Encyclopedia of Optimization

2001 Edition
| Editors: Christodoulos A. Floudas, Panos M. Pardalos

Bilevel Programming: Optimality Conditions and Duality

  • S. Zlobec
Reference work entry
DOI: https://doi.org/10.1007/0-306-48332-7_39
The bilevel programming problem (abbreviation: BPP) is a mathematical program in two variables x and θ, where x = x°(θ) is an optimal solution of another program. Specifically, BPP can be formulated in terms of two ordered objective functions φ and Ψ as follows:
90C25 90C29 90C30 90C31 
bilevel programming optimality conditions duality stability parametric programming 
This is a preview of subscription content, log in to check access.

References

  1. [1]
    Bank, B., Guddat, J., Klatte, D., Kummer, B., and Tammer, K.: Nonlinear parametric optimization, Akad. Verlag 1982.Google Scholar
  2. [2]
    Bard, J.: ‘Optimality conditions for the bilevel programming problem’, Naval Res. Logist. Quart.31 (1984), 13–26.MathSciNetzbMATHGoogle Scholar
  3. [3]
    Bard, J.: ‘Convex two-level optimization’, Math. Program.40 (1988), 15–27.MathSciNetzbMATHGoogle Scholar
  4. [4]
    Ben-Ayed, O., and Blair, C.: ‘Computational difficulties of bilevel linear programming’, Oper. Res.38 (1990), 556–560.MathSciNetzbMATHGoogle Scholar
  5. [5]
    Ben-Israel, A., Ben-Tal, A., and Zlobec, S.: Optimality in nonlinear programming: A feasible directions approach, Wiley/Interscience 1981.Google Scholar
  6. [6]
    Berge, C.: Topological spaces, Oliver and Boyd 1963.Google Scholar
  7. [7]
    Bi, Z., and Calamai, P.: ‘Optimality conditions for a class of bilevel programming problems’, Techn. Report Dept. Systems Design Engin. Univ. Waterloo, no. 191-O-191291 (1991).Google Scholar
  8. [8]
    Bracken, J., Falk, J., and McGill, J.: ‘Equivalence of two mathematical programs with optimization problems in the constraints’, Oper. Res.22 (1974), 1102–1104.MathSciNetzbMATHGoogle Scholar
  9. [9]
    Calamai, P., and Vicente, L.N.: ‘Generating linear and linear-quadratic bilevel programming problems’, SIAM J. Sci. Statist. Comput.14 (1993), 770–782.MathSciNetzbMATHGoogle Scholar
  10. [10]
    Candler, W.: ‘A linear bilevel programming algorithm: A comment’, Computers Oper. Res.15 (1988), 297–298.MathSciNetzbMATHGoogle Scholar
  11. [11]
    Chen, Y., and Florian, M.: ‘The nonlinear bilevel programming problem: formulations, regularity and optimality conditions’, Optim.32 (1995), 193–209.MathSciNetzbMATHGoogle Scholar
  12. [12]
    Clarke, P., and Westerberg, A.: ‘A note on the optimality conditions for the bilevel programming problem’, Naval Res. Logist. Quart.35 (1988), 413–418.Google Scholar
  13. [13]
    Dempe, S.: ‘A necessary and sufficient optimality condition for bilevel programming problems’, Optim.25 (1992), 341–354.MathSciNetzbMATHGoogle Scholar
  14. [14]
    Dempe, S.: ‘On the leader’s dilemma and a new idea for attacking bilevel programming problems’, Preprint Techn. Univ. Chemnitz (1997).Google Scholar
  15. [15]
    Floudas, C.A., and Zlobec, S.: ‘Optimality and duality in parametric convex lexicographic programming’, in P.M. Pardalos A. Migdalas and P. Värbrand (eds.): Multilevel Optimization: Algorithms and Applications, Kluwer Acad. Publ. 1998, pp. 359–379.Google Scholar
  16. [16]
    Harker, P., and Pang, J.-S.: ‘Existence of optimal solutions to mathematical programs with equilibrium constraints’, Oper. Res. Lett.7 (1988), 61–64.MathSciNetzbMATHGoogle Scholar
  17. [17]
    Haurie, A., Savard, G., and White, D.: ‘A note on: An efficient point algorithm for a linear two-stage optimization problem’, Oper. Res.38 (1990), 553–555.zbMATHGoogle Scholar
  18. [18]
    Hettich, R., and Jongen, H.Th.: ‘Semi-infinite programming: conditions of optimality and applications’, in J. Stoer (ed.): Optimization Techniques, Part 2, Vol. 7 of Lecture Notes Control Inform. Sci., Springer, 1978, pp. 1–11.Google Scholar
  19. [19]
    Ishizuka, Y.: ‘Optimality conditions for quasi-differentiable programs with applications to two-level optimization’, SIAM J. Control Optim.26 (1988), 1388–1398.MathSciNetzbMATHGoogle Scholar
  20. [20]
    Jongen, H.Th., Rückmann, J.-J, and Stein, O.: ‘Generalized semi-infinite optimization: A first order optimality condition and examples’, Math. Program.83 (1998), 145–158.Google Scholar
  21. [21]
    Jongen, H.Th., and Rückmann, J.-J.: ‘On stability and deformation in semi-infinite optimization’, in R. Reemtsen and J.-J. Rückmann (eds.): Semi-Infinite Programming, Kluwer Acad. Publ., 1998, pp. 29–67.Google Scholar
  22. [22]
    Kolstad, C.D.: ‘A review of the literature on bilevel mathematical programming’, Techn. Report Los Alamos Nat. Lab. no. LA-10284-MS, UC-32 (Oct. 1985).Google Scholar
  23. [23]
    Lignola, M.B., and Morgan, J.: ‘Topological existence and stability for Stackelberg problems’, J. Optim. Th. Appl.84 (1995), 145–169.MathSciNetzbMATHGoogle Scholar
  24. [24]
    Lignola, M.B., and Morgan, J.: ‘Existence of solutions to generalized bilevel programing problem’, in A. Migdalas P.M. Pardalos and P. Värbrand (eds.): Multilevel Optimization: Algorithms and Applications, Kluwer Acad. Publ. 1998, pp. 315–332.Google Scholar
  25. [25]
    Liu, Y., and Hart, S.: ‘Characterizing an optimal solution to the linear bilevel programming problem’, Europ. J. Oper. Res.166 (1994), 164–166.Google Scholar
  26. [26]
    Loridan, P., and Morgan, J.: ‘New results on approximate solutions in two-level optimization’, Optim.20 (1989), 819–836.MathSciNetzbMATHGoogle Scholar
  27. [27]
    Mallozzi, L., and Morgan, J.: ‘Weak Stackelberg problem and mixed solutions under data perturbations’, Optim.32 (1995), 269–290.MathSciNetzbMATHGoogle Scholar
  28. [28]
    Marcotte, P., and Savard, G.: ‘A note on Pareto optimality of solutions to the linear bilevel programming problem’, Comput. Oper. Res.18 (1991), 355–359.MathSciNetzbMATHGoogle Scholar
  29. [29]
    Migdalas, A.: ‘When is a Stackelberg equilibrium Pareto optimum?’, in P.M. Pardalos, Y. Siskos and C. Zopounidis (eds.): Advances in Multicriteria Analysis, Kluwer Acad. Publ. 1995, pp. 175–181.Google Scholar
  30. [30]
    Migdalas, A., and Pardalos, P.M.: ‘Editorial: Hierarchical and bilevel programming’, J. Global Optim.8 (1996), 209–215.MathSciNetGoogle Scholar
  31. [31]
    A. Migdalas, P.M. Pardalos and Värbrand P. (eds.): Multilevel optimization: Algorithms and applications, Kluwer Acad. Publ. 1998, pp. 29–67.Google Scholar
  32. [32]
    Outrata, J.: ‘Necessary optimality conditions for Stackelberg problems’, J. Optim. Th. Appl.76 (1993), 305–320.MathSciNetzbMATHGoogle Scholar
  33. [33]
    Savard, G., and Gauvin, J.: ‘The steepest descent direction for the nonlinear bilevel programming problem’, Oper. Res. Lett.15 (1994), 275–282.MathSciNetGoogle Scholar
  34. [34]
    Tammer, K., and Rückmann, J.-J.: ‘Relations between the Karush-Kuhn-Tucker points of a nonlinear optimization problem and of a generalized Lagrange dual’, in H.-J Sebastian and K. Tammer (eds.): System Modelling and Optimization, 143 of Lecture Notes Control Inform. Sci., Springer 1990.Google Scholar
  35. [35]
    Trujillo-Cortez, R.: ‘LFS functions in stable bilevel programming’, PhD Thesis Dept. Math. and Statist. McGill Univ.July (1997).Google Scholar
  36. [36]
    Trujillo-Cortez, R.: ‘Stable bilevel programming and applications’, PhD Thesis McGill Univ. (2000), in preparation.Google Scholar
  37. [37]
    Tuy, H.: ‘Bilevel linear programming, multiobjective programming, and monotonic reverse convex programming’, in A. Migdalas, P.M. Pardalos and P. VÄrbrand (eds.): Multilevel Optimization: Algorithms and Applications, Kluwer Acad. Publ. 1998, 295–314.Google Scholar
  38. [38]
    Ünlü, G.: ‘A linear bilevel programming algorithm based on bicriteria programming’, Computers Oper. Res.14 (1987), 173–179.zbMATHGoogle Scholar
  39. [39]
    Vicente, L.N., and Calamai, P.H.: ‘Bilevel and multilevel programming: A bibliography review’, J. Global Optim.5 (1994), 291–306.MathSciNetzbMATHGoogle Scholar
  40. [40]
    Vicente, L.N., and Calamai, P.H.: ‘Geometry and local optimality conditions for bilevel programs with quadratic strictly convex lower levels’, in D.-Z Dhu and P.M. Pardalos (eds.): Minimax and Applications, Kluwer Acad. Publ. 1995, 141–151.Google Scholar
  41. [41]
    Vicente, L.N., Savard, G., and Judice, J.: ‘Descent approaches for quadratic bilevel programming’, J. Optim. Th. Appl.81 (1994), 379–399.MathSciNetzbMATHGoogle Scholar
  42. [42]
    Visweswaran, V., Floudas, C.A., Ierapetritou, M.G., and Pistikopoulos, E.N.: ‘A decomposition-based global optimization approach for solving bilevel linear and quadratic programs’, in C.A. Floudas and P.M. Pardalos (eds.): State of the Art in Global Optimization, Kluwer Acad. Publ. 1996, 139–162.Google Scholar
  43. [43]
    Wen, U.-P, and Hsu, S.-T.: ‘A note on a linear bilevel programming algorithm based on bicriteria programming’, Comput. Oper. Res.16 (1989), 79–83.MathSciNetzbMATHGoogle Scholar
  44. [44]
    Wetterling, W.: ‘Definitheitsbedingungen für relative Extrema bei Optimierungs-und Aproximation saufgaben’, Numerische Math.15 (1970), 122–136.MathSciNetzbMATHGoogle Scholar
  45. [45]
    Ye, J.: ‘Necessary conditions for bilevel dynamic optimization problems’, SIAM J. Control Optim.33 (1995), 1208–1223.MathSciNetzbMATHGoogle Scholar
  46. [46]
    Ye, J.J., and Zhu, D.L.: ‘Optimality conditions for bilevel programming problems’, Optim.33 (1995), 9–27.MathSciNetzbMATHGoogle Scholar
  47. [47]
    Zlobec, S.: ‘Lagrange duality in partly convex programming’, in C.A. Floudas and P.M. Pardalos (eds.): State of the Art in Global Optimization, Kluwer Acad. Publ. 1996, pp. 1–18.Google Scholar
  48. [48]
    Zlobec, S.: ‘Stable parametric programming’, Optim.45 (1998), 387–416.MathSciNetGoogle Scholar
  49. [49]
    Zlobec, S.: ‘Parametric programming: An illustrative mini-encyclopedia’, Math. Commun.5 (2000), 1–39.MathSciNetzbMATHGoogle Scholar

Copyright information

© Kluwer Academic Publishers 2001

Authors and Affiliations

  • S. Zlobec
    • 1
  1. 1.Dept. Math. Statist. McGill Univ.MontrealCanada