Skip to main content
SpringerLink
Log in

Bilevel Programming

  • Chapter
Essays and Surveys in Global Optimization

Abstract

Bilevel programming problems are hierarchical optimization problems where an objective function is minimized over the graph of the solution set mapping of a parametric optimization problem. In this paper, a selective survey for this living research area is given. Focus is on main recent approaches to solve such problems, on optimality conditions as well as on essential features of this problem class.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 129.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  • Anandalingam, G. and Friesz, T. (eds.). (1992). Hierarchical Optimization. Annals of Operations Research, vol. 24.

    Google Scholar 

  • Anitescu, M. (2002). On solving mathematical programs with complementarity constraints as nonlinear programs. Technical Report No. ANL/NCS-P864-1200, Department of Mathematics, University of Pittsburgh,.

    Google Scholar 

  • Audet, C., Hansen, P., Jaumard, B., and Savard, G. (1997). Links between linear bilevel and mixed 0–1 programming problems. Journal of Optimization Theory and Applications, 93:273–300.

    Article  MathSciNet  Google Scholar 

  • Bard, J.F. (1998). Practical Bilevel Optimization: Algorithms and Applications. Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  • Clarke, F.H. (1983). Optimization and Nonsmooth Analysis. John Wiley & Sons, New York.

    Google Scholar 

  • Dempe, S. (1992). A necessary and a sufficient optimality condition for bilevel programming problems. Optimization, 25:341–354.

    MATH  MathSciNet  Google Scholar 

  • Dempe, S. (2002). Foundations of Bilevel Programming. Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  • Dempe, S. (2003). Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization, 52:333–359.

    Article  MATH  MathSciNet  Google Scholar 

  • Dempe, S. and Kalashnikov, V. (2002). Discrete bilevel programming: Application to a gas shipper's problem. Preprint No. 2002-02, TU Bergakademie Freiberg, Fakultdt für Mathematik und Informatik.

    Google Scholar 

  • Dempe, S. and Pallaschke, D. (1997). Quasidifferentiability of optimal solutions in parametric nonlinear optimization. Optimization, 40:1–24.

    MathSciNet  Google Scholar 

  • Deng, X. (1998). Complexity issues in bilevel linear programming. In: Multilevel Optimization: Algorithms and Applications (A. Migdalas, P.M. Pardalos, and P. Värbrand, eds.), pp. 149–164, Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  • Fiacco, A.V. (1983). Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Academic Press, New York.

    Google Scholar 

  • Flegel, M.L. and Kanzow, C. (2002). Optimality conditions for mathematical programs with equilibrium constraints: Fritz John and Abadie-Type approaches. Report, Universität Würzburg, Germany.

    Google Scholar 

  • Flegel, M.L. and Kanzow, C. (2003). A Fritz John approach to first order optimality conditions for mathematical programs with equilibrium constraints. Optimization, 52:277–286.

    Article  MathSciNet  Google Scholar 

  • Fletcher, R. and Leyffer, S. (2002). Numerical experience with solving MPECs as NLPs. Numerical Analysis Report NA/210, Department of Mathematics, University of Dundee, UK.

    Google Scholar 

  • Fletcher, R., Leyffer, S., Ralph, D., and Scholtes, S. (2002). Local Convergence of SQP Methods for Mathematical Programs with Equilibrium Constraints. Numerical Analysis Report NA/209, Department of Mathematics, University of Dundee, UK.

    Google Scholar 

  • Fliege, J. and Vicente, L.N. (2003). A Bicriteria Approach to Bilevel Optimization, Technical Report, Fachbereich Mathematik, Universität Dortmund, Germany.

    Google Scholar 

  • Frangioni, A.(1995). On a new class of bilevel programming problems and its use for reformulating mixed integer problems. European Journal of Operational Research, 82:615–646.

    Google Scholar 

  • Fukushima, M. and Pang, J.-S. (1999). Convergence of a smoothing continuation method for mathematical programs with complementarity constraints. In: Ill-posed Variational Problems and Regularization Techniques (M. Thera and R. Tichatschke, eds.). Lecture Notes in Economics and Mathematical Systems, No. 477, Springer-Verlag, Berlin.

    Google Scholar 

  • Fülöp., J. (1993). On the Equivalence between a Linear Bilevel Programming Problem and Linear Optimization over the Efficient Set. Working Paper, No. WP 93-1, Laboratory of Operations Research and Decision Systems, Computer and Automation Institute, Hungarian Academy of Sciences.

    Google Scholar 

  • Geiger, C. and Kanzow, C. (2003). Theorie und Numerik restringierter Optimierungsaufgaben. Springer-Verlag, Berlin.

    Google Scholar 

  • Guddat, J., Guerra Vasquez, F., and Jongen, H.Th. (1990). Parametric Optimization: Singularities, Pathfollowing and Jumps. John Wiley & Sons, Chichester and B.G. Teubner, Stuttgart.

    Google Scholar 

  • Hansen, P., Jaumard, B., and Savard, G. (1992). New branch-and-bound rules for linear bilevel programming. SIAM Journal on Scientific and Statistical Computing, 13:1194–1217.

    Article  MathSciNet  Google Scholar 

  • Harker, P.T. and Pang, J.-S. (1988). Existence of optimal solutions to mathematical programs with equilibrium constraints. Operations Research Letters, 7:61–64.

    Article  MathSciNet  Google Scholar 

  • Haurie, A., Savard, G., and White, D. (1990). A note on: An efficient point algorithm for a linear two-stage optimization problem. Operations Research, 38:553–555.

    Google Scholar 

  • Hirsch, M.W. (1994). Differential Topology. Springer-Verlag, Berlin.

    Google Scholar 

  • Klatte, D. and Kummer, B. (2002). Nonsmooth Equations in Optimization; Regularity, Calculus, Methods and Applications. Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  • Kojima, M. (1980). Strongly stable stationary solutions in nonlinear programs. In: Analysis and Computation of Fixed Points (S.M. Robinson, ed.), pp. 93–138, Academic Press, New York.

    Google Scholar 

  • Lignola, M.B. and Morgan, J. (1997). Stability of regularized bilevel programming problems. Journal of Optimization Theory and Applications, 93:575–596.

    Article  MathSciNet  Google Scholar 

  • Loridan, P, and Morgan, J. (1989). New results on approximate solutions in two-level optimization. Optimization, 20:819–836.

    MathSciNet  Google Scholar 

  • Loridan, P. and Morgan, J. (1996). Weak via strong Stackelberg problem: New results. Journal of Global Optimization, 8:263–287.

    Article  MathSciNet  Google Scholar 

  • Lucchetti, R., Mignanego, F., and Pieri, G. (1987). Existence theorem of equilibrium points in Stackelberg games with constraints. Optimization, 18:857–866.

    MathSciNet  Google Scholar 

  • Luo, Z.-Q., Pang, J.-S., and Ralph, D. (1996). Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge.

    Google Scholar 

  • Macal, C.M. and Hurter, A.P. (1997). Dependence of bilevel mathematical programs on irrelevant constraints. Computers and Operations Research, 24:1129–1140.

    Article  MathSciNet  Google Scholar 

  • Marcotte, P. (1986). Network design problem with congestion effects: A case of bilevel programming. Mathematical Programming, 34:142–162.

    Article  MATH  MathSciNet  Google Scholar 

  • Marcotte, P. and Savard, G. (1991). A note on the Pareto optimality of solutions to the linear bilevel programming problem. Computers and Operations Research, 18:355–359.

    Article  MathSciNet  Google Scholar 

  • Multilevel optimization: algorithms and applications. Nonconvex Optimization and its Applications (Athanasios Migdalas, Panos M. Pardalos and Peter Värbrand, eds.), vol. 20, Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  • Mikhalevich, V.S., Gupal, A.M., and Norkin, V.I. (1987). Methods of Nonconvex Optimization. Nauka, Moscow (in Russian).

    Google Scholar 

  • Mirrlees, J.A. (1999). The theory of moral hazard and unobservable bevaviour: Part I. Review of Economic Studies, 66:3–21.

    Article  MATH  Google Scholar 

  • Moore, J. and Bard, J.F. (1990). The mixed integer linear bilevel programming problem. Operations Research, 38:911–921.

    Article  MathSciNet  Google Scholar 

  • Muu, L.D. (2000). On the construction of initial polyhedral convex set for optimization problems over the efficient set and bilevel linear programs. Vietnam Journal of Mathematics, 28:177–182.

    MATH  MathSciNet  Google Scholar 

  • Outrata, J., Kočvara, M., and Zowe, J. (1998). Nonsmooth Approach to Optimization Problems with Equilibrium Constraints. Kluwer Academic Publishers, Dordrecht.

    Google Scholar 

  • Pardalos, P.M., Siskos, Y., and Zopounidis, C., eds. (1995). Advances in Multicriteria Analysis. Kiuwer Academic Publishers, Dordrecht.

    Google Scholar 

  • Ralph, D. and Dempe, S. (1995). Directional derivatives of the solution of a parametric nonlinear program. Mathematical Programming, 70:159–172.

    MathSciNet  Google Scholar 

  • Scheel, H. and Scholtes, S. (2000). Mathematical programs with equilibrium constraints: stationarity, optimality, and sensitivity. Mathematics of Operations Research, 25:1–22.

    Article  MathSciNet  Google Scholar 

  • Scholtes, S. and Stöhr, M. (2001). How stringent is the linear independence assumption for mathematical programs with stationarity constraints? Mathematics of Operations Research, 26:851–863.

    Article  MathSciNet  Google Scholar 

  • Schramm, H. (1989). Eine Kombination von bundle-und trust-region-Verfahren zur Lösung nichtdifferenzierbarer Optimierungsprobleme, No. 30, Bayreuther Mathematische Schriften, Bayreuth,.

    Google Scholar 

  • Shapiro, A. (1988). Sensitivity analysis of nonlinear programs and differentiability properties of metric projections. SIAM Journal Control Optimization, 26:628–645.

    Article  MATH  Google Scholar 

  • Vicente, L.N., Savard, G., and Judice, J.J. (1996). The discrete linear bilevel programming problem. Journal of Optimization Theory and Applications, 89:597–614.

    Article  MathSciNet  Google Scholar 

  • Vogel, S. (2002). Zwei-Ebenen-Optimierungsaufgaben mit nichtkonvexer Zielfunktion in der unteren Ebene: Pfadverfolgung und Sprünge. Ph.D thesis, Technische Universitit Bergakademie Freiberg.

    Google Scholar 

  • White, D.J. and Anandalingam, G. (1993). A penalty function approach for solving bi-level linear programs. Journal of Global Optimization, 3:397–419.

    Article  MathSciNet  Google Scholar 

Download references

Authors
  1. Stephan Dempe
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. École Polytechnique de Montréal and GERAD, Montréal, Canada

    Charles Audet  & Gilles Savard  & 

  2. HEC Montréal & GERAD, Montréal, Canada

    Pierre Hansen

Rights and permissions

Reprints and permissions

Copyright information

© 2005 Springer Science+Business Media, Inc.

About this chapter

Cite this chapter

Dempe, S. (2005). Bilevel Programming. In: Audet, C., Hansen, P., Savard, G. (eds) Essays and Surveys in Global Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-387-25570-2_6

Download citation

Publish with us

Policies and ethics

Search

Navigation