Computational Economics

, Volume 35, Issue 4, pp 301–329 | Cite as

A Benders Decomposition Method for Solving Stochastic Complementarity Problems with an Application in Energy

  • S. A. Gabriel
  • J. D. Fuller


In this paper we present a new Benders decomposition method for solving stochastic complementarity problems based on the work by Fuller and Chung (Comput Econ 25:303–326, 2005; Eur J Oper Res 185(1):76–91, 2007). A master and subproblem are proposed both of which are in the form of a complementarity problem or an equivalent variational inequality. These problems are solved iteratively until a certain convergence gap is sufficiently close to zero. The details of the method are presented as well as an extension of the theory from Fuller and Chung (2005, 2007). In addition, extensive numerical results are provided based on an electric power market model of Hobbs (IEEE Trans Power Syst 16(2):194–202, 2001) but for which stochastic elements have been added. These results validate the approach and indicate dramatic improvements in solution times as compared to solving the extensive form of the problem directly.


Game theory Optimization Stochastic programming Energy 


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  1. Ahn B. H., Hogan W. W. (1982) On convergence of the PIES algorithm for computing equilibria. Operations Research 30(2): 281–300CrossRefGoogle Scholar
  2. Belknap, M. H., Chen, C. H., & Harker, P. T. (2000). A gradient-based method for analyzing stochastic variational inequalities with one uncertain parameter. OPIM Working Paper 00-03-13. Department of Operations and Information Management, Wharton School, March.Google Scholar
  3. Benders J. F. (1962) Partitioning procedures for solving mixed-variables programming problems. Numerische Mathematik 4: 238–252CrossRefGoogle Scholar
  4. Birge J. R., Louveaux F. (1997) Introduction to stochastic programming. Springer, New YorkGoogle Scholar
  5. Brooke, A., Kendrick, D., Meeraus, A., & Raman, R. (2008). GAMS—a user’s guide. GAMS Development Corporation, Washington, DC.
  6. Cabero, J., Baillo, A., Cerisola, S., & Ventosa, M. (2005a). Electricity market equilibrium model with risk constraints via Benders decomposition. In INFORMS annual conference, San Francisco, November.Google Scholar
  7. Cabero, J., Baillo, A., Ventosa, M., & Cerisola, S. (2005b). Application of Benders decomposition to an equilibrium problem. In 15th PSCC, Liege, August.Google Scholar
  8. Chen Y., Hobbs B. F. (2005) An oligopolistic power market model with tradable Nox permits. IEEE Transactions on Power Systems 20(1): 119–129CrossRefGoogle Scholar
  9. Dantzig G., Wolfe P. (1961) The decomposition algorithm for linear programs. Econometrica 29(4): 767–778CrossRefGoogle Scholar
  10. De Wolf D., Smeers Y. (1997) A stochastic version of a Stackelberg–Nash–Cournot equilibrium model. Management Science 43(2): 190–197CrossRefGoogle Scholar
  11. Dupačová J., Grőwe-Kuska N., Rőmisch W. (2003) Scenario reduction in stochastic programming. Mathematical Programming 93(3): 493–511CrossRefGoogle Scholar
  12. Egging R., Gabriel S. A. (2006) Examining market power in the European natural gas market. Energy Policy 34(17): 2762–2778CrossRefGoogle Scholar
  13. Egging R., Gabriel S. A., Holz F., Zhuang J. (2008) A complementarity model for the European natural gas market. Energy Policy 36: 2385–2414CrossRefGoogle Scholar
  14. Facchinei F., Pang J.-S. (2003) Finite-dimensional variational inequalities and complementarity problems. Springer, New YorkGoogle Scholar
  15. Ferris, M. C., & Munson, T. S. (2005). PATH 4.6. GAMS Development Corporation, Washington, DC.
  16. Fuller J. D., Chung W. (2005) Dantzig–Wolfe decomposition of variational inequalities. Computational Economics 25: 303–326CrossRefGoogle Scholar
  17. Fuller J. D., Chung W. (2007) Benders decomposition for a class of variational inequalities. European Journal of Operational Research 185(1): 76–91CrossRefGoogle Scholar
  18. Gabriel, S. A., & Fuller, J. D. (2008). A Benders method for solving stochastic complementarity problems with an application in energy. In CORS/optimization days joint conference, Quebec City, May.Google Scholar
  19. Gabriel S. A., Kiet S., Zhuang J. (2005a) A mixed complementarity-based equilibrium model of natural gas markets. Operations Research 53(5): 799–818CrossRefGoogle Scholar
  20. Gabriel S. A., Zhuang J., Egging R. (2009) Solving stochastic complementarity problems in energy market modeling using scenario reduction. European Journal of Operational Research 197(3): 1028–1040CrossRefGoogle Scholar
  21. Gabriel S. A., Zhuang J., Kiet S. (2005b) A large-scale complementarity model of the North American natural gas market. Energy Economics 27: 639–665CrossRefGoogle Scholar
  22. García-Bertrand R., Conejo A. J., Gabriel S. A. (2006) Electricity market near-equilibrium under locational marginal pricing and minimum profit conditions. European Journal of Operational Research 174: 457–479CrossRefGoogle Scholar
  23. Genc T. S., Reynolds S. S., Sen S. (2007) Dynamic oligopolistic games under uncertainty: A stochastic programming approach. Journal of Economic Dynamics and Control 31(1): 55–80CrossRefGoogle Scholar
  24. Global Competition Review. (2003). Gas regulation in 26 jurisdictions worldwide.
  25. Gröwe-Kuska, N., Heitsch, H., & Römisch, W. (2003). Scenario reduction and scenario tree construction for power management problems. In A. Borghetti, C. A. Nucci, & M. Paolone (Eds.), IEEE Bologna Power tech proceedings.Google Scholar
  26. Gürkan G., Özge A. Y., Robinson S. M. (1999) Sample-path solution of stochastic variational inequalities. Mathematical Programming 84: 313–333CrossRefGoogle Scholar
  27. Haurie A., Moresino F. (2002) S-Adapted oligopoly equilibria and approximations in stochastic variational inequalities. Annals of Operations Research 114: 183–201CrossRefGoogle Scholar
  28. Haurie, A. & Zaccour, G. (2005). S-Adapted equilibria in games played over event trees: An overview. In A.S. Nowak et al. (Eds.), Advances in dynamic games. Annals of the International Society of Dynamic Games, 7, 417–444.Google Scholar
  29. Haurie, A., Zaccour, G., Legrand, J., & Smeers, Y. (1987). A stochastic dynamic Nash–Cournot model for the European gas market. Technical Report G-86-24, GERAD, Ecole des Hautes Etudes Commerciales, Montréal, Québec, Canada.Google Scholar
  30. Heitsch H., Römisch W. (2003) Scenario reduction algorithms in stochastic programming. Computational Optimization and Applications 24: 187–206CrossRefGoogle Scholar
  31. Hobbs B. F. (2001) Linear complementarity models of Nash–Cournot competition in bilateral and POOLCO power markets. IEEE Transactions on Power Systems 16(2): 194–202CrossRefGoogle Scholar
  32. Metzler C., Hobbs B. F., Pang J.-S. (2003) Nash–Cournot equilibria in power markets on a linearized DC network with arbitrage: Formulations and properties. Networks & Spatial Economics 3(2): 123–150CrossRefGoogle Scholar
  33. Nocedal J., Wright S. J. (1999) Numerical optimization. Springer, New YorkCrossRefGoogle Scholar
  34. Römisch, W., Dupačová, J., Gröwe-Kuska, N., & Heitsch, H. (2003). Approximations of stochastic programs. Scenario tree reduction and construction. GAMS Workshop, Heidelberg, September 1–3, Berlin: DFG Research Center.Google Scholar
  35. Shanbhag, U., Glynn, P., & Infanger, G. (2005). A complementarity framework for forward contracting under uncertainty. In INFORMS annual conference, San Francisco, November.Google Scholar
  36. Shanbhag, U., Infanger, G., & Glynn, P. (2008). A complementarity framework for forward contracting under uncertainty (under review).Google Scholar
  37. Waller, S. T. (2000). Optimization and control of stochastic dynamic transportation systems: Formulations, solution methodologies, and computational experience. Ph.D. Dissertation, Northwestern University, Chicago.Google Scholar
  38. Yao J., Adler I., Oren S.S. (2008) Modeling and computing two-settlement oligopolistic equilibrium in a congested electricity network. Operations Research 56(1): 34–47CrossRefGoogle Scholar
  39. Zhuang J., Gabriel S. A. (2008) A complementarity model for solving stochastic natural gas market equilibria. Energy Economics 30(1): 113–147CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2010

Authors and Affiliations

  1. 1.Department of Civil and Environmental EngineeringUniversity of MarylandCollege ParkUSA
  2. 2.Department of Management Sciences, Faculty of EngineeringUniversity of WaterlooWaterlooCanada

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