Skip to main content
SpringerLink
Log in

Multiobjective bilevel optimization

  • FULL LENGTH PAPER
  • Series A
  • Published:
Mathematical Programming Submit manuscript

Abstract

In this work nonlinear non-convex multiobjective bilevel optimization problems are discussed using an optimistic approach. It is shown that the set of feasible points of the upper level function, the so-called induced set, can be expressed as the set of minimal solutions of a multiobjective optimization problem. This artificial problem is solved by using a scalarization approach by Pascoletti and Serafini combined with an adaptive parameter control based on sensitivity results for this problem. The bilevel optimization problem is then solved by an iterative process using again sensitivity theorems for exploring the induced set and the whole efficient set is approximated. For the case of bicriteria optimization problems on both levels and for a one dimensional upper level variable, an algorithm is presented for the first time and applied to two problems: a theoretical example and a problem arising in applications.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abo-Sinna M.: A bi-level non-linear multi-objective decision making under fuzziness. Opsearch 38(5), 484–495 (2001)

    MATH  MathSciNet  Google Scholar 

  2. Alt W.: Parametric optimization with applications to optimal control and sequential quadratic programming. Bayreuther Math. Schr. 35, 1–37 (1991)

    MATH  MathSciNet  Google Scholar 

  3. Bard, J.F.: Practical Bilevel Optimization. Algorithms and Applications. Nonconvex Optimization and Its Applications, vol. 30. Kluwer, Dordrecht (1998)

  4. Bonnel H., Morgan J.: Semivectorial bilevel optimization problem: Penalty approach. J. Optim. Theory Appl. 131(3), 365–382 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  5. Das I., Dennis J.: A closer look at drawbacks of minimizing weighted sums of objectives for Pareto set generation in multicriteria optimization problems. Struct. Optim. 14, 63–69 (1997)

    Article  Google Scholar 

  6. Das I., Dennis J.: Normal-boundary intersection: A new method for generating the Pareto surface in nonlinear multicriteria optimization problems. SIAM J. Optim. 8(3), 631–657 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  7. Dempe S.: Foundations of Bilevel Programming. Nonconvex Optimization and Its Applications, vol. 61. Kluwer, Dordrecht (2002)

  8. Dempe S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52(3), 333–359 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  9. Dempe, S.: Bilevel programming—a survey. Preprint 2003-11, Fakultät für Mathematik und Informatik, TU Bergakademie Freiberg, Germany (2003)

  10. Dinkelbach, W., Dürr, W.: Effizienzaussagen bei Ersatzprogrammen zum Vektormaximumproblem, iv. Oberwolfach-Tag. Operations Res 1971. Oper. Res. Verf. 12:69–77 (1972)

    Google Scholar 

  11. Ehrgott, M.: Multicriteria optimisation. Lect. Notes Econ. math. Syst, vol. 491. Springer, Berlin (2000)

  12. Eichfelder, G.: Parametergesteuerte Lösung nichtlinearer multikriterieller Optimierungsprobleme. PhD Thesis, Univ. Erlangen-Nürnberg, Germany (2006)

  13. Eichfelder, G.: Scalarizations for adaptively solving multi-objective optimization problems. Comput. Optim. Appl. (2007) doi:10.1007/s10589-007-9155-4

  14. Eichfelder, G.: An adaptive scalarization method in multi-objective optimization. SIAM J. Optim. (to appear)

  15. Fiacco, A.V.: Introduction to Sensitivity and Stability Analysis in Nonlinear Programming. Mathematics in Science and Engineering, vol. 165. Academic Press, London (1983)

  16. Fliege J.: Gap-free computation of Pareto-points by quadratic scalarizations. Math. Methods Oper. Res. 59(1), 69–89 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  17. Fliege J., Vicente LN: Multicriteria approach to bilevel optimization. J. Optim. Theory Appl. 131(2), 209–225 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  18. Haimes Y., Lasdon L., Wismer D.: On a bicriterion formulation of the problems of integrated system identification and system optimization. IEEE Trans. Syst. Man Cybern. 1, 296–297 (1971)

    Article  MATH  MathSciNet  Google Scholar 

  19. Hillermeier C., Jahn J.: Multiobjective optimization: survey of methods and industrial applications. Surv. Math. Ind. 11, 1–42 (2005)

    Article  MATH  Google Scholar 

  20. Jahn J.: Vector Optimization: Theory, Applications and Extensions. Springer, Berlin (2004)

    MATH  Google Scholar 

  21. Jahn J.: Multiobjective search algorithm with subdivision technique. Comput. Optim. Appl. 35, 161–175 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  22. Jahn J., Merkel A.: Reference point approximation method for the solution of bicriterial nonlinear optimization problems. J. Optim. Theory Appl. 74(1), 87–103 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  23. Jittorntrum K.: Solution point differentiability without strict complementarity in nonlinear programming. Math. Program. Study 21, 127–138 (1984)

    MATH  MathSciNet  Google Scholar 

  24. Kim I., de Weck O.: Adaptive weighted sum method for bi-objective optimization. Struct. Multidisciplinary Optim. 29, 149–158 (2005)

    Article  Google Scholar 

  25. Lin J.G.: On min-norm and min-max methods of multi-objective optimization. Math. Program. 103(1), 1–33 (2005)

    Article  MathSciNet  Google Scholar 

  26. Loridan P.: ε-solutions in vector minimization problems. J. Optim. Theory Appl. 43, 265–276 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  27. Marglin S.: Public Investment Criteria. MIT Press, Cambridge (1967)

    Google Scholar 

  28. Miettinen K.M.: Nonlinear Multiobjective Optimization. Kluwer, Boston (1999)

    MATH  Google Scholar 

  29. Nishizaki I., Sakawa M.: Stackelberg solutions to multiobjective two-level linear programming problems. J. Optim. Theory Appl. 103(1), 161–182 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  30. Osman M., Abo-Sinna M., Amer A., Emam O.: A multi-level nonlinear multi-objective decision-making under fuzziness. Appl. Math. Comput. 153(1), 239–252 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  31. Pascoletti A., Serafini P.: Scalarizing vector optimization problems. J. Optim. Theory Appl. 42(4), 499–524 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  32. Prohaska, J.: Optimierung von Spulenkonfigurationen zur Bewegung magnetischer Sonden. Diplomarbeit, Univ. Erlangen-Nürnberg, Germany (2005)

  33. Ruzika S., Wiecek M.: Approximation methods in multiobjective programming. J. Optim. Theory Appl. 126(3), 473–501 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  34. Sawaragi Y., Nakayama H., Tanino T.: Theory of Multiobjective Optimization. Number 176 in Mathematics in science and engineering. Academic Press, London (1985)

    Google Scholar 

  35. Schandl B., Klamroth K., Wiecek M.M.: Norm-based approximation in bicriteria programming. Comput. Optim. Appl. 20(1), 23–42 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  36. Shi X., Xia H.: Interactive bilevel multi-objective decision making. J. Oper. Res. Soc. 48(9), 943–949 (1997)

    MATH  Google Scholar 

  37. Shi X., Xia H.: Model and interactive algorithm of bi-level multi-objective decision-making with multiple interconnected decision makers. J. Multi-Criteria Decis. Anal. 10, 27–34 (2001)

    Article  MATH  Google Scholar 

  38. Staib T.: On two generalizations of Pareto minimality. J. Optim. Theory Appl. 59(2), 289–306 (1988)

    MATH  MathSciNet  Google Scholar 

  39. Teng C.-X., Li L., Li H.-B.: A class of genetic algorithms on bilevel multi-objective decision making problem. J. Syst. Sci. Syst. Eng. 9(3), 290–296 (2000)

    MathSciNet  Google Scholar 

  40. Tuy H., Migdalas A., Hoai-Phuong N.: A novel approach to bilevel nonlinear programming. J. Global Optim. 38(4), 527–554 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  41. Vicente L.N., Calamai P.H.: Bilevel and multilevel programming: A bibliography review. J. Glob. Optim. 5(3), 291–306 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  42. Yin Y.: Multiobjective bilevel optimization for transportation planning and management problems. J. Adv. Transp. 36(1), 93–105 (2000)

    Article  Google Scholar 

  43. Younes, Y.: Studies on discrete vector optimization. Dissertation, University of Demiatta, Egypt (1993)

Download references

Author information

Authors and Affiliations

  1. Institute of Applied Mathematics, University of Erlangen-Nürnberg, Martensstr. 3, 91058, Erlangen, Germany

    Gabriele Eichfelder

Authors
  1. Gabriele Eichfelder
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gabriele Eichfelder.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Eichfelder, G. Multiobjective bilevel optimization. Math. Program. 123, 419–449 (2010). https://doi.org/10.1007/s10107-008-0259-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10107-008-0259-0

Keywords

Mathematics Subject Classification (2000)

Search

Navigation