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Convexification Method for Bilevel Programs with a Nonconvex Follower’s Problem

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Abstract

A new numerical method is presented for bilevel programs with a nonconvex follower’s problem. The basic idea is to piecewise construct convex relaxations of the follower’s problems, replace the relaxed follower’s problems equivalently by their Karush–Kuhn–Tucker conditions and solve the resulting mathematical programs with equilibrium constraints. The convex relaxations and needed parameters are constructed with ideas of the piecewise convexity method of global optimization. Under mild conditions, we show that every accumulation point of the optimal solutions of the sequence approximate problems is an optimal solution of the original problem. The convergence theorems of this method are presented and proved. Numerical experiments show that this method is capable of solving this class of bilevel programs.

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References

  1. Bard, J.F.: Practical bilevel optimization: Algorithms and applications. (Nonconvex optimization and its applications). Kluwer Academic Publishers, Dordrecht (1998)

  2. Dempe, S.: Foundations of bilevel programming. (Nonconvex optimization and its applications). Kluwer Academic Publishers, Dordrecht (2002)

  3. Shimizu, K., Ishizuka, Y., Bard, J.F.: Nondifferentiable and Two-Level Mathematical Programming. Kluwer Academic Publishers, New York (1997)

    Book  Google Scholar 

  4. Outrata, J.V.: On the numerical solution of a class of Stackelberg problems. Z. Oper. Res. 34(4), 255–277 (1990)

    MathSciNet  MATH  Google Scholar 

  5. Ye, J.J., Zhu, D.L.: Optimality conditions for bilevel programming problems. Optimization 39(4), 361–366 (1997)

    Article  MathSciNet  Google Scholar 

  6. Lin, G.H., Xu, M., Ye, J.J.: On solving simple bilevel programs with a nonconvex lower level program. Math. Program. 144(1–2), 277–305 (2014)

    Article  MathSciNet  Google Scholar 

  7. Xu, M., Ye, J., Zhang, L.: Smoothing SQP methods for solving degenerate nonsmooth constrained optimization problems with applications to bilevel programs. SIAM J. Optim. 25(3), 1388–1410 (2014)

    Article  MathSciNet  Google Scholar 

  8. Li, G., Wan, Z.: On bilevel programs with a convex lower-level problem violating Slater’s constraint qualification. J. Optim. Theory Appl. 179(3), 820–837 (2018)

    Article  MathSciNet  Google Scholar 

  9. Dempe, S., Mehlitz, P.: Semivectorial bilevel programming versus scalar bilevel programming. Optimization 69(4), 657–679 (2020)

    Article  MathSciNet  Google Scholar 

  10. Mirrlees, J.A.: The theory of moral hazard and unobservable behaviour: Part i. Rev. Econ. Stud. 66(1), 3–21 (1999)

    Article  Google Scholar 

  11. 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 

  12. Allende, G.B., Still, G.: Solving bilevel programs with the KKT-approach. Math. Program. 138(1–2), 309–332 (2013)

    Article  MathSciNet  Google Scholar 

  13. Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Ann. Oper. Res. 153(1), 235–256 (2007)

    Article  MathSciNet  Google Scholar 

  14. Dempe, S., Zemkoho, A.B.: The generalized Mangasarian–Fromowitz constraint qualification and optimality conditions for bilevel programs. J. Optim. Theory Appl. 148(1), 46–68 (2011)

    Article  MathSciNet  Google Scholar 

  15. Henrion, R., Surowiec, T.: On calmness conditions in convex bilevel programming. Appl. Anal. 90(6), 951–970 (2011)

    Article  MathSciNet  Google Scholar 

  16. Ye, J.J., Zhu, D.: New necessary optimality conditions for bilevel programs by combining the MPEC and value function approaches. SIAM J. Optim. 20(4), 1885–1905 (2010)

    Article  MathSciNet  Google Scholar 

  17. Facchinei, F., Jiang, H., Qi, L.: A smoothing method for mathematical programs with equilibrium constraints. Math. Program. 85(1), 107–134 (1999)

    Article  MathSciNet  Google Scholar 

  18. Guo, L., Lin, G.H., Ye, J.J.: Solving mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 166(1), 234–256 (2015)

    Article  MathSciNet  Google Scholar 

  19. Guo, L., Lin, G.H., Ye, J.J., Zhang, J.: Sensitivity analysis of the value function for parametric mathematical programs with equilibrium constraints. SIAM J. Optim. 24(3), 1206–1237 (2014)

    Article  MathSciNet  Google Scholar 

  20. Luo, Z.: Mathematical programs with equilibrium constraints. Cambridge University Press, Cambridge (1996)

    Book  Google Scholar 

  21. Ye, J.J.: Optimality conditions for optimization problems with complementarity constraints. SIAM J. Optim. 9(2), 374–387 (1999)

    Article  MathSciNet  Google Scholar 

  22. Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307(1), 350–369 (2004)

    Article  MathSciNet  Google Scholar 

  23. Adjiman, C.S., Dallwig, S., Floudas, C.A., Neumaier, A.: A global optimization method, \(\alpha \)BB, for general twice-differentiable constrained NLPs-I. Theoretical advances. Comput. Chem. Eng. 22(9), 1137–1158 (1998)

    Article  Google Scholar 

  24. Floudas, C.A.: Deterministic Global Optimization. Theory, Methods and Applications. Springer, Berlin (2000)

    Book  Google Scholar 

  25. Floudas, C.A., Stein, O.: The adaptive convexification algorithm: a feasible point method for semi-infinite programming. SIAM J. Optim. 18(4), 1187–1208 (2007)

    Article  MathSciNet  Google Scholar 

  26. Stein, O., Steuermann, P.: The adaptive convexification algorithm for semi-infinite programming with arbitrary index sets. Math. Program. 136(1), 183–207 (2012)

    Article  MathSciNet  Google Scholar 

  27. Lignola, M.B., Morgan, J.: Stability of regularized bilevel programming problems. J. Optim. Theory Appl. 93(3), 575–596 (1997)

    Article  MathSciNet  Google Scholar 

  28. Hogan, W.W.: Point-to-set maps in mathematical programming. SIAM Rev. 15(3), 591–603 (1973)

    Article  MathSciNet  Google Scholar 

  29. Bonnans, J.F., Shapiro, A.: Perturbation Analysis of Optimization Problems. Springer, Berlin (2000)

    Book  Google Scholar 

  30. Floudas, C.A.: Deterministic global optimization: theory, methods and applications, vol. 37. Springer, Berlin (2013)

    Google Scholar 

  31. Hansen, E., Walster, G.W.: Global Optimization Using Interval Analysis: Revised and Expanded, vol. 264. CRC Press, Boca Raton (2003)

    Book  Google Scholar 

  32. Floudas, C.A.: Deterministic global optimization: Theory, methods and applications, Kluwer Academic Publishers, Dordrecht (2000)

  33. Kanzow, C., Yamashita, N., Fukushima, M.: New NCP-functions and their properties. J. Optim. Theory Appl. 94(1), 115–135 (1997)

    Article  MathSciNet  Google Scholar 

  34. Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. 137(1), 257–288 (2013)

    Article  MathSciNet  Google Scholar 

  35. Kanzow, C.: Some noninterior continuation methods for linear complementarity problems. SIAM J. Matrix Anal. Appl. 17(4), 851–868 (1996)

    Article  MathSciNet  Google Scholar 

  36. Mitsos, A., Barton, P.I.: A test set for bilevel programs. Technical Report, Massachusetts Institute of Technology (2006). http://yoric.mit.edu/download/Reports/ bileveltestset.pdf

Download references

Acknowledgements

This work was supported by the Natural Science Foundation of China (11901068), China Postdoctoral Science Foundation(2020M673167), Natural Science Foundation of Chongqing (cstc2019jcyj-msxmX0456), The Project of Chongqing Technology and Business University (1952034,ZDPTTD201908,1856009)

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Authors and Affiliations

  1. School of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing, 400067, China

    Gaoxi Li

  2. Chongqing Key Laboratory of Social Economy and Applied Statistics, Chongqing, China

    Gaoxi Li

  3. School of Mathematics Science, Chongqing Normal University, Chongqing, 401331, China

    Xinmin Yang

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  1. Gaoxi Li
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  2. Xinmin Yang
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Correspondence to Xinmin Yang.

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Communicated by Guang-Ya Chen.

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Li, G., Yang, X. Convexification Method for Bilevel Programs with a Nonconvex Follower’s Problem. J Optim Theory Appl 188, 724–743 (2021). https://doi.org/10.1007/s10957-020-01804-9

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  • DOI: https://doi.org/10.1007/s10957-020-01804-9

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