Analytics-Based Decomposition of a Class of Bilevel Problems

Conference paper
Part of the Advances in Intelligent Systems and Computing book series (AISC, volume 991)


This paper proposes a new class of multi-follower bilevel problems. In this class the followers may be nonlinear, do not share constraints or variables, and are at most weakly constrained. This allows the leader variables to be partitioned among the followers. The new class is formalised and compared with existing problems in the literature. We show that approaches currently in use for solving multi-follower problems are unsuitable for this class. Evolutionary algorithms can be used, but these are computationally intensive and do not scale up well. Instead we propose an analytics-based decomposition approach. Two example problems are solved using our approach and two evolutionary algorithms, and the decomposition approach produces much better and faster results as the problem size increases.


Bilevel Analytics Clustering Decomposition 



This publication has emanated from research conducted with the financial support of Science Foundation Ireland (SFI) under Grant Number SFI/12/RC/2289.


  1. 1.
    de Amorim, R., Fenner, T.: Weighting Features for Partition Around Medoids Using the Minkowski Metric, pp. 35–44. Springer, Heidelberg (2012)Google Scholar
  2. 2.
    Angelo, J., Barbosa, H.: Differential evolution to find Stackelberg-Nash equilibrium in bilevel problems with multiple followers. In: IEEE Congress on Evolutionary Computation, CEC 2015, Sendai, Japan, May 25–28, 2015, pp. 1675–1682 (2015)Google Scholar
  3. 3.
    Bard, J.: Convex two-level optimization. Math. Program. 40(1), 15–27 (1988)Google Scholar
  4. 4.
    Calvete, H., Galé, C.: Linear bilevel multi-follower programming with independent followers. J. Glob. Optim. 39(3), 409–417 (2007)Google Scholar
  5. 5.
    Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Ann. Oper. Res. 153(1), 235–256 (2007)Google Scholar
  6. 6.
    Deb, K.: An efficient constraint handling method for genetic algorithms. Comput. Methods Appl. Mech. Eng. 186(2–4), 311–338 (2000)Google Scholar
  7. 7.
    DeMiguel, V., Xu, H.: A stochastic multiple-leader Stackelberg model: analysis, computation, and application. Oper. Res. 57(5), 1220–1235 (2009)Google Scholar
  8. 8.
    IBM: User’s manual of IBM CPLEX optimizer for z/OS: what is an indicator constraint? (2017). https://ibmco/2ErnDynGoogle Scholar
  9. 9.
    Islam, M., Singh, H., Ray, T.: A memetic algorithm for solving bilevel optimization problems with multiple followers. In: IEEE Congress on Evolutionary Computation, CEC 2016, Vancouver, BC, Canada, July 24–29, 2016, pp. 1901–1908 (2016)Google Scholar
  10. 10.
    Kaufman, L., Rousseeuw, P.: Finding Groups in Data: An Introduction to Cluster Analysis, vol. 344. Wiley (2009)Google Scholar
  11. 11.
    Liu, B.: Stackelberg-Nash equilibrium for multilevel programming with multiple followers using genetic algorithms. Comput. Math. Appl. 36(7), 79–89 (1998)Google Scholar
  12. 12.
    Lu, J., Han, J., Hu, Y., Zhang, G.: Multilevel decision-making: a survey. Inf. Sci. 346–347(Supplement C), 463 – 487 (2016).,
  13. 13.
    Lu, J., Shi, C., Zhang, G.: On bilevel multi-follower decision making: general framework and solutions. Inf. Sci. 176(11), 1607–1627 (2006)Google Scholar
  14. 14.
    Lu, J., Shi, C., Zhang, G., Dillon, T.: Model and extended Kuhn-Tucker approach for bilevel multi-follower decision making in a referential-uncooperative situation. J. Glob. Optim. 38(4), 597–608 (2007)Google Scholar
  15. 15.
    Lu, J., Shi, C., Zhang, G., Ruan, D.: Multi-follower linear bilevel programming: model and Kuhn-Tucker approach. In: AC 2005, Proceedings of the IADIS International Conference on Applied Computing, Algarve, Portugal, February 22–25, 2005, vol. 2, pp. 81–88 (2005)Google Scholar
  16. 16.
    Lu, J., Shi, C., Zhang, G., Ruan, D.: An extended branch and bound algorithm for bilevel multi-follower decision making in a referential-uncooperative situation. Int. J. Inf. Technol. Decis. Mak. 6(2), 371–388 (2007)Google Scholar
  17. 17.
    Maechler, M., Rousseeuw, P., Struyf, A., Hubert, M., Hornik, K.: cluster: Cluster Analysis Basics and Extensions (2017). R package version 2.0.6—for new features, see the ‘Changelog’ file (in the package source)Google Scholar
  18. 18.
    Marsaglia, G.: Choosing a point from the surface of a sphere. Ann. Math. Statist. 43(2), 645–646 (1972).
  19. 19.
    Muller, M.: A note on a method for generating points uniformly on n-dimensional spheres. Commun. ACM 2(4), 19–20 (1959)Google Scholar
  20. 20.
    Prestwich, S., Fajemisin, A., Climent, L., O’Sullivan, B.: Solving a Hard Cutting Stock Problem by Machine Learning and Optimisation, pp. 335–347. Springer International Publishing, Cham (2015)Google Scholar
  21. 21.
    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(Supplement C), 190–207 (2016).,
  22. 22.
    Shi, C., Lu, J., Zhang, G., Zhou, H.: An extended Kuhn-Tucker approach for linear bilevel multifollower programming with partial shared variables among followers. In: Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, Waikoloa, Hawaii, USA, October 10–12, 2005, pp. 3350–3357 (2005)Google Scholar
  23. 23.
    Shi, C., Zhang, G., Lu, J.: The Kth-best approach for linear bilevel multi-follower programming. J. Glob. Optim. 33(4), 563–578 (2005)Google Scholar
  24. 24.
    Shi, C., Zhou, H., Lu, J., Zhang, G., Zhang, Z.: The Kth-best approach for linear bilevel multifollower programming with partial shared variables among followers. Appl. Math. Comput. 188(2), 1686–1698 (2007)Google Scholar
  25. 25.
    Sinha, A., Malo, P., Frantsev, A., Deb, K.: Finding optimal strategies in a multi-period multi-leader-follower Stackelberg game using an evolutionary algorithm. Comput. Oper. Res. 41, 374–385 (2014)Google Scholar
  26. 26.
    Wei, C.P., Lee, Y.H., Hsu, C.M.: Empirical comparison of fast clustering algorithms for large data sets. In: Proceedings of the 33rd Annual Hawaii International Conference on System Sciences, pp. 10-pp. IEEE (2000)Google Scholar
  27. 27.
    Zhang, G., Lu, J.: Fuzzy bilevel programming with multiple objectives and cooperative multiple followers. J. Glob. Optim. 47(3), 403–419 (2010)Google Scholar
  28. 28.
    Zhang, G., Shi, C., Lu, J.: An extended Kth-best approach for referential-uncooperative bilevel multi-follower decision making. Int. J. Comput. Intell. Syst. 1(3), 205–214 (2008)Google Scholar

Copyright information

© Springer Nature Switzerland AG 2020

Authors and Affiliations

  1. 1.School of ComputingNational College of IrelandDublinIreland
  2. 2.Computer Science DepartmentCork Institute of TechnologyCorkIreland
  3. 3.Insight Centre for Data AnalyticsUniversity College CorkCorkIreland

Personalised recommendations