Skip to main content
SpringerLink
Log in

A Nonconvex Optimization Approach to Quadratic Bilevel Problems

  • Conference paper
  • First Online:
Learning and Intelligent Optimization (LION 2017)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 10556))

Included in the following conference series:

Abstract

This paper addresses one of the classes of bilevel optimization problems in their optimistic statement. The reduction of the bilevel problem to a series of nonconvex mathematical optimization problems, together with the specialized Global Search Theory, is used for developing methods of local and global searches to find optimistic solutions. Illustrative examples show that the approach proposed is prospective and performs well.

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

Access this chapter

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 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
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

Institutional subscriptions

References

  1. Bard, J.F.: Convex two-level optimization. Math. Prog. 40, 15–27 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bazara, M.S., Shetty, C.M.: Nonlinear Programming. Theory and Algorithms. Wiley, New York (1979)

    Google Scholar 

  3. Bonnans, J.-F., Gilbert, J.C., Lemarechal, C., Sagastizabal, C.A.: Numerical Optimization: Theoretical and Practical Aspects. Springer, Heidelberg (2006)

    MATH  Google Scholar 

  4. Calamai, P., Vicente, L.: Generating quadratic bilevel programming test problems. ACM Trans. Math. Softw. 20, 103–119 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Colson, B., Marcotte, P., Savard, G.: A trust-region method for nonlinear bilevel programming: algorithm and computational experience. Comput. Optim. Appl. 30, 211–227 (2005)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    MATH  Google Scholar 

  8. Dempe, S.: Bilevel programming. In: Audet, C., Hansen, P., Savard, G. (eds.) Essays and Surveys in Global Optimization, pp. 165–193. Springer, Boston (2005)

    Chapter  Google Scholar 

  9. Dempe, S., Kalashnikov, V.V., Perez-Valdes, G.A., Kalashnykova, N.: Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks. Springer, Heidelberg (2015)

    Book  MATH  Google Scholar 

  10. Etoa, J.B.E.: Solving quadratic convex bilevel programming problems using a smoothing method. Appl. Math. Comput. 217, 6680–6690 (2011)

    MATH  MathSciNet  Google Scholar 

  11. Gruzdeva, T.V., Petrova, E.G.: Numerical solution of a linear bilevel problem. Comp. Math. Math. Phys. 50, 1631–1641 (2010)

    Article  MATH  Google Scholar 

  12. Gumus, Z.H., Floudas, C.A.: Global optimization of nonlinear bilevel programming problems. J. Glob. Optim. 20, 1–31 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  13. MATLAB—The language of technical computing. http://www.mathworks.com/products/matlab/

  14. Muu, L.D., Quy, N.V.: A global optimization method for solving convex quadratic bilevel programming problems. J. Glob. Optim. 26, 199–219 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  15. Orlov, A.V.: Numerical solution of bilinear programming problems. Comput. Math. Math. Phys. 48, 225–241 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  16. Orlov, A.V., Strekalovsky, A.S.: Numerical search for equilibria in bimatrix games. Comput. Math. Math. Phys. 45, 947–960 (2005)

    MathSciNet  Google Scholar 

  17. Pang, J.-S.: Three modeling paradigms in mathematical programming. Math. Prog. Ser. B. 125, 297–323 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  18. Pistikopoulos, E.N., Dua, V., Ryu, J.-H.: Global optimization of bilevel programming problems via parametric programming. In: Floudas, C.A., Pardalos, P.M. (eds.) Frontiers in Global Optimization, pp. 457–476. Kluwer Academic Publishers, Dordrecht (2004)

    Chapter  Google Scholar 

  19. Saboia, C.H., Campelo, M., Scheimberg, S.: A computational study of global algorithms for linear bilevel programming. Numer. Algorithms 35, 155–173 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Strekalovsky, A.S.: Elements of Nonconvex Optimization. Nauka, Novosibirsk (2003). [in Russian]

    Google Scholar 

  21. Strekalovsky, A.S.: On solving optimization problems with hidden nonconvex structures. In: Rassias, T.M., Floudas, C.A., Butenko, S. (eds.) Optimization in Science and Engineering, pp. 465–502. Springer, New York (2014). doi:10.1007/978-1-4939-0808-0_23

    Chapter  Google Scholar 

  22. Strekalovsky, A.S., Orlov, A.V.: Bimatrix Games and Bilinear Programming. FizMatLit, Moscow (2007). [in Russian]

    Google Scholar 

  23. Strekalovsky, A.S., Orlov, A.V., Malyshev, A.V.: On computational search for optimistic solution in bilevel problems. J. Glob. Optim. 48, 159–172 (2010)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Acknowledgments

This work has been supported by the Russian Science Foundation (Project no. 15-11-20015).

Author information

Authors and Affiliations

  1. Matrosov Institute for System Dynamics and Control Theory of Siberian Branch of RAS, Irkutsk, Russia

    Andrei Orlov

Authors
  1. Andrei Orlov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Andrei Orlov .

Editor information

Editors and Affiliations

  1. University of Trento, Trento, Italy

    Roberto Battiti

  2. University of Calabria, Rende, Italy

    Dmitri E. Kvasov

  3. University of Calabria, Rende, Italy

    Yaroslav D. Sergeyev

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer International Publishing AG

About this paper

Cite this paper

Orlov, A. (2017). A Nonconvex Optimization Approach to Quadratic Bilevel Problems. In: Battiti, R., Kvasov, D., Sergeyev, Y. (eds) Learning and Intelligent Optimization. LION 2017. Lecture Notes in Computer Science(), vol 10556. Springer, Cham. https://doi.org/10.1007/978-3-319-69404-7_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-69404-7_16

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-69403-0

  • Online ISBN: 978-3-319-69404-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation