The bilevel programming problem is a hierarchical problem in the sense that its constraints are defined in part by a second parametric optimization problem. Let Ψ(x) be the solution set of this second problem (the so-called lower level problem):
where f, g i ∈ C 2(R n × R m, R), i = 1,..., p. Then, the bilevel programming problem is defined as
with F ∈ C 1(R n × R m, R) and X ⊆ R n is closed. Problem (2) is also called the upper level problem. The inclusion of equality constraints in the problem (1) is possible without difficulties. If inequalities and/or equations in both x and y appear in the problem (2), this problem becomes even more difficult since these constraints restrict the set Ψ(x) after a solution y out of it has been chosen. This can make the selection of y ∈ Ψ(x) a posteriori infeasible [6].
The bilevel programming problem can easily be interpreted in terms of Stackelberg gameswhich are a special case of them widely used in economics. In Stackelberg...
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Clarke, F.H.: Optimization and nonsmooth analysis, Wiley, 1983.
Dempe, S.: ‘A necessary and a sufficient optimality condition for bilevel programming problems’, Optim.25 (1992), 341–354.
Dempe, S.: ‘Directional differentiability of optimal solutions under Slater’s condition’, Math. Program.59 (1993), 49–69.
Dempe, S.: ‘An implicit function approach to bilevel programming problems’, in A. Migdalas, P.M. Pardalos, and P. Värbrand (eds.): Multilevel Optimization: Algorithms, Complexity and Applications, Kluwer Acad. Publ., 1997.
Dempe, S., and Pallaschke, D.: ‘Quasidifferentiability of optimal solutions in parametric nonlinear optimization’, Optim.40 (1997), 1–24.
Dempe, S., and Schmidt, H.: ‘On an algorithm solving two-level programming problems with nonunique lower level solutions’, Comput. Optim. Appl.6 (1996), 227–249.
Fiacco, A.V., and McCormic, G.P.: Nonlinear programming: Sequential unconstrained minimization techniques, Wiley, 1968.
Gauvin, J., and Savard, G.: ‘The steepest descent direction for the nonlinear bilevel programming problem’, Oper. Res. Lett.15 (1994), 265–272.
Gfrerer, H.: ‘Hölder continuity of solutions of perturbed optimization problems under Mangasarian-Fromowitz constraint qualification’, in J. Guddat et al. (eds.): Parametric Optimization and Related Topics, Akad. Verlag, 1987, pp. 113–127.
Hager, W.W.: ‘Lipschitz continuity for constrained processes’, SIAM J. Control Optim.17 (1979), 321–328.
Harker, P.T., and Pang, J.-S.: ‘Existence of optimal solutions to mathematical programs with equilibrium constraints’, Oper. Res. Lett.7 (1988), 61–64.
Kiwiel, K.C.: Methods of descent for nondifferentiable optimization, Springer 1985.
Kojima, M.: ‘Strongly stable stationary solutions in nonlinear programs’, in S.M. Robinson (ed.): Analysis and Computation of Fixed Points, Acad. Press, 1980 p. 93–138.
Kummer, B.: ‘Newton’s method for non-differentiable functions’, Adv. Math. Optim., Vol. 45 Math. Res., Akad. Verlag, 1988.
Loridan, P., and Morgan, J.: ‘ε-regularized two-level optimization problems: Approximation and existence results’: Optimization: Fifth French-German Conf. (Varez), Vol. 1405 of Lecture Notes Math., Springer, 1989, pp. 99–113.
Lucchetti, R., Mignanego, F., and Pieri, G.: ‘Existence theorem of equilibrium points in Stackelberg games with constraints’, Optim.18 (1987), 857–866.
Luo, Z.-Q, Pang, J.-S, and Ralph, D.: Mathematical programs with equilibrium constraints, Cambridge Univ. Press, 1996.
Outrata, J.: ‘On the numerical solution of a class of Stackelberg problems’, ZOR: Methods and Models of Oper. Res.34 (1990), 255–277.
Outrata, J.V.: ‘Necessary optimality conditions for Stackelberg problems’, J. Optim. Th. Appl.76 (1993), 305–320.
Outrata, J., and Zowe, J.: ‘A numerical approach to optimization problems with variational inequality constraints’, Math. Program.68 (1995), 105–130.
Ralph, D., and Dempe, S.: ‘Directional derivatives of the solution of a parametric nonlinear program’, Math. Program.70 (1995), 159–172.
Robinson, S.M.: ‘Generalized equations and their solutions, Part II: Applications to nonlinear programming’, Math. Program. Stud.19 (1982), 200–221.
Schramm, H., and Zowe, J.: ‘A version of the bundle idea for minimizing a nonsmooth function: conceptual idea, convergence analysis, numerical results’, SIAM J. Optim.2 (1992), 121–152.
Shapiro, A.: ‘Sensitivity analysis of nonlinear programs and differentiability properties of metric projections’, SIAM J. Control Optim.26 (1988), 628–645.
Vicente, L.N., and Calamai, P.H.: ‘Bilevel and multilevel programming: A bibliography review’, J. Global Optim.5, no. 3 (1994).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2001 Kluwer Academic Publishers
About this entry
Cite this entry
Dempe, S. (2001). Bilevel Programming: Implicit Function Approach . In: Floudas, C.A., Pardalos, P.M. (eds) Encyclopedia of Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-306-48332-7_36
Download citation
DOI: https://doi.org/10.1007/0-306-48332-7_36
Publisher Name: Springer, Boston, MA
Print ISBN: 978-0-7923-6932-5
Online ISBN: 978-0-306-48332-5
eBook Packages: Springer Book Archive