Skip to main content
SpringerLink
Log in
Book cover

Operations Research Proceedings 2014 pp 575–580Cite as

Global Solution of Bilevel Programming Problems

  • Conference paper
  • First Online:

Part of the book series: Operations Research Proceedings ((ORP))

Abstract

We discuss the global solution of Bilevel Programming Problems using their reformulations as Mathematical Programs with Complementarity Constraints and/or Mixed Integer Nonlinear Programs. We show that under suitable assumptions the Bilevel Program can be reformulated and globally solved via MINLP refomulation. We also briefly discuss some simplifications and suitable additional constraints.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD   169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

References

  1. Bard, J.F.: Practical Bilevel Optimization: Algorithms and Applications. Kluwer Academic Publishers, Dordrecht (1998)

    Book  Google Scholar 

  2. Belotti, P., Kirches, Ch., Leyffer, S., Linderoth, J., Luedtke, J., Mahajan, A.: Mixed-Integer nonlinear optimization. Acta Numerica 22, 1–131 (2013)

    Article  Google Scholar 

  3. Bonami, P., Biegler, L.T., Conn, A.R., Cornujols, G., Grossmann, I.E., Laird, C.D., Lee, J., Lodi, A., Margot, F., Sawaya, N., Wchter, A.: An algorithmic framework for convex mixed integer nonlinear programs. Discret. Optim. 5(2), 186–204 (2008)

    Article  Google Scholar 

  4. Bracken, J., McGill, J.: Mathematical programs with optimization problems in the constraints. Oper. Res. 21, 3744 (1973)

    Google Scholar 

  5. Bracken, J., McGill, J.: Defense applications of mathematical programs with optimization problems in the constraints. Oper. Res. 22, 1086–1096 (1974)

    Google Scholar 

  6. Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Ann. Oper. Res. 153, 235–256 (2007)

    Article  Google Scholar 

  7. Ct, J.-P., Marcotte, P., Savard, G.: A bilevel modeling approach to pricing and fare optimization in the airline industry. J. Revenue Pricing Manage. 2, 23–36 (2003). Dempe, S.: A necessary and a sufficient optimality condition for bilevel programming (1992a)

    Google Scholar 

  8. Dempe, S.: Foundations of Bilevel Programming. Kluwer Academic Publishers, Dordrecht (2002)

    Google Scholar 

  9. Dempe, S., Dutta, J.: Is Bilevel programming a special case of mathematical programs with complementarity constraints?. Math. Program. Series A, 131, 37–48 (2012)

    Google Scholar 

  10. de Miguel, A.V., Friedlander, M.P., Nogales, F.J., Scholtes, S.: A two-sided relaxation scheme for mathematical programs with equilibrium constraints. SIAM J. Optim. 16, 587–609 (2005)

    Article  Google Scholar 

  11. Fletcher, R., Leyffer, S., Ralph, D., Scholtes, S.: Local convergence of SQP-methods for mathematical programs with equilibrium constraints. SIAM J. Optim. 17(1), 259–286 (2006)

    Article  Google Scholar 

  12. Garcia, R., Marin, A.: Parking Capacity and pricing in parkn ride trips: a continuous equilibrium network design problem. Ann. Oper. Res. 116, 153–178 (2002)

    Google Scholar 

  13. Grossmann, I.E.: Review of nonlinear mixed-integer and disjunctive programming techniques. Optim. Eng. 3, 227–252 (2002)

    Article  Google Scholar 

  14. Hu, X.M., Ralph, D.: Convergence of a penalty method for mathematical programming with complementarity constraints. J. Optim. Theory Appl. 123(2), 365–390 (2004)

    Article  Google Scholar 

  15. Mitsos, A., Lemonidis, P., Barton, P.I.: Globla solution of bilevel programs with a nonconvex inner program. J. Optim. Theory Appl. 123(2), 365–390 (2004)

    Article  Google Scholar 

  16. Raghunathan, A.U., Biegler, L.T.: Interior point methods for Mathematical Programs with Complementarity Constraints (MPCCs). SIAM J. Optim. 15(3), 720–750 (2005)

    Article  Google Scholar 

  17. Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity. Optim. Sensit. Math. Oper. Res. 25, 1–22 (2000)

    Article  Google Scholar 

  18. Scholtes, S.: Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J. Optim. 11(4), 918–936 (2001)

    Article  Google Scholar 

  19. Steffensen, S., Ulbrich, M.: A new relaxation scheme for mathematical programs with equilibrium constraints. SIAM J. Optim. (2010)

    Google Scholar 

  20. Tawarmalani, M., Sahinidis, N.V.: Global optimization of mixed-integer nonlinear programs: a theoretical and computational study. Math. Program. 99(3), 563–591 (2004)

    Article  Google Scholar 

  21. Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307, 350–369 (2005)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, RWTH Aachen University, Templergraben 55, D-52062, Aachen, Germany

    Sonja Steffensen

Authors
  1. Sonja Steffensen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Sonja Steffensen .

Editor information

Editors and Affiliations

  1. Operations Research, RWTH Aachen University, Aachen, Germany

    Marco Lübbecke

  2. Lehrstuhl II für Mathematik, RWTH Aachen University, Aachen, Germany

    Arie Koster

  3. RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Peter Letmathe

  4. E.ON Energy Research Center, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Reinhard Madlener

  5. Management Science, RWTH Aachen University, Aachen, Nordrhein-Westfalen, Germany

    Britta Peis

  6. Chair of Operations Management, RWTH Aachen University, Aachen, Germany

    Grit Walther

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Steffensen, S. (2016). Global Solution of Bilevel Programming Problems. In: Lübbecke, M., Koster, A., Letmathe, P., Madlener, R., Peis, B., Walther, G. (eds) Operations Research Proceedings 2014. Operations Research Proceedings. Springer, Cham. https://doi.org/10.1007/978-3-319-28697-6_80

Download citation

Publish with us

Policies and ethics

Search

Navigation