Journal of Global Optimization

, Volume 44, Issue 1, pp 29–51 | Cite as

Resolution method for mixed integer bi-level linear problems based on decomposition technique

Article

Abstract

In this article, we propose a new algorithm for the resolution of mixed integer bi-level linear problem (MIBLP). The algorithm is based on the decomposition of the initial problem into the restricted master problem (RMP) and a series of problems named slave problems (SP). The proposed approach is based on Benders decomposition method where in each iteration a set of variables are fixed which are controlled by the upper level optimization problem. The RMP is a relaxation of the MIBLP and the SP represents a restriction of the MIBLP. The RMP interacts in each iteration with the current SP by the addition of cuts produced using Lagrangian information from the current SP. The lower and upper bound provided from the RMP and SP are updated in each iteration. The algorithm converges when the difference between the upper and lower bound is within a small difference ε. In the case of MIBLP Karush–Kuhn–Tucker (KKT) optimality conditions could not be used directly to the inner problem in order to transform the bi-level problem into a single level problem. The proposed decomposition technique, however, allows the use of KKT conditions and transforms the MIBLP into two single level problems. The algorithm, which is a new method for the resolution of MIBLP, is illustrated through a modified numerical example from the literature. Additional examples from the literature are presented to highlight the algorithm convergence properties.

Keywords

Bi-level optimization Mixed integer linear programming Benders decomposition Active constraints 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bard J.F. (1983). An efficient point algorithm for a linear two-stage optimization problem. Oper. Res. 31(4): 670–684 CrossRefGoogle Scholar
  2. 2.
    Bard J.F. (1984). An investigation of the linear tree level programming problem. IEEE Trans. Syst. Man Cybern. 14: 711–717 Google Scholar
  3. 3.
    Bard J.F. and Moore J.T. (1990). A branch and bound algorithm for the bilevel programming problem. SIAM J. Sci. Stat. Comp. 11: 281–292 CrossRefGoogle Scholar
  4. 4.
    Benders J.F. (1962). Partitioning procedures for solving mixed-variables programming problems. Numer. Math. 4: 238–252 CrossRefGoogle Scholar
  5. 5.
    Bialas, W., Karwan, M.: Multilevel linear programming. Technical Report 78-1, State University of New York at Buffalo, Operations Research Program (1978)Google Scholar
  6. 6.
    Bilias, W., Karwan, M., Shaw, J.: A parametric complementary pivot approach for two-level linear programming. Technical Report 80-2, State University of New York at Buffalo, Operations Research Program (1980)Google Scholar
  7. 7.
    Candler W. and Townsley R. (1982). A linear two-level programming problem. Comput. Oper. Res. 9: 59–76 CrossRefGoogle Scholar
  8. 8.
    Chen, Y., Florian, M.: On the geometry structure of linear bilevel programs: a dual approach. Technical Report CRT-867, Centre de recherche sur les Transports (1992)Google Scholar
  9. 9.
    Dempe, S.: Discrete bi-level optimization problems. http://www.mathe.tufreiberg.de/dempe, TU Chemnizt (1995)
  10. 10.
    Faisca N., Dua V., Rustem B., Saraiva P.M. and Pistikopoulos E.N. (2007). Parametric global optimization for bi-level programming. J. Glob. Optim. 38(4): 609–623 CrossRefGoogle Scholar
  11. 11.
    Floudas C.A. (1995). Nonlinear and Mixed-Integer Optimization, Fundamentals and Applications. Oxford University Press, Oxford, New York Google Scholar
  12. 12.
    Floudas C.A., Gümus Z.H. and Ierapetritou M.G. (2001). Global optimization in design under uncertainty feasibility test and flexibility index problems. Ind. Eng. Chem. Res. 40: 4267–4282 CrossRefGoogle Scholar
  13. 13.
    Foteinou, P., Yang, E., Sacharidis, G.K., Ierapetritou, M.G., Androulakis, I.P.: A mixed integer optimization framework for the synthesis and analysis of regulatory networks. To appear in JOGO (2008)Google Scholar
  14. 14.
    Grossmann I.E. and Floudas C.A. (1987). Active constraint strategy for flexibility analysis in chemical processes. Comput. Chem. Eng. 11: 675 CrossRefGoogle Scholar
  15. 15.
    Gümus Z.H. and Floudas C.A. (2005). Global optimization of mixed-integer bilevel programming problems. Comput. Manag. Sci. 2: 181–212 CrossRefGoogle Scholar
  16. 16.
    Hansen P., Jaumard B. and Savard G. (1992). New branch-and-bound rules for linear bi-level programming. SIAM J. Sci. Stat. Comput. 13: 1194–1217 CrossRefGoogle Scholar
  17. 17.
    Haurie A., Savard G. and White D. (1990). A note on: an efficient point algorithm for a linear two stage optimization problem. Oper. Res. 38: 553–555 CrossRefGoogle Scholar
  18. 18.
    Marcotte P. and Zhu D.L. (1996). Exact and inexact penalty methods for the generalized bi-level programming problem. Math. Program. 74: 141–157 Google Scholar
  19. 19.
    Minoux, M.: Mathematical Programming Theory and Algorithms. Wiley-Interscience Series in Discrete Mathematics and Optimization (1986)Google Scholar
  20. 20.
    Moore J.T. and Bard J.F. (1990). The mixed integer linear bi-level programming problem. Oper. Res. 38: 5 CrossRefGoogle Scholar
  21. 21.
    Papavassilopoulos, G.: Algorithms for static Stachelberg games with linear costs and polyhedral constraints. In: Proceeding of the 21st IEEE Conference on Decisions and control, pp. 647–652 (1982)Google Scholar
  22. 22.
    Sacharidis, G.K., Minoux, M., Ierapetritou, M.: Accelerating Benders decomposition using covering cut bundles generation. submitted (2008)Google Scholar
  23. 23.
    Savard, G.: Contributions à la programmation mathématique a deux niveaux. PhD thesis, Universite de Montreal, Ecole Polytechnique (1989)Google Scholar
  24. 24.
    Shi C., Lu J. and Zhang G. (2005). An extended Kuhn–Tucker approach for linear bi-level programming. Appl. Math. Comput. 162: 51–63 CrossRefGoogle Scholar
  25. 25.
    Shi C., Lu J., Zhang G. and Zhou H. (2006). An extended branch and bound algorithm for linear bilevel programming. Appl. Math. Comput. 180: 529–537 CrossRefGoogle Scholar
  26. 26.
    Tuy, H.: Handbook of applied optimization Panos M. Pardalos and Mauricion G.C. Resende chapter 12 Hierarchical optimization. Oxford University Press (2002)Google Scholar
  27. 27.
    Tuy H., Migdalas A. and Varbrand P. (1993). A global optimization approach for the linear two-level programs. J. Glob. Optim. 3: 1–23 CrossRefGoogle Scholar
  28. 28.
    Vincente L., Savard G. and Judice J. (1996). The discrete linear bi-level programming problem. J. Optim. Theory Appl. 89: 597–614 CrossRefGoogle Scholar
  29. 29.
    Visweswaran V., Floudas C.A., Ierapetritou M.G., Pistikopoulos E.N. (1996). A decomposition based global optimization approach for bi-level convex programming problems. In: Floudas C.A., Pardalos P.M. (eds). State of the Art in Global Optimization: Computational Methods and Applications, Book Series on Nonconvex Optimization and Its Applications. Kluwer Academic Publisher Printed in the Netherlands, pp. 139–162Google Scholar
  30. 30.
    Wang, G., Wan, Z., Wang, X.: Solving method for a class of bi-level linear programming based on genetic algorithms. Proceedings of PDCAT conference (2005)Google Scholar
  31. 31.
    Wen U.P. and Yang Y.H. (1990). Algorithms for solving the mixed integer two level linear programming problem. Comput. Op. Res. 17: 133–142 CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC. 2008

Authors and Affiliations

  1. 1.Department of Chemical and Biochemical EngineeringRutgers—The State University of New JerseyPiscatawayUSA

Personalised recommendations