Nothing Special   »   [go: up one dir, main page]
Skip to main content
Springer Nature Link
Log in

On Bilevel Programs with a Convex Lower-Level Problem Violating Slater’s Constraint Qualification

  • Published:
Journal of Optimization Theory and Applications Aims and scope Submit manuscript

Abstract

This paper focuses on bilevel programs with a convex lower-level problem violating Slater’s constraint qualification. We relax the constrained domain of the lower-level problem. Then, an approximate solution of the original bilevel program can be obtained by solving this perturbed bilevel program. As the lower-level problem of the perturbed bilevel program satisfies Slater’s constraint qualification, it can be reformulated as a mathematical program with complementarity constraints which can be solved by standard algorithms. The lower convergence properties of the constraint set mapping and the solution set mapping of the lower-level problem of the perturbed bilevel program are expanded. We show that the solutions of a sequence of the perturbed bilevel programs are convergent to that of the original bilevel program under some appropriate conditions. And this convergence result is applied to simple trilevel programs.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

    Book  Google Scholar 

  2. Castro, S.L., Chela, J.L.: An inexact-restoration method for nonlinear bilevel programming problems. Comput. Optim. Appl. 43, 307–328 (2009)

    Article  MathSciNet  Google Scholar 

  3. Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Ann. Oper. Res. 153, 235–256 (2007)

    Article  MathSciNet  Google Scholar 

  4. Dempe, S., Kalashnikov, V., Prez-Valds, G.A., Kalashnykova, N.: Bilevel Programming Problems: Theory, Algorithms and Applications to Energy Networks. Springer, Berlin (2015)

    Book  Google Scholar 

  5. Li, G., Wan, Z., Zhao, X.: Optimality conditions for bilevel optimization problem with both levels programs being multiobjective. Pac. J. Optim. 14, 421–441 (2017)

    Google Scholar 

  6. Lv, Y., Hu, T., Wan, Z.: A penalty function method for solving weak price control problem. Appl. Math. Comput. 186, 1520–1525 (2007)

    MathSciNet  MATH  Google Scholar 

  7. Mitsos, A., Lemonidis, P., Barton, P.I.: Global solution of bilevel programs with a nonconvex inner program. J. Global Optim. 42, 475–513 (2008)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. Wang, G., Wang, X., Wan, Z., Lv, Y.: A globally convergent algorithm for a class of bilevel nonlinear programming problem. Appl. Math. Comput. 188, 166–172 (2007)

    MathSciNet  MATH  Google Scholar 

  10. Xu, M., Ye, J.J.: A smoothing augmented Lagrangian method for solving simple bilevel programs. Comput. Optim. Appl. 59, 353–377 (2014)

    Article  MathSciNet  Google Scholar 

  11. Ye, J.J.: Nondifferentiable multiplier rules for optimization and bilevel optimization problems. SIAM J. Optim. 15, 252–274 (2004)

    Article  MathSciNet  Google Scholar 

  12. Ye, J.J., Zhu, D., Zhu, Q.: Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. 7, 481–507 (1997)

    Article  MathSciNet  Google Scholar 

  13. Outrata, J.V.: On the numerical solution of a class of Stackelberg problems. Zeitschrift Fur Oper. Res. 34, 255–277 (1990)

    MathSciNet  MATH  Google Scholar 

  14. Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 39, 361–366 (1997)

    Article  MathSciNet  Google Scholar 

  15. Lin, G.H., Xu, M., Ye, J.J.: On solving simple bilevel programs with a nonconvex lower level program. Math. Program. Ser. A 144, 277–305 (2014)

    Article  MathSciNet  Google Scholar 

  16. Allende, G.B., Still, G.: Solving bilevel programs with the KKT-approach. Math. Program. Ser. A 138, 309–332 (2013)

    Article  MathSciNet  Google Scholar 

  17. Dempe, S., Zemkoho, A.B.: The generalized Mangasarian–Fromowitz constraint qualification and optimality conditions for bilevel programs. J. Optim. Theory Appl. 148, 46–68 (2011)

    Article  MathSciNet  Google Scholar 

  18. Henrion, R., Surowiec, T.: On calmness conditions in convex bilevel programming. Appl. Anal. 90, 951–970 (2011)

    Article  MathSciNet  Google Scholar 

  19. Dempe, S., Dutta, J.: Is bilevel programming a special case of a mathematical program with complementarity constraints? Math. Program. Ser. A 131, 37–48 (2012)

    Article  MathSciNet  Google Scholar 

  20. Ye, J.J.: Necessary optimality conditions for multiobjective bilevel programs. Math. Oper. Res. 36, 165–184 (2011)

    Article  MathSciNet  Google Scholar 

  21. Lignola, M.B., Morgan, J.: Stability of regularized bilevel programming problems. J. Optim. Theory Appl. 93, 575–596 (1997)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  23. Dempe, S., Mordukhovich, B.S., Zemkoho, A.B.: Sensitivity analysis of two-level value functions with applications to bilevel programming. SIAM J. Optim. 24, 1309–1343 (2012)

    Article  MathSciNet  Google Scholar 

  24. Rockafellar, R.T., Wets, R.J.B.: Variational Analysis. Springer, New York (2009)

    MATH  Google Scholar 

  25. Lignola, M.B., Morgan, J.: Convergences of marginal functions with dependent constraints. Optimization 23, 189–213 (1992)

    Article  MathSciNet  Google Scholar 

  26. Bard, J.F.: An investigation of the linear three level programming problem. IEEE Trans. Syst. Man Cybernet. 14, 711–717 (1984)

    Article  MathSciNet  Google Scholar 

  27. Guo, L., Lin, G.H., Ye, J.J.: Solving mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 166, 234–256 (2015)

    Article  MathSciNet  Google Scholar 

  28. Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. Ser. A 137, 257–288 (2013)

    Article  MathSciNet  Google Scholar 

  29. Kanzow, C., Schwartz, A.: A new regularization method for mathematical programs with complementarity constraints with strong convergence properties. SIAM J. Optim. 23, 770–798 (2013)

    Article  MathSciNet  Google Scholar 

  30. Outrata, J., Kocvara, M., Zowe, J.: Nonsmooth approach to optimization problems with equilibrium constraints: theory, applications and numerical results. Springer, New York (2013)

    MATH  Google Scholar 

  31. Chen, B., Harker, P.T.: A non-interior-point continuation method for linear complementarity problems. SIAM J. Matrix Anal. Appl. 14(4), 1168–1190 (1993)

    Article  MathSciNet  Google Scholar 

  32. Fischer, A.: A special newton-type optimization method. Optimization 24, 269–284 (1992)

    Article  MathSciNet  Google Scholar 

  33. Floudas, C.A., Stein, O.: The adaptive convexification algorithm: a feasible point method for semi-infinite programming. SIAM J. Optim. 18, 1187–1208 (2007)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

This work was supported by the Natural Science Foundation of China (71471140,11871383), High Level Introduction of Talent Research Start-up Fund No. 1856009, and Scientific and Technological Research Program of Chongqing Municipal Education Commission No. KJQN201800810.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, China

    Gaoxi Li

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, China

    Gaoxi Li & Zhongping Wan

Authors
  1. Gaoxi Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhongping Wan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gaoxi Li.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, G., Wan, Z. On Bilevel Programs with a Convex Lower-Level Problem Violating Slater’s Constraint Qualification. J Optim Theory Appl 179, 820–837 (2018). https://doi.org/10.1007/s10957-018-1392-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10957-018-1392-4

Keywords

Mathematics Subject Classification

Search

Navigation