A projection-based reformulation and decomposition algorithm for global optimization of a class of mixed integer bilevel linear programs

  • Dajun Yue
  • Jiyao Gao
  • Bo Zeng
  • Fengqi You


We propose an extended variant of the reformulation and decomposition algorithm for solving a special class of mixed-integer bilevel linear programs (MIBLPs) where continuous and integer variables are involved in both upper- and lower-level problems. In particular, we consider MIBLPs with upper-level constraints that involve lower-level variables. We assume that the inducible region is nonempty and all variables are bounded. By using the reformulation and decomposition scheme, an MIBLP is first converted into its equivalent single-level formulation, then computed by a column-and-constraint generation based decomposition algorithm. The solution procedure is enhanced by a projection strategy that does not require the relatively complete response property. To ensure its performance, we prove that our new method converges to the global optimal solution in a finite number of iterations. A large-scale computational study on random instances and instances of hierarchical supply chain planning are presented to demonstrate the effectiveness of the algorithm.


Mixed-integer bilevel linear program Global optimization Single-level reformulation Reformulation and decomposition method Projection Hierarchical supply chain planning 



We greatly appreciate the helpful discussions with Professor Andreas Wächter at Department of Industrial Engineering and Management Sciences at Northwestern University. The paper has been greatly improved by the insightful and constructive feedback from the associate editor and three anonymous reviewers. The authors acknowledge financial support from National Science Foundation (NSF) CAREER Award (CBET-1643244).


  1. 1.
    Bard, J.F.: Practical Bilevel Optimization: Algorithm and Applications. Kluwer Academic Publishers, Dordrecht (1998)CrossRefzbMATHGoogle Scholar
  2. 2.
    Talbi, E.-G.: A taxonomy of metaheuristics for bi-level optimization. In: Talbi, E.-G. (ed.) Metaheuristics for Bi-level Optimization, pp. 1–39. Springer, Berlin (2013)CrossRefGoogle Scholar
  3. 3.
    Wiesemann, W., Tsoukalas, A., Kleniati, P.-M., Rustem, B.: Pessimistic bilevel optimization. SIAM J. Optim. 23(1), 353–380 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Audet, C., Hansen, P., Jaumard, B., Savard, G.: Links between linear bilevel and mixed 0–1 programming problems. J. Optim. Theory Appl. 93(2), 273–300 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    von Stackelberg, H.: Marktform und Gleichgewicht. Springer, Berlin (1934)Google Scholar
  6. 6.
    Bracken, J., McGill, J.T.: Mathematical programs with optimization problems in the constraints. Oper. Res. 21(1), 37–44 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Xu, P., Wang, L.: An exact algorithm for the bilevel mixed integer linear programming problem under three simplifying assumptions. Comput. Oper. Res. 41, 309–318 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Moore, J.T., Bard, J.F.: The mixed integer linear bilevel programming problem. Oper. Res. 38(5), 911–921 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Tang, Y., Richard, J.-P.P., Smith, J.C.: A class of algorithms for mixed-integer bilevel min–max optimization. J. Global Optim. 66(2), 225–262 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gümüş, Z.H., Floudas, C.A.: Global optimization of mixed-integer bilevel programming problems. CMS 2(3), 181–212 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Domínguez, L.F., Pistikopoulos, E.N.: Multiparametric programming based algorithms for pure integer and mixed-integer bilevel programming problems. Comput. Chem. Eng. 34(12), 2097–2106 (2010)CrossRefGoogle Scholar
  12. 12.
    Kleniati, P.-M., Adjiman, C.S.: A generalization of the branch-and-sandwich algorithm: from continuous to mixed-integer nonlinear bilevel problems. Comput. Chem. Eng. 72, 373–386 (2015)CrossRefGoogle Scholar
  13. 13.
    Mitsos, A.: Global solution of nonlinear mixed-integer bilevel programs. J. Global Optim. 47(4), 557–582 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Fliscounakis, S., Panciatici, P., Capitanescu, F., Wehenkel, L.: Contingency ranking with respect to overloads in very large power systems taking into account uncertainty, preventive, and corrective actions. IEEE Trans. Power Syst. 28(4), 4909–4917 (2013)CrossRefGoogle Scholar
  15. 15.
    De Negre, S.T., Ralphs, T.K.: A Branch-and-cut algorithm for integer bilevel linear programs. In: Chinneck, J.W., Kristjansson, B., Saltzman, M.J. (eds.) Operations research and cyber-infrastructure, pp. 65–78. Springer, Boston (2009). Google Scholar
  16. 16.
    Hemmati, M., Smith, J.C.: A mixed-integer bilevel programming approach for a competitive prioritized set covering problem. Discrete Optim. 20, 105–134 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Wen, U.P., Yang, Y.H.: Algorithms for solving the mixed integer two-level linear programming problem. Comput. Oper. Res. 17(2), 133–142 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lozano, L., Smith, J.C.: A value-function-based exact approach for the bilevel mixed-integer programming problem. Oper. Res. (2017). MathSciNetzbMATHGoogle Scholar
  19. 19.
    Dempe, S.: Discrete Bilevel Optimization Problems. Citeseer (2001)Google Scholar
  20. 20.
    Köppe, M., Queyranne, M., Ryan, C.T.: Parametric integer programming algorithm for bilevel mixed integer programs. J. Optim. Theory Appl. 146(1), 137–150 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Fischetti, M., Ljubić, I., Monaci, M., Sinnl, M.: Intersection cuts for bilevel optimization. In: Louveaux, Q., Skutella, M. (eds.) Integer Programming and Combinatorial Optimization: 18th International Conference, IPCO 2016, Liège, Belgium, June 1–3, 2016, Proceedings, pp. 77–88. Springer, Cham (2016)Google Scholar
  22. 22.
    Fischetti, M., Ljubic, I., Monaci, M., Sinnl, M.: A New General-Purpose Algorithm for Mixed-Integer Bilevel Linear Programs. (2016)
  23. 23.
    Zeng, B., An, Y.: Solving Bilevel Mixed Integer Program by Reformulations and Decomposition. (2014)
  24. 24.
    Florensa, C., Garcia-Herreros, P., Misra, P., Arslan, E., Mehta, S., Grossmann, I.E.: Capacity planning with competitive decision-makers: trilevel MILP formulation, degeneracy, and solution approaches. Eur. J. Oper. Res. 262(2), 449–463 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Mersha, A.G., Dempe, S.: Linear bilevel programming with upper level constraints depending on the lower level solution. Appl. Math. Comput. 180(1), 247–254 (2006)MathSciNetzbMATHGoogle Scholar
  26. 26.
    Bard, J.F., Moore, J.T.: An algorithm for the discrete bilevel programming problem. NRL 39(3), 419–435 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Saharidis, G.K., Ierapetritou, M.G.: Resolution method for mixed integer bi-level linear problems based on decomposition technique. J. Global Optim. 44(1), 29–51 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Poirion, P.-L., Toubaline, S., Ambrosio, C.D., Liberti, L.: Bilevel mixed-integer linear programs and the zero forcing set. Optimization (2015) (online)Google Scholar
  29. 29.
    Edmunds, T., Bard, J.: An algorithm for the mixed-integer nonlinear bilevel programming problem. Ann. Oper. Res. 34(1), 149–162 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Faísca, N., Dua, V., Rustem, B., Saraiva, P., Pistikopoulos, E.: Parametric global optimisation for bilevel programming. J. Global Optim. 38(4), 609–623 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Mitsos, A., Lemonidis, P., Barton, P.I.: Global solution of bilevel programs with a nonconvex inner program. J. Global Optim. 42(4), 475–513 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Kleniati, P.-M., Adjiman, C.: Branch-and-Sandwich: a deterministic global optimization algorithm for optimistic bilevel programming problems. Part I: theoretical development. J. Global Optim. 60(3), 425–458 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Falk, J.E., Hoffman, K.: A nonconvex max–min problem. Nav. Res. Log. Q. 24(3), 441–450 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zuhe, S., Neumaier, A., Eiermann, M.C.: Solving minimax problems by interval methods. BIT Numer. Math. 30(4), 742–751 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Bhattacharjee, B., Lemonidis, P., Green Jr., W.H., Barton, P.I.: Global solution of semi-infinite programs. Math. Program. 103(2), 283–307 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  36. 36.
    Blankenship, J.W., Falk, J.E.: Infinitely constrained optimization problems. J. Optim. Theory Appl. 19(2), 261–281 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Floudas, C.A., Stein, O.: The adaptive convexification algorithm: a feasible point method for semi-infinite programming. SIAM J. Optim. 18(4), 1187–1208 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Mitsos, A., Tsoukalas, A.: Global optimization of generalized semi-infinite programs via restriction of the right hand side. J. Global Optim. 61(1), 1–17 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Stein, O., Still, G.: On generalized semi-infinite optimization and bilevel optimization. Eur. J. Oper. Res. 142(3), 444–462 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  40. 40.
    Jongen, H.T., Rückmann, J.J., Stein, O.: Generalized semi-infinite optimization: a first order optimality condition and examples. Math. Program. 83(1–3), 145–158 (1998)MathSciNetzbMATHGoogle Scholar
  41. 41.
    Talbi, E.-G.: Metaheuristics for Bi-level Optimization. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  42. 42.
    Smith, J.C., Lim, C., Alptekinoglu, A.: Optimal mixed-integer programming and heuristic methods for a bilevel Stackelberg product introduction game. NRL 56(8), 714–729 (2009)CrossRefGoogle Scholar
  43. 43.
    Vicente, L., Savard, G., Judice, J.: Discrete linear bilevel programming problem. J. Optim. Theory Appl. 89(3), 597–614 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Bank, B.: Non-linear Parametric Optimization. Akademie Verlag, Berlin (1982)CrossRefGoogle Scholar
  45. 45.
    Ishizuka, Y., Aiyoshi, E.: Double penalty method for bilevel optimization problems. Ann. Oper. Res. 34(1), 73–88 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Chen, Y., Florian, M.: The nonlinear bilevel programming problem: formulations, regularity and optimality conditions. Optimization 32(3), 193–209 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Dempe, S.: Foundations of Bilevel Programming. Springer, Berlin (2002)zbMATHGoogle Scholar
  48. 48.
    Vicente, L., Calamai, P.: Bilevel and multilevel programming: a bibliography review. J. Global Optim. 5(3), 291–306 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Dewez, S., Labbé, M., Marcotte, P., Gilles, S.: New formulations and valid inequalities for a bilevel pricing problem. Oper. Res. Lett. 36(2), 141–149 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  50. 50.
    Lodi, A., Ralphs, T., Woeginger, G.: Bilevel programming and the separation problem. Math. Program. 146(1–2), 437–458 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Takeda, A., Taguchi, S., Tütüncü, R.H.: Adjustable robust optimization models for a nonlinear two-period system. J. Optim. Theory Appl. 136(2), 275–295 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  52. 52.
    Zeng, B., Zhao, L.: Solving two-stage robust optimization problems using a column-and-constraint generation method. Oper. Res. Lett. 41(5), 457–461 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    GAMS: GAMS/CPLEX Indicator Constraints. (2015)
  54. 54.
    Floudas, C.A., Pardalos, P.M.: Recent Advances in Global Optimization. Princeton University Press, Princeton (2014)Google Scholar
  55. 55.
    Ferris, M.C., Mangasarian, O.L., Pang, J.S.: Complementarity: Applications, Algorithms and Extensions, vol. 50. Springer, Berlin (2013)Google Scholar
  56. 56.
    Hu, J., Mitchell, J., Pang, J.S., Yu, B.: On linear programs with linear complementarity constraints. J. Global Optim. 53(1), 29–51 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. 137(1–2), 257–288 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  58. 58.
    Ferris, M.C., Munson, T.S.: Complementarity problems in GAMS and the PATH solver1. J. Econ. Dyn. Control 24(2), 165–188 (2000)CrossRefzbMATHGoogle Scholar
  59. 59.
    Rosenthal, R.E.: GAMS—a user’s guide. (2004)Google Scholar
  60. 60.
    Cao, D., Chen, M.: Capacitated plant selection in a decentralized manufacturing environment: a bilevel optimization approach. Eur. J. Oper. Res. 169(1), 97–110 (2006)MathSciNetCrossRefzbMATHGoogle Scholar

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Authors and Affiliations

  1. 1.Northwestern UniversityEvanstonUSA
  2. 2.Cornell UniversityIthacaUSA
  3. 3.University of PittsburghPittsburghUSA

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