Conejo

    (Universidad de Castilla-La Mancha)

  • S de la Torre

    (University of Málaga)

  • R García-Bertrand

    (Universidad de Castilla-La Mancha)

Abstract

We present a new methodology to solve discretely-constrained mathematical programs with equilibrium constraints (DC-MPECs). Typically these problems include an upper planning-level optimization with some discrete decision variables (eg, build/don’t build) as well as a lower operations-level problem often described by an optimization or nonlinear complementarity problem. This lower-level problem may also include some discrete variables. MPECs are very challenging problems to solve and the inclusion of integrality constraints makes this class of problems even more computationally difficult. We develop a new variant of the Benders algorithm combined with a heuristic procedure that decomposes the domain of the upper-level discrete variables to solve the resulting DC-MPECs. We provide convergence theory as well as a number of numerical examples, some derived from energy applications, to validate the new method. It should be noted that the convergence theory applies if the heuristic procedure correctly identifies a decomposition of the domain so that the lower-level problem's optimal value function is convex. This is challenging but our numerical results are positive.

Suggested Citation

  • S A Gabriel & Y Shim & A J Conejo & S de la Torre & R García-Bertrand, 2010. "A Benders decomposition method for discretely-constrained mathematical programs with equilibrium constraints," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(9), pages 1404-1419, September.
  • Handle: RePEc:pal:jorsoc:v:61:y:2010:i:9:d:10.1057_jors.2009.84
    DOI: 10.1057/jors.2009.84
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    1. 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.
    2. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(4), pages 691-705, August.
    3. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(5), pages 879-883, October.
    4. X. M. Hu & D. Ralph, 2004. "Convergence of a Penalty Method for Mathematical Programming with Complementarity Constraints," Journal of Optimization Theory and Applications, Springer, vol. 123(2), pages 365-390, November.
    5. Gao, Ziyou & Sun, Huijun & Shan, Lian Long, 2004. "A continuous equilibrium network design model and algorithm for transit systems," Transportation Research Part B: Methodological, Elsevier, vol. 38(3), pages 235-250, March.
    6. Steven A. Gabriel & Andy S. Kydes & Peter Whitman, 2001. "The National Energy Modeling System: A Large-Scale Energy-Economic Equilibrium Model," Operations Research, INFORMS, vol. 49(1), pages 14-25, February.
    7. Bard, Jonathan F. & Plummer, John & Claude Sourie, Jean, 2000. "A bilevel programming approach to determining tax credits for biofuel production," European Journal of Operational Research, Elsevier, vol. 120(1), pages 30-46, January.
    8. Jonathan F. Bard, 1983. "An Efficient Point Algorithm for a Linear Two-Stage Optimization Problem," Operations Research, INFORMS, vol. 31(4), pages 670-684, August.
    9. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(6), pages 1195-1198, December.
    10. Wen, U. P. & Huang, A. D., 1996. "A simple Tabu Search method to solve the mixed-integer linear bilevel programming problem," European Journal of Operational Research, Elsevier, vol. 88(3), pages 563-571, February.
    11. Gabriel, Steven A. & Vikas, Shree & Ribar, David M., 2000. "Measuring the influence of Canadian carbon stabilization programs on natural gas exports to the United States via a 'bottom-up' intertemporal spatial price equilibrium model," Energy Economics, Elsevier, vol. 22(5), pages 497-525, October.
    12. M. Seetharama Gowda & Jong-Shi Pang, 1992. "On Solution Stability of the Linear Complementarity Problem," Mathematics of Operations Research, INFORMS, vol. 17(1), pages 77-83, February.
    13. Wayne F. Bialas & Mark H. Karwan, 1984. "Two-Level Linear Programming," Management Science, INFORMS, vol. 30(8), pages 1004-1020, August.
    14. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(1), pages 225-228, February.
    15. Garci'a-Bertrand, Raquel & Conejo, Antonio J. & Gabriel, Steven, 2006. "Electricity market near-equilibrium under locational marginal pricing and minimum profit conditions," European Journal of Operational Research, Elsevier, vol. 174(1), pages 457-479, October.
    16. ,, 2003. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 19(2), pages 411-413, April.
    17. James T. Moore & Jonathan F. Bard, 1990. "The Mixed Integer Linear Bilevel Programming Problem," Operations Research, INFORMS, vol. 38(5), pages 911-921, October.
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    Cited by:

    1. Yohan Shim & Marte Fodstad & Steven Gabriel & Asgeir Tomasgard, 2013. "A branch-and-bound method for discretely-constrained mathematical programs with equilibrium constraints," Annals of Operations Research, Springer, vol. 210(1), pages 5-31, November.
    2. Sarhadi, Hassan & Tulett, David M. & Verma, Manish, 2017. "An analytical approach to the protection planning of a rail intermodal terminal network," European Journal of Operational Research, Elsevier, vol. 257(2), pages 511-525.
    3. Murphy, Frederic & Pierru, Axel & Smeers, Yves, 2019. "Measuring the effects of price controls using mixed complementarity models," European Journal of Operational Research, Elsevier, vol. 275(2), pages 666-676.
    4. Steven Gabriel & Sauleh Siddiqui & Antonio Conejo & Carlos Ruiz, 2013. "Solving Discretely-Constrained Nash–Cournot Games with an Application to Power Markets," Networks and Spatial Economics, Springer, vol. 13(3), pages 307-326, September.
    5. Bai, Yun & Ouyang, Yanfeng & Pang, Jong-Shi, 2016. "Enhanced models and improved solution for competitive biofuel supply chain design under land use constraints," European Journal of Operational Research, Elsevier, vol. 249(1), pages 281-297.
    6. Losada, Chaya & Scaparra, M. Paola & O’Hanley, Jesse R., 2012. "Optimizing system resilience: A facility protection model with recovery time," European Journal of Operational Research, Elsevier, vol. 217(3), pages 519-530.
    7. Paul Berglund & Changhyun Kwon, 2014. "Solving a Location Problem of a Stackelberg Firm Competing with Cournot-Nash Firms," Networks and Spatial Economics, Springer, vol. 14(1), pages 117-132, March.
    8. Nima Rabiei & Jose Muñoz, 2015. "AAR-based decomposition algorithm for non-linear convex optimisation," Computational Optimization and Applications, Springer, vol. 62(3), pages 761-786, December.
    9. Bai, Yun & Ouyang, Yanfeng & Pang, Jong-Shi, 2012. "Biofuel supply chain design under competitive agricultural land use and feedstock market equilibrium," Energy Economics, Elsevier, vol. 34(5), pages 1623-1633.

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