The bilevel programming (BP) problem is a hierarchical optimization problem where a subset of the variables is constrained to be a solution of a given optimization problem parameterized by the remaining variables. The BP problem is a multilevel programming problem with two levels. The hierarchical optimization structure appears naturally in many applications when lower level actions depend on upper level decisions. The applications of bilevel and multilevel programming include transportation (taxation, network design, trip demand estimation), management (coordination of multidivisional firms, network facility location, credit allocation), planning (agricultural policies, electric utility), and optimal design.
In mathematical terms, the BP problem consists of finding a solution to the upper level problem
where y, for each value of x, is the solution of the lower level problem:
with x ∈ R nx, y ∈ R ny, F, f : R nx + ny → R, g : R nx + ny → R nu, and h : R nx + ny → R nl (nx, ny, nu, and nl...
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References
Aiyoshi, E., and Shimizu, K.: ‘A solution method for the static constrained Stackelberg problem via penalty method’, IEEE Trans. Autom. Control29 (1984), 1111–1114.
Anandalingam, G., and Friesz, T.: ‘Hierarchical optimization: An introduction’, Ann. Oper. Res.34 (1992), 1–11.
Bard, J.F.: Practical bilevel optimization: Algorithms and applications, Kluwer Acad. Publ., 1998.
Ben-Ayed, O.: ‘Bilevel linear programming’, Comput. Oper. Res.20 (1993), 485–501.
Bracken, J., and McGill, J.: ‘Mathematical programs with optimization problems in the constraints’, Oper. Res.21 (1973), 37–44.
Calamai, P., and Vicente, L.N.: ‘Generating linear and linear-quadratic bilevel programming problems’, SIAM J. Sci. Statist. Comput.14 (1993), 770–782.
Candler, W., and Norton, R.: ‘Multilevel programming’, Techn. Report World Bank Developm. Res. Center, Washington D.C.20 (1977).
Dempe, S.: ‘A necessary and a sufficient optimality condition for bilevel programming problems’, Optim.25 (1992), 341–354.
Edmunds, T., and Bard, J.: ‘Algorithms for nonlinear bilevel mathematical programming’, IEEE Trans. Syst., Man Cybern.21 (1991), 83–89.
Hansen, P., Jaumard, B., and Savard, G.: ‘New branch-and-bound rules for linear bilevel programming’, SIAM J. Sci. Statist. Comput.13 (1992), 1194–1217.
Hsu, S., and Wen, U.: ‘A review of linear bilevel programming problems’: Proc. Nat. Sci. Council Report China, Part A: Physical Sci. and Engin., Vol. 13, 1989, pp. 53–61.
Júdice, J., and Faustino, A.: ‘The linear-quadratic bilevel programming problem’, INFOR32 (1994), 87–98.
Kolstad, C.: ‘A review of the literature on bi-level mathematical programming’, Techn. Report Los Alamos Nat. Lab.LA-10284-MS/US-32 (1985).
Kolstad, C., and Lasdon, L.: ‘Derivative evaluation and computational experience with large bilevel mathematical programs’, J. Optim. Th. Appl.65 (1990), 485–499.
Marcotte, P., and Zhu, D.: ‘Exact and inexact penalty methods for the generalized bilevel programming problem’, Math. Program.74 (1996), 142–157.
Outrata, J.: ‘On optimization problems with variational inequality constraints’, SIAM J. Optim.4 (1994), 340–357.
Savard, G.: ‘Contributions à la programmation mathématique à deux niveaux’, PhD Thesis Ecole Polytechn. Univ. Montréal (1989).
Savard, G., and Gauvin, J.: ‘The steepest descent direction for the nonlinear bilevel programming problem’, Oper. Res. Lett.15 (1994), 275–282.
Shimizu, K., Ishizuka, Y., and Bard, J.F.: Nondifferentiable and two-level mathematical programming, Kluwer Acad. Publ., 1997.
Stackelberg, H.: The theory of the market economy, Oxford Univ. Press, 1952.
Vicente, L.N., and Calamai, P.H.: ‘Bilevel and multilevel programming: A bibliography review’, Jogo5 (1994), 291–306.
Vicente, L.N., Savard, G., and Júdice, J.: ‘The discrete linear bilevel programming problem’, J. Optim. Th. Appl.89 (1996), 597–614.
Wen, U., and Hsu, S.: ‘Linear bi-level programming problems: A review’, J. Oper. Res. Soc.42 (1991), 125–133.
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Vicente, L.N. (2001). Bilevel Programming: Introduction, History and Overview . In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-306-48332-7_38
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DOI: https://doi.org/10.1007/0-306-48332-7_38
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