A Benders Decomposition Method for Solving Stochastic Complementarity Problems with an Application in Energy
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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.
KeywordsGame theory Optimization Stochastic programming Energy
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- 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
- Birge J. R., Louveaux F. (1997) Introduction to stochastic programming. Springer, New YorkGoogle Scholar
- Brooke, A., Kendrick, D., Meeraus, A., & Raman, R. (2008). GAMS—a user’s guide. GAMS Development Corporation, Washington, DC. http://www.gams.com/docs/document.htm.
- 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
- Cabero, J., Baillo, A., Ventosa, M., & Cerisola, S. (2005b). Application of Benders decomposition to an equilibrium problem. In 15th PSCC, Liege, August.Google Scholar
- Facchinei F., Pang J.-S. (2003) Finite-dimensional variational inequalities and complementarity problems. Springer, New YorkGoogle Scholar
- Ferris, M. C., & Munson, T. S. (2005). PATH 4.6. GAMS Development Corporation, Washington, DC. http://www.gams.com/docs/document.htm.
- 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
- Global Competition Review. (2003). Gas regulation in 26 jurisdictions worldwide. http://www.globalcompetitionreview.com.
- 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
- 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
- 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
- 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
- Shanbhag, U., Glynn, P., & Infanger, G. (2005). A complementarity framework for forward contracting under uncertainty. In INFORMS annual conference, San Francisco, November.Google Scholar
- Shanbhag, U., Infanger, G., & Glynn, P. (2008). A complementarity framework for forward contracting under uncertainty (under review).Google Scholar
- 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