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Bilevel Programming: A Combinatorial Perspective

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Graph Theory and Combinatorial Optimization

Abstract

Bilevel programming is a branch of optimization where a subset of variables is constrained to lie in the optimal set of an auxiliary mathematical prograri. This chapter presents an overview of two specific classes cf bilevel programs, and in particular their relationship to well-known combinatorial problems.

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References

  • Alarie, S., Audet, C., Jaumard, B., and Savard, G. (2001). Concavity cuts for disjoint bilinear programming. Mathematical Programming, 90:373–398.

    Article  MathSciNet  Google Scholar 

  • Audet, C. (1997). Optimisation globale structurée: propriétés, équivalences et résolution. Ph.D. thesis, École Polytechnique de Montréal.

    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 

  • Audet, C., Hansen, P., Jaumard, B., and Savard, G. (1999). A Symmetrical linear maxmin approach to disjoint bilinear programming. Mathematical Programming, 85:573–592.

    Article  MathSciNet  Google Scholar 

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

    Google Scholar 

  • Beale, E. and Small, R. (1965). Mixed integer programming by a branch and bound technique. In: Proceedings of the 3rd IFIP Congress. pp. 450–451.

    Google Scholar 

  • Candler, W. and Norton, R. (1977). Multilevel programing. Technical Report 20, World Bank Development Research Center, Washington, DC.

    Google Scholar 

  • Côté, J.-P., Marcotte, P., and Savard, G. (2003). A bilevel modeling approach to pricing and fare optimization in the airline industry. Journal of Revenue and Pricing Management, 2:23–36.

    Google Scholar 

  • Dantzig, G., Fulkerson, D., and Johnson, S. (1954). Solution of a largescale traveling salesman problem. Operations Research, 2:393–410.

    Article  MathSciNet  Google Scholar 

  • Dempe, S. (2002). Foundations of Bilevel Programming. Nonconvex Optirnization and Its Applications, vol. 61. Kluwer Academic Publishers, Dordrecht, The Netherlands.

    Google Scholar 

  • Desrochers, M. and Laporte, G. (1991). Improvements and extensions to the Miller—Tucker-Zemlin subtour eliminatiori constraints. Operations Research Letters, 10:27–36.

    Article  MathSciNet  Google Scholar 

  • Flood, M. (1956). The traveling salesman problem. Operations Research, 4:61–75.

    MathSciNet  Google Scholar 

  • Fukushima, M. (1992). Equivalent differentiable optimization problems and descent methods for asymmetric variational inequality problems. Mathematical Programming, 53:99–110.

    Article  MATH  MathSciNet  Google Scholar 

  • Fukushima, M. and Pang, J. (1999). Complementarity constraint qualifications and simplified B-stationarity conditions for mathernatical programs with equilibrium constraints. Computational Optimization and Applications, 13:111–136.

    Article  MathSciNet  Google Scholar 

  • Garey, M. and Johnson, D. (1979). Computers and Intractability: A Guide to the Theory of NP-Completeness. W.H. Freeman, New York.

    Google Scholar 

  • Grigoriev, A., van Hoesel, S., van der Kraaij, A., Uetz, M., and Bouhtou, M. (2004). Pricing network edges to cross a river. Technical Report RM04009, Maastricht Economic Research School on Technology and Organisation, Maastricht, The Netherlands.

    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 

  • Hogan, W. (1973). Point-to-set maps in mathematical programming. SIAM Review, 15:591–603.

    Article  MATH  MathSciNet  Google Scholar 

  • Jeroslow, R. (1985). The polynomial hierarchy and a simple model for competitive analysis. Mathematical Programming, 32:146–164.

    Article  MATH  MathSciNet  Google Scholar 

  • Konno, H. (1971). Bilinear Programming. II. Applications of Bilinear Programming. Technical Report Technical Report No. 71-10, Operations Research House, Department of Operations Research, Stanford University, Stanford.

    Google Scholar 

  • Kočvara, M. and Outrata, J. (1994). On optimization of systems governed by implicit complementarity problems. Numerical Functional Analysis and Optimization, 15:869–887.

    MathSciNet  Google Scholar 

  • Labbé, M., Marcotte, P., and Savard, G. (1998). A bilevel niodel of taxation and its applications to optimal highway pricing. Management Science, 44:1608–1622.

    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 

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

    Google Scholar 

  • Marcotte, P., Savard, G., and Semet, F. (2003). A bilevel programming approach to the travelling salesman problem. Operations Research Letters, 32:240–248.

    Article  MathSciNet  Google Scholar 

  • Marcotte, P., Savard, G., and Zhu, D. (2001). A trust region algorithm for nonlinear bilevel programming. Operations Research Letters, 29:171–179.

    Article  MathSciNet  Google Scholar 

  • Mills, H. (1960). Equilibrium points in finite games. Journal of the Society for Industrial and Applied Mathematics, 8:397–402.

    Article  MATH  MathSciNet  Google Scholar 

  • Nash, J. (1951). Noncooperative games. Annals of Mathematics, 14:286–295.

    Article  MathSciNet  Google Scholar 

  • Pardalos, P. and Schnitger, G. (1988). Checking local optimality in constrained qu adratic programming is NP-hard. Operations Research Letters, 7:33–35.

    Article  MathSciNet  Google Scholar 

  • Roch, S., Savard, G., and Marcotte, P. (2004). An Approximation Algorithm for Stackelberg Network Pricing.

    Google Scholar 

  • Savard, G. (1989). Contribution à la programmation mathématique à deux niveaux. Ph.D. thesis, Ecole Polytechnique de Montréal, Université de Montréal.

    Google Scholar 

  • Scholtes, S. (2004). Nonconvex structures in nonlinear programming. Operations Research, 52:368–383.

    Article  MathSciNet  Google Scholar 

  • Scholtes, S. and Stòhr, M. (1999). Exact penalization of mathematical programs with equilibriurm constraints. SIAM Journal on Control and Optimization, 37:(2):617–652.

    Article  MathSciNet  Google Scholar 

  • Shimizu, K., Ishizuka, Y., and Bard, J. (1997). Nondifferentiable and Two-Level Mathematical Programming. Kluwer Academic Publishers.

    Google Scholar 

  • Stackelberg, H. (1952). The Theory of Market Economy. Oxford University Press, Oxford.

    Google Scholar 

  • TSPLIB. A library of traveling salesman problems. Available at http://www.iwr.uni-heidelberg.de/groups/comopt/software/ TSPLIB95.

    Google Scholar 

  • van Hoesel, S., van der Kraaij, A., Mannino, C., Oriolo, G., and Bouhtou. M. (2003). Polynomial cases of the tarification problem. Technical Report RM03053, Maastricht Economic Research School on Technology and Organisation, Maastricht, The Netherlands.

    Google Scholar 

  • Vicente, L., Savard, G., and Júdice, J. (1994). Descent approaches for quadratic bilevel programming. Journal of Optimization Theory and Applications, 81:379–399.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

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Authors
  1. Patrice Marcotte
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  2. Gilles Savard
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Editor information

Editors and Affiliations

  1. McGill University & GERAD, Montréal, Canada

    David Avis

  2. École Polytechnique de Montréal & GERAD, Montréal, Canada

    Alain Hertz

  3. Université du Québec a Montréal and GERAD, Montréal, Canada

    Odile Marcotte

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© 2005 Springer Science+Business Media, Inc.

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Marcotte, P., Savard, G. (2005). Bilevel Programming: A Combinatorial Perspective. In: Avis, D., Hertz, A., Marcotte, O. (eds) Graph Theory and Combinatorial Optimization. Springer, Boston, MA. https://doi.org/10.1007/0-387-25592-3_7

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