Advertisement

Journal of Systems Science and Complexity

, Volume 26, Issue 6, pp 1019–1027 | Cite as

A penalty function method for solving ill-posed bilevel programming problem via weighted summation

  • Shihui JiaEmail author
  • Zhongping Wan
Article
  • 133 Downloads

Abstract

For ill-posed bilevel programming problem, the optimistic solution is always the best decision for the upper level but it is not always the best choice for both levels if the authors consider the model’s satisfactory degree in application. To acquire a more satisfying solution than the optimistic one to realize the two levels’ most profits, this paper considers both levels’ satisfactory degree and constructs a minimization problem of the two objective functions by weighted summation. Then, using the duality gap of the lower level as the penalty function, the authors transfer these two levels problem to a single one and propose a corresponding algorithm. Finally, the authors give an example to show a more satisfying solution than the optimistic solution can be achieved by this algorithm.

Key words

Bilevel programming duality gap penalty function satisfactory degree weighted summation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    Nie P, Another bilevel optimization problems, International Journal of Applied Mathematical Sciences, 2005, 2(1): 31–38.MathSciNetzbMATHGoogle Scholar
  2. [2]
    Bonnel H and Morgan J, Semivectorial bilevel optimization problems: Penalty approach, Journal of Optimization Theory and Applications, 2006, 131(3): 365–382.MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Ankhili Z and Mansouri A, An exact penalty on bilevel programs with linear vector optimization lower level, European Journal of Operational Research, 2009, 197(1): 36–41.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    Calvete H I and Gale C, On linear bilevel problems with multiple objectives at the lower level, Omega, 2011, 39(1): 33–40.CrossRefGoogle Scholar
  5. [5]
    Loridan P and Morgan J, Weak via strong stackelberg problem: New results, Journal of Global Optimization, 1996, 8(8): 263–287.MathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    Aboussoror A and Mansouri A, Weak lineae bilevel programming problems: Existence of solutions via a penalty method, Journal of Mathematical Analysis and Applications, 2005, 304(1): 399–408.MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Lü Y B, Hu T S, and Wan Z P, A penalty function method for solving weak price control problem, Applied Mathematics and Computations, 2007, 186(2): 1520–1525.CrossRefGoogle Scholar
  8. [8]
    Tsoukalas A, Wiesemann W, and Rustem B, Global optimization of pessimistic bi-level problems, Fields Institute Communications, 2009, 55: 1–29.MathSciNetGoogle Scholar
  9. [9]
    Cao D and Leung L, A partial cooperation model for non-unique linear two-level decision problems, European Journal of Operation Research, 2002, 140: 134–141.CrossRefzbMATHGoogle Scholar
  10. [10]
    Zheng Y, Wan Z P, and Wang G M, A fuzzy interactive method for a class of bilevel multiobjective programming problem, Expert Systems and Applications, 2011, 38: 10384–10388.CrossRefGoogle Scholar
  11. [11]
    Wang G M, Wang X J, and Wan Z P, A globally convergent algorithm for a class of bilevel nonlinear programming problem, Applied Mathematics and Computation, 2007, 188: 166–172.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Lü Y B, Chen Z, Wan Z P, and Wang G M, A penalty function method for solving nonlinear-linear bilevel programming problem, Journal of System Science and Mathematica Science, 2009, 29(5): 630–636 (in Chinese).zbMATHGoogle Scholar
  13. [13]
    Wan Z P, Wang G M, and Lü Y B, A dual-relax penalty function approach for solving nonlinear bilevel programming with linear lower level problem, Acta Mathematica Scientia, 2011, 31B(2): 652–660.Google Scholar
  14. [14]
    Zheng Y, Wan Z P, and Lü Y B, A global convergent method for nonlinear bilevel programming problem, Journal of System Science and Mathematica Science, 2012, 32(5): 513–521 (in Chinese).zbMATHGoogle Scholar
  15. [15]
    Zheng Y and Wan Z P, A solution method for semivectorial bilevel programming problem via penalty method, Journal of Applied Mathematics and Computing, 2011, 37(1–2): 207–219.MathSciNetCrossRefGoogle Scholar
  16. [16]
    Pramanik S and Kumar R, Fuzzy goal programming approach to multilevel programming problems, European Journal of Operational Research, 2007, 176: 1151–1166.CrossRefzbMATHGoogle Scholar
  17. [17]
    Shimizu K, Ishizuk Y, and Bard J F, Nondifferentiable and Two Level Mathematical Programming, Kluwer Academic Publishers, Boston, 1997.CrossRefzbMATHGoogle Scholar
  18. [18]
    Rockafellar R, Convex Analysis, Princeton University Press, Princeton, 1970.zbMATHGoogle Scholar

Copyright information

© Institute of Systems Science, Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Math Department, School of ScienceWuhan University of Science and TechnologyWuhanChina
  2. 2.Hubei Province Key Laboratory of Systems Science in Metallurgical ProcessWuhan University of Science and TechnologyWuhanChina
  3. 3.School of Mathematics and StatisticsWuhan UniversityWuhanChina

Personalised recommendations