Optimization Problems Constrained by Complementarity and Other Optimization Problems

  • Steven A. Gabriel
  • Antonio J. Conejo
  • J. David Fuller
  • Benjamin F. Hobbs
  • Carlos Ruiz
Part of the International Series in Operations Research & Management Science book series (ISOR, volume 180)


This chapter provides a friendly introduction to the analysis of optimization problems constrained by complementarity and other optimization problems. These problems are also known as bilevel problems [3], and the field of study to which they belong, hierarchical optimization. Throughout this chapter, we refer to them using the acronym OPcOPs, Optimization Problems constrained by other Optimization Problems, which explicitly indicates a hierarchy.


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  1. 1.
    D.P. Bertsekas. Nonlinear Programming. Athena Scientific, Belmont, Massachusetts, second edition, 1999.Google Scholar
  2. 2.
    G. Brown, M. Carlyle, J. Salmerón, and K. Wood. Defending critical infrastructure. Interfaces, 36(6):530–544, November 2006.CrossRefGoogle Scholar
  3. 3.
    B. Colson, P. Marcotte, and G. Savard. Bilevel programming: A survey. 4OR, 3(2):87–108, June 2005.Google Scholar
  4. 4.
    A.J. Conejo, E. Castillo, R. Mínguez, and R. García Bertrand. Decomposition Techniques in Mathematical Programming. Engineering and Science Applications. Springer, Heidelberg, Germany, 2006.Google Scholar
  5. 5.
    CPLEX. GAMS. The Solver Manuals. GAMS/CPLEX, 2010. Available at www.gams.com.
  6. 6.
    A.S. Drud. GAMS. The Solver Manuals. GAMS/CONOPT. ARKI Consulting and Development, Bagsvaerdvej 246A, DK-2880 Bagsvaerd, Denmark, 2010. Available at www.gams.com.
  7. 7.
    F. Facchinei and C. Kanzow. Generalized Nash equilibrium problems. 4OR, 5(3):173–210, September 2007.Google Scholar
  8. 8.
    J. Fortuny-Amat and B. McCarl. A representation and economic interpretation of a two-level programming problem. The Journal of the Operational Research Society, 32(9):783–792, September 1981.Google Scholar
  9. 9.
    L. Garcés, A.J. Conejo, R. García-Bertrand, and R. Romero. A bilevel approach to transmission expansion planning within a market environment. IEEE Transactions on Power Systems, 24(3):1513–1522, August 2009.CrossRefGoogle Scholar
  10. 10.
    P.E. Gill, W. Murray, M.A. Saunders, and A. Drud. GAMS. The Solver Manuals. GAMS/SNOPT, 2010. Available at www.gams.com.
  11. 11.
    D.G. Luenberger and Y. Ye. Linear and Nonlinear Programming. Springer, New York, third edition, 2008.Google Scholar
  12. 12.
    A. Mas-Colell, M.D. Whinston, and J.R. Green. Microeconomic Theory. Oxford University Press, New York, 1995.Google Scholar
  13. 13.
    A.L. Motto, J.M. Arroyo, and F.D. Galiana. A mixed-integer LP procedure for the analysis of electric grid security under disruptive threat. IEEE Transactions onPower Systems, 20(3):1357–1365, August 2005.CrossRefGoogle Scholar
  14. 14.
    J. Nash. Equilibrium points in n-person games. Proceedings of the National Academy of Sciences of the United States of America, 36(1):48–49, 1950.CrossRefGoogle Scholar
  15. 15.
    J. Nash. Non-cooperative games. The Annals of Mathematics. Second Series, 54(2):286–295, September 1951.CrossRefGoogle Scholar
  16. 16.
    C. Ruiz and A.J. Conejo. Pool strategy of a producer with endogenous formation of locational marginal prices. IEEE Transactions on Power Systems, 24(4):1855–1866, November 2009.CrossRefGoogle Scholar
  17. 17.
    J. Salmerón, K. Wood, and R. Baldick. Analysis of electric grid security under terrorist threat. IEEE Transactions on Power Systems, 19(2):905–912, May 2004.CrossRefGoogle Scholar
  18. 18.
    S. Siddiqui and S.A. Gabriel. An SOS1-based approach for solving MPECs with a natural gas market application, accepted at Networks and Spatial Economics, May 2012.Google Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Steven A. Gabriel
    • 1
  • Antonio J. Conejo
    • 2
  • J. David Fuller
    • 3
  • Benjamin F. Hobbs
    • 4
  • Carlos Ruiz
    • 5
  1. 1.Department of Civil and Environmental EngineeringUniversity of MarylandCollege ParkUSA
  2. 2.University of Castilla – La ManchaCiudad RealSpain
  3. 3.Department of Management SciencesUniversity of WaterlooWaterlooCanada
  4. 4.Department of Geography and Environmental EngineeringThe Johns Hopkins UniversityBaltimoreUSA
  5. 5.European Foundation for New Energy – EDF École Centrale Paris and SupélecChâtenay-MalabryFrance

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