Skip to main content
Springer Nature Link
Log in

A primal nonsmooth reformulation for bilevel optimization problems

  • Full Length Paper
  • Series B
  • Published:
Mathematical Programming Submit manuscript

Abstract

The solution of bilevel optimization problems with possibly nondifferentiable upper objective functions and with smooth and convex lower-level problems is discussed. A new approximate one-level reformulation for the original problem is introduced. An algorithm based on this reformulation is developed that is proven to converge to a solution of the bilevel problem. Each iteration of the algorithm depends on the solution of a nonsmooth optimization problem and its implementation leverages recent advances on nonsmooth optimization algorithms, which are fundamental to obtain a practical method. Experimental work is performed in order to demonstrate some characteristics of the algorithm in practice.

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

Access this article

Log in via an institution

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

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7

Similar content being viewed by others

References

  1. Aliprantis, C.D., Border, K.C.: Correspondences. In: Infinite Dimensional Analysis: A Hitchhiker’s Guide, 2 edn., chap. 16, pp. 523–556. Springer, Berlin, Heidelberg (1999)

  2. Allende, G.B., Still, G.: Solving bilevel programs with the KKT-approach. Math. Program. 138(1–2), 309–332 (2013)

    MathSciNet  MATH  Google Scholar 

  3. Bard, J.F.: Practical Bilevel Optimization: Algorithms and Applications, vol. 30. Springer, US (1998)

    MATH  Google Scholar 

  4. Bard, J.F., Plummer, J., Sourie, J.C.: A bilevel programming approach to determining tax credits for biofuel production. Eur. J. Oper. Res. 120(1), 30–46 (2000)

    MATH  Google Scholar 

  5. Burke, J.V., Curtis, F.E., Lewis, A.S., Overton, M.L., Simões, L.E.A.: Gradient Sampling Methods for Nonsmooth Optimization, pp. 201–225. Springer International Publishing, Cham, Switzerland (2020)

  6. Burke, J.V., Lewis, A.S., Overton, M.L.: A robust gradient sampling algorithm for nonsmooth, nonconvex optimization. SIAM J. Optim. 15(3), 751–779 (2005)

    MathSciNet  MATH  Google Scholar 

  7. Cabot, A.: Proximal point algorithm controlled by a slowly vanishing term: applications to hierarchical minimization. SIAM J. Optim. 15(2), 555–572 (2005)

    MathSciNet  MATH  Google Scholar 

  8. Clarke, F.H.: Optimization and nonsmooth analysis, vol. 5. SIAM, Montreal, Canada (1990)

    MATH  Google Scholar 

  9. Colson, B.: BIPA – BIlevel Programming with Approximation methods – software guide and test problems. Tech. rep., Les Cahiers du GERAD, G-2002-37, Montreal, Canada (2002)

  10. Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Ann. Oper. Res. 153(1), 235–256 (2007)

    MathSciNet  MATH  Google Scholar 

  11. Constantin, I., Florian, M.: Optimizing frequencies in a transit network: a nonlinear bi-level programming approach. Int. Trans. Oper. Res. 2(2), 149–164 (1995)

    MATH  Google Scholar 

  12. Côté, J.P., Marcotte, P., Savard, G.: A bilevel modelling approach to pricing and fare optimisation in the airline industry. J. Rev. Pricing. Manag. 2(1), 23–36 (2003)

    Google Scholar 

  13. Curtis, F.E., Mitchell, T., Overton, M.L.: A BFGS-SQP method for nonsmooth, nonconvex, constrained optimization and its evaluation using relative minimization profiles. Optim. Method. Softw. 32(1), 148–181 (2017)

    MathSciNet  MATH  Google Scholar 

  14. Dempe, S.: Foundations of bilevel programming. Kluwer Academic Publishers, Dordrecht (2002)

    MATH  Google Scholar 

  15. Dempe, S., Dutta, J.: Is bilevel programming a special case of a mathematical program with complementarity constraints? Math. Program. 131(1–2), 37–48 (2012)

    MathSciNet  MATH  Google Scholar 

  16. Dempe, S., Franke, S.: On the solution of convex bilevel optimization problems. Comput. Optim. Appl. 63(3), 685–703 (2016)

    MathSciNet  MATH  Google Scholar 

  17. Dempe, S., Zemkoho, A.: Bilevel Optimization: Advances and Next Challenges. Springer International Publishing, Cham, Switzerland (2020)

    MATH  Google Scholar 

  18. Dempe, S., Zemkoho, A.B.: The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math. Program. 138(1–2), 447–473 (2013)

    MathSciNet  MATH  Google Scholar 

  19. Dutta, J., Dempe, S.: Bilevel programming with convex lower level problems. In: Dempe, S., Kalashnikov, V. (eds.) Optimization with Multivalued Mappings: Theory, Applications, and Algorithms, pp. 51–71. Springer, US, Boston, MA (2006)

    MATH  Google Scholar 

  20. Fliege, J., Tin, A., Zemkoho, A.: Gauss-Newton-type methods for bilevel optimization. Comput. Optim. Appl. 78(3), 793–824 (2021)

    MathSciNet  MATH  Google Scholar 

  21. Gibbons, R.: A primer in game theory. Harvester Wheatsheaf, New York (1992)

    MATH  Google Scholar 

  22. Guo, L., Lin, G.H., Jane, J.Y.: Solving mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 166(1), 234–256 (2015)

    MathSciNet  MATH  Google Scholar 

  23. Harder, F., Mehlitz, P., Wachsmuth, G.: Reformulation of the M-stationarity conditions as a system of discontinuous equations and its solution by a semismooth Newton method. SIAM J. Optim. 31(2), 1459–1488 (2021)

    MathSciNet  MATH  Google Scholar 

  24. Helou, E.S., Santos, S.A., Simões, L.E.A.: On the local convergence analysis of the gradient sampling method for finite max-functions. J. Optim. Theory Appl. 175(1), 137–157 (2017)

    MathSciNet  MATH  Google Scholar 

  25. Helou, E.S., Santos, S.A., Simões, L.E.A.: A fast gradient and function sampling method for finite-max functions. Comput. Optim. Appl. 71(3), 673–717 (2018)

    MathSciNet  MATH  Google Scholar 

  26. Helou, E.S., Santos, S.A., Simões, L.E.A.: A new sequential optimality condition for constrained nonsmooth optimization. SIAM J. Optim. 30(2), 1610–1637 (2020)

    MathSciNet  MATH  Google Scholar 

  27. Helou, E.S., Simões, L.E.A.: \(\epsilon \)-subgradient algorithms for bilevel convex optimization. Inverse Prob. 33(5), 055020 (2017)

    MathSciNet  MATH  Google Scholar 

  28. Helou Neto, E.S., De Pierro, A.R.: On perturbed steepest descent methods with inexact line search for bilevel convex optimization. Optimization 60(8–9), 991–1008 (2011)

    MathSciNet  MATH  Google Scholar 

  29. Ishizuka, Y., Aiyoshi, E.: Double penalty method for bilevel optimization problems. Ann. Oper. Res. 34(1), 73–88 (1992)

    MathSciNet  MATH  Google Scholar 

  30. Kunisch, K., Pock, T.: A bilevel optimization approach for parameter learning in variational models. SIAM J. Imag. Sci. 6(2), 938–983 (2013)

    MathSciNet  MATH  Google Scholar 

  31. Labbé, M., Marcotte, P., Savard, G.: A bilevel model of taxation and its application to optimal highway pricing. Manage. Sci. 44(12), 1608–1622 (1998)

    MATH  Google Scholar 

  32. Lampariello, L., Sagratella, S.: A bridge between bilevel programs and Nash games. J. Optim. Theory Appl. 174(2), 613–635 (2017)

    MathSciNet  MATH  Google Scholar 

  33. Lampariello, L., Sagratella, S.: Numerically tractable optimistic bilevel problems. Comput. Optim. Appl. 76, 277–303 (2020)

    MathSciNet  MATH  Google Scholar 

  34. Lewis, A.S., Overton, M.L.: Nonsmooth optimization via quasi-Newton methods. Math. Program. 141(1–2), 135–163 (2013)

    MathSciNet  MATH  Google Scholar 

  35. Ochs, P., Ranftl, R., Brox, T., Pock, T.: Bilevel optimization with nonsmooth lower level problems. In: International Conference on Scale Space and Variational Methods in Computer Vision, pp. 654–665. Springer (2015)

  36. Outrata, J.V.: A note on the usage of nondifferentiable exact penalties in some special optimization problems. Kybernetika 24(4), 251–258 (1988)

    MathSciNet  MATH  Google Scholar 

  37. Sabach, S., Shtern, S.: A first order method for solving convex bilevel optimization problems. SIAM J. Optim. 27(2), 640–660 (2017)

    MathSciNet  MATH  Google Scholar 

  38. Savard, G., Gauvin, J.: The steepest descent direction for the nonlinear bilevel programming problem. Oper. Res. Lett. 15(5), 265–272 (1994)

    MathSciNet  MATH  Google Scholar 

  39. Shehu, Y., Vuong, P.T., Zemkoho, A.: An inertial extrapolation method for convex simple bilevel optimization. Optim. Method. Softw. 36(1), 1–19 (2021)

    MathSciNet  MATH  Google Scholar 

  40. Solodov, M.V.: A bundle method for a class of bilevel nonsmooth convex minimization problems. SIAM J. Optim. 18(1), 242–259 (2007)

    MathSciNet  MATH  Google Scholar 

  41. Vicente, L., Savard, G., Júdice, J.: Descent approaches for quadratic bilevel programming. J. Optim. Theory Appl. 81(2), 379–399 (1994)

    MathSciNet  MATH  Google Scholar 

  42. Wu, J., Zhang, L., Zhang, Y.: An inexact Newton method for stationary points of mathematical programs constrained by parameterized quasi-variational inequalities. Num. Algor. 69(4), 713–735 (2015)

    MathSciNet  MATH  Google Scholar 

  43. Ye, J.J., Zhu, D.: New necessary optimality conditions for bilevel programs by combining the MPEC and value function approaches. SIAM J. Optim. 20(4), 1885–1905 (2010)

    MathSciNet  MATH  Google Scholar 

  44. Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 33(1), 9–27 (1995)

    MathSciNet  MATH  Google Scholar 

  45. Zemkoho, A.B., Zhou, S.: Theoretical and numerical comparison of the Karush-Kuhn-Tucker and value function reformulations in bilevel optimization. Comput. Optim. Appl. 78(2), 625–674 (2021)

    MathSciNet  MATH  Google Scholar 

  46. Zhou, S., Zemkoho, A.B., Tin, A.: Bolib: Bilevel Optimization LIBrary of test problems. In: Dempe, S., Zemkoho, A. (eds.) Bilevel Optimization: Advances and Next Challenges, pp. 563–580. Springer International Publishing, Cham (2020)

    MATH  Google Scholar 

Download references

Acknowledgements

This work was partially funded by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq) grants 310893/2019-4 and 305010/2020-4 and Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP) grants 2018/24293-0, 2016/22989-2 and 2013/07375-0. The authors are thankful to the anonymous referees, who provided insightful comments that improved the presentation of this work.

Author information

Authors and Affiliations

  1. University of São Paulo, São Carlos, SP, Brazil

    Elias S. Helou

  2. University of Campinas, Campinas, SP, Brazil

    Sandra A. Santos & Lucas E. A. Simões

Authors
  1. Elias S. Helou
    View author publications

    You can also search for this author inPubMed Google Scholar

  2. Sandra A. Santos
    View author publications

    You can also search for this author inPubMed Google Scholar

  3. Lucas E. A. Simões
    View author publications

    You can also search for this author inPubMed Google Scholar

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Helou, E.S., Santos, S.A. & Simões, L.E.A. A primal nonsmooth reformulation for bilevel optimization problems. Math. Program. 198, 1381–1409 (2023). https://doi.org/10.1007/s10107-021-01764-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-021-01764-6

Keywords

Mathematics Subject Classification