Advertisement

Mathematical Methods of Operations Research

, Volume 86, Issue 2, pp 255–275 | Cite as

Approaches to four types of bilevel programming problems with nonconvex nonsmooth lower level programs and their applications to newsvendor problems

  • Xide Zhu
  • Peijun GuoEmail author
Original Article

Abstract

This paper concentrates on solving bilevel programming problems where the lower level programs are max–min optimization problems and the upper level programs have max–max or max–min objective functions. Because these bilevel programming problems include nonconvex and nonsmooth lower level program problems, it is a challenging undone work. Giving some assumptions, we translate these problems into general single level optimization problems or min–max optimization problems. To deal with these equivalent min–max optimization problems, we propose a class of regularization methods which approximate the maximum function by using a family of maximum entropy functions. In addition, we examine the limit situations of the proposed regularization methods and show that any limit points of the global optimal solutions obtained by the approximation methods are the same as the ones of the original problems. Finally, we apply the proposed methods to newsvendor problems and use a numerical example to show their effectiveness.

Keywords

Bilevel programming Min–max optimization Nonconvex Nonsmooth Newsvendor problem One-shot decision theory 

References

  1. Allende GB, Still G (2013) Solving bilevel programs with the KKT-approach. Math Program 138(1–2):309–332CrossRefzbMATHMathSciNetGoogle Scholar
  2. Bard JF (1998) Practical bilevel optimization: algorithms and applications, vol 30. Springer, BerlinzbMATHGoogle Scholar
  3. Colson B, Marcotte P, Savard G (2005) Bilevel programming: a survey. 4OR-Q J Oper Res 3(2):87–107CrossRefzbMATHMathSciNetGoogle Scholar
  4. Dempe S (2002) Foundations of bilevel programming. Springer, BerlinzbMATHGoogle Scholar
  5. Dempe S, Zemkoho AB (2013) The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math Program 138(1–2):447–473CrossRefzbMATHMathSciNetGoogle Scholar
  6. Dempe S, Zemkoho AB (2014) KKT reformulation and necessary conditions for optimality in nonsmooth bilevel optimization. SIAM J Optim 24(4):1639–1669CrossRefzbMATHMathSciNetGoogle Scholar
  7. Dempe S, Mordukhovich BS, Zemkoho AB (2012) Sensitivity analysis for two-level value functions with applications to bilevel programming. SIAM J Optim 22(4):1309–1343CrossRefzbMATHMathSciNetGoogle Scholar
  8. Facchinei F, Jiang H, Qi L (1999) A smoothing method for mathematical programs with equilibrium constraints. Math Program 85(1):107–134CrossRefzbMATHMathSciNetGoogle Scholar
  9. Fletcher R, Leyffer S, Ralph D, Scholtes S (2006) Local convergence of SQP methods for mathematical programs with equilibrium constraints. SIAM J Optim 17(1):259–286CrossRefzbMATHMathSciNetGoogle Scholar
  10. Guo P (2010a) One-shot decision approach and its application to duopoly market. Int J Inf Decis Sci 2(3):213–232Google Scholar
  11. Guo P (2010b) Private real estate investment analysis within one-shot decision framework. Int Real Estate Rev 13(3):238–260Google Scholar
  12. Guo P (2011) One-shot decision theory. IEEE Trans On Syst Man Cybern A Syst Hum 41(5):917–926CrossRefGoogle Scholar
  13. Guo P, Li Y (2014) Approaches to multistage one-shot decision making. Eur J Oper Res 236(2):612–623CrossRefzbMATHMathSciNetGoogle Scholar
  14. Guo P, Ma X (2014) Newsvendor models for innovative products with one-shot decision theory. Eur J Oper Res 239(2):523–536CrossRefzbMATHMathSciNetGoogle Scholar
  15. Guo L, Lin GH, Ye JJ (2015) Solving mathematical programs with equilibrium constraints. J Optim Theory Appl 166(1):234–256CrossRefzbMATHMathSciNetGoogle Scholar
  16. Li XS, Fang SC (1997) On the entropic regularization method for solving min–max problems with applications. Math Methods Oper Res 46(1):119–130CrossRefzbMATHMathSciNetGoogle Scholar
  17. Li Y, Guo P (2015) Possibilistic individual multi-period consumption-investment models. Fuzzy Sets Syst 274:47–61CrossRefMathSciNetzbMATHGoogle Scholar
  18. Lin GH, Fukushima M (2005) A modified relaxation scheme for mathematical programs with complementarity constraints. Ann Oper Res 133(1–4):63–84CrossRefzbMATHMathSciNetGoogle Scholar
  19. Lin GH, Xu M, Ye JJ (2014) On solving simple bilevel programs with a nonconvex lower level program. Math Program 144(1–2):277–305CrossRefzbMATHMathSciNetGoogle Scholar
  20. Luo ZQ, Pang JS, Ralph D (1996) Mathematical programs with equilibrium constraints. Cambridge University Press, CambridgeCrossRefzbMATHGoogle Scholar
  21. Outrata JV (1990) On the numerical solution of a class of Stackelberg problems. Z Oper Res 34(4):255–277zbMATHMathSciNetGoogle Scholar
  22. Rockafellar RT, Wets RJB (1998) Variational analysis. Springer, BerlinCrossRefzbMATHGoogle Scholar
  23. Scholtes S (2001) Convergence properties of a regularization scheme for mathematical programs with complementarity constraints. SIAM J Optim 11(4):918–936CrossRefzbMATHMathSciNetGoogle Scholar
  24. Stephen B, Vandenberghe L (2004) Convex optimization. Cambridge University Press, CambridgezbMATHGoogle Scholar
  25. Vicente LN, Calamai PH (1994) Bilevel and multilevel programming: a bibliography review. J Global Optim 5(3):291–306CrossRefzbMATHMathSciNetGoogle Scholar
  26. Von Stackelberg H (1952) The theory of the market economy. Oxford University Press, OxfordGoogle Scholar
  27. Wang C, Guo P (2017) Behavioral models for first-price sealed-bid auctions with the one-shot decision theory. Eur J Oper Res 261(3):994–1000CrossRefMathSciNetGoogle Scholar
  28. Xu M, Ye JJ (2014) A smoothing augmented Lagrangian method for solving simple bilevel programs. Comput Optim Appl 59(1–2):353–377CrossRefzbMATHMathSciNetGoogle Scholar
  29. Ye JJ (2005) Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J Math Anal Appl 307(1):350–369CrossRefzbMATHMathSciNetGoogle Scholar
  30. Ye JJ, Zhu DL (1995) Optimality conditions for bilevel programming problems. Optimization 33(1):9–27CrossRefzbMATHMathSciNetGoogle Scholar
  31. Ye JJ, Zhu DL (2010) New necessary optimality conditions for bilevel programs by combining the MPEC and value function approaches. SIAM J Optim 20(4):1885–1905CrossRefzbMATHMathSciNetGoogle Scholar
  32. Zhu X, Lin GH (2016) Improved convergence results for a modified Levenberg–Marquardt method for nonlinear equations and applications in MPCC. Optim Methods Softw 31(4):791–804CrossRefzbMATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Faculty of Business AdministrationYokohama National UniversityHodogaya-ku, YokohamaJapan

Personalised recommendations