Skip to main content

On Solving Bilevel Optimization Problems with a Nonconvex Lower Level: The Case of a Bimatrix Game

  • Conference paper
  • First Online:
Mathematical Optimization Theory and Operations Research (MOTOR 2021)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12755))

Abstract

This paper addresses the optimistic statement of one class of bilevel optimization problems (BOPs) with a nonconvex lower level. Namely, we study BOPs with a convex quadratic objective function at the upper level and with a bimatrix game at the lower level. It is known that the problem of finding a Nash equilibrium point in a bimatrix game is equivalent to the special nonconvex optimization problem with a bilinear structure. Nevertheless, we can replace such a lower level with its optimality conditions and transform the original bilevel problem into a single-level nonconvex optimization problem. Then we apply the original Global Search Theory (GST) for general D.C. optimization problems and the Exact Penalization Theory to the resulting problem. After that, a special method of local search, which takes into account the structure of the problem under consideration, is developed.

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

Access this chapter

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Similar content being viewed by others

References

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

    MATH  Google Scholar 

  2. Dempe, S., Zemkoho, A. (eds.): Bilevel Optimization: Advances and Next Challenges. Springer International Publishing, New York (2020). https://doi.org/10.1007/978-3-030-52119-6

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

    Article  MathSciNet  Google Scholar 

  4. Dempe, S.: Bilevel programming. In: Audet, C., Hansen, P., Savard, G. (eds.) Essays and Surveys in Global Optimization, pp. 165–193. Springer, Boston (2005). https://doi.org/10.1007/0-387-25570-2_6

    Chapter  Google Scholar 

  5. Dempe, S., Kalashnikov, V.V., Perez-Valdes, G.A., Kalashnykova, N.: Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks. Springer-Verlag, Berlin-Heidelberg (2015). https://doi.org/10.1007/978-3-662-45827-3

    Book  MATH  Google Scholar 

  6. Stackelberg, H.F.V.: Marktform und Gleichgewicht. Springer, Wien (1934). (in german)

    MATH  Google Scholar 

  7. Mitsos, A., Lemonidis, P., Barton, P.I.: Global solution of bilevel programs with a nonconvex inner program. J. Global Optim. 42, 475–513 (2008)

    Article  MathSciNet  Google Scholar 

  8. Lin, G.-H., Xu, M., Ye, J.J.: On solving simple bilevel programs with a nonconvex lower level program. Math. Program. 144, 277–305 (2013). https://doi.org/10.1007/s10107-013-0633-4

    Article  MathSciNet  MATH  Google Scholar 

  9. Zhu, X., Guo, P.: Approaches to four types of bilevel programming problems with nonconvex nonsmooth lower level programs and their applications to newsvendor problems. Math. Methods Oper. Res. 86(2), 255–275 (2017). https://doi.org/10.1007/s00186-017-0592-2

    Article  MathSciNet  MATH  Google Scholar 

  10. Hu, M., Fukushima, M.: Existence, uniqueness, and computation of robust Nash equilibria in a class of multi-leader-follower games. SIAM J. Optim. 23(2), 894–916 (2013)

    Article  MathSciNet  Google Scholar 

  11. Ramos, M., Boix, M., Aussel, D., Montastruc, L., Domenech, S.: Water integration in eco-industrial parks using a multi-leader-follower approach. Comput. Chem. Eng. 87, 190–207 (2016)

    Article  Google Scholar 

  12. Yang, Z., Ju, Y.: Existence and generic stability of cooperative equilibria for multi-leader-multi-follower games. J. Global Optim. 65(3), 563–573 (2015). https://doi.org/10.1007/s10898-015-0393-1

    Article  MathSciNet  MATH  Google Scholar 

  13. Mazalov, V.: Mathematical Game Theory and Applications. John Wiley & Sons, New York (2014)

    MATH  Google Scholar 

  14. Strekalovsky, A.S., Orlov, A.V.: Bimatrix Games and Bilinear Programming. FizMatLit, Moscow (2007). (in russian)

    Google Scholar 

  15. Orlov, A.V., Gruzdeva, T.V.: The local and global searches in bilevel problems with a matrix game at the lower level. In: Khachay, M., Kochetov, Y., Pardalos, P. (eds.) MOTOR 2019. LNCS, vol. 11548, pp. 172–183. Springer, Cham (2019). https://doi.org/10.1007/978-3-030-22629-9_13

    Chapter  MATH  Google Scholar 

  16. Törn, A., Žilinskas, A. (eds.): Global Optimization. LNCS, vol. 350. Springer, Heidelberg (1989). https://doi.org/10.1007/3-540-50871-6

    Book  MATH  Google Scholar 

  17. Strongin, R.G., Sergeyev, Ya.D.: Global Optimization with Non-convex Constraints. Sequential and Parallel Algorithms. Springer-Verlag, New York (2000). https://doi.org/10.1007/978-1-4615-4677-1

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

    Google Scholar 

  19. Nocedal, J., Wright, S.J.: Numerical Optimization. Springer-Verlag, New York (2000). https://doi.org/10.1007/978-0-387-40065-5

    Book  MATH  Google Scholar 

  20. Bonnans, J.-F., Gilbert, J.C., Lemarechal, C., Sagastizabal, C.A.: Numerical Optimization: Theoretical and Practical Aspects. Springer, Berlin-Heidelberg (2006). https://doi.org/10.1007/978-3-540-35447-5

    Book  MATH  Google Scholar 

  21. Strekalovsky, A.S.: Global optimality conditions and exact penalization. Optim. Lett. 13(3), 597–615 (2017). https://doi.org/10.1007/s11590-017-1214-x

    Article  MathSciNet  MATH  Google Scholar 

  22. Strekalovsky, A.S.: On a global search in D.C. optimization problems. In: Jaćimović, M., Khachay, M., Malkova, V., Posypkin, M. (eds.) OPTIMA 2019. CCIS, vol. 1145, pp. 222–236. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-38603-0_17

    Chapter  Google Scholar 

  23. Strekalovsky, A.S., Orlov, A.V.: Linear and Quadratic-linear Problems of Bilevel Optimization. SB RAS Publishing, Novosibirsk (2019). (in russian)

    Google Scholar 

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

    MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Strekalovsky, A.S., Orlov, A.V., Malyshev, A.V.: On computational search for optimistic solutions in bilevel problems. J. Global Optim. 48(1), 159–172 (2010)

    Article  MathSciNet  Google Scholar 

  28. Orlov, A.V., Strekalovsky, A.S., Batbileg, S.: On computational search for Nash equilibrium in hexamatrix games. Optim. Lett. 10(2), 369–381 (2014). https://doi.org/10.1007/s11590-014-0833-8

    Article  MathSciNet  MATH  Google Scholar 

  29. Orlov, A.V.: The global search theory approach to the bilevel pricing problem in telecommunication networks. In: Kalyagin, V.A., Pardalos, P.M., Prokopyev, O., Utkina, I. (eds.) NET 2016. SPMS, vol. 247, pp. 57–73. Springer, Cham (2018). https://doi.org/10.1007/978-3-319-96247-4_5

    Chapter  Google Scholar 

  30. Orlov, A.V.: On a solving bilevel D.C.-convex optimization problems. In: Kochetov, Y., Bykadorov, I., Gruzdeva, T. (eds.) MOTOR 2020. CCIS, vol. 1275, pp. 179–191. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-58657-7_16

    Chapter  Google Scholar 

  31. Strekalovsky, A.S., Orlov, A.V.: Global search for bilevel optimization with quadratic data. In: Dempe, S., Zemkoho, A. (eds.) Bilevel Optimization. SOIA, vol. 161, pp. 313–334. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-52119-6_11

    Chapter  Google Scholar 

  32. Mangasarian, O.L., Stone, H.: Two-person nonzero games and quadratic programming. J. Math. Anal. Appl. 9, 348–355 (1964)

    Article  MathSciNet  Google Scholar 

  33. Strekalovsky, A.S.: On local search in D.C. optimization problems. Appl. Math. Comput. 255, 73–83 (2015)

    Google Scholar 

  34. Tao, P.D., Souad, L.B.: Algorithms for solving a class of non convex optimization. Methods of subgradients. In: Hiriart-Urruty J.-B. (ed.) Fermat Days 85, pp. 249–271. Elservier Sience Publishers B.V., North Holland (1986)

    Google Scholar 

  35. Strekalovsky, A.S.: Local search for nonsmooth DC optimization with DC equality and inequality constraints. Accepted for publication. In: Bagirov, A. et al. (Eds.) Numerical Nonsmooth Optimization - State of the Art Algorithms (2020)

    Google Scholar 

  36. Ben-Tal, A., Nemirovski, A.: Non-Euclidean restricted memory level method for large-scale convex optimization. Math. Program. 102, 407–456 (2005)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgement

The research was funded by the Ministry of Education and Science of the Russian Federation within the framework of the project “Theoretical foundations, methods and high-performance algorithms for continuous and discrete optimization to support interdisciplinary research” (No. of state registration: 121041300065-9).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to A. V. Orlov .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2021 Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Orlov, A.V. (2021). On Solving Bilevel Optimization Problems with a Nonconvex Lower Level: The Case of a Bimatrix Game. In: Pardalos, P., Khachay, M., Kazakov, A. (eds) Mathematical Optimization Theory and Operations Research. MOTOR 2021. Lecture Notes in Computer Science(), vol 12755. Springer, Cham. https://doi.org/10.1007/978-3-030-77876-7_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-030-77876-7_16

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-77875-0

  • Online ISBN: 978-3-030-77876-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics