Abstract
We study a bilevel programming problem (BLPP) with a maximin lower level problem which is non-smooth and sometimes even non-convex. We reformulate such a non-smooth BLPP as a tractable single-level optimization problem and provide a verifiable sufficient condition to guarantee that the reformulated model is equivalent to the original one. We show that using the proposed approach, some type of BLPP can be reformulated as a simple convex optimization problem. It significantly reduces the computational complexity of the BLPP. The proposed approaches can be widely used for decision-making problems. We apply them to assembly and newsvendor problems in this paper.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Allende, G.B., Still, G.: Solving bilevel programs with the KKT-approach. Math. Program. 138(1–2), 309–332 (2013)
Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Ann. Oper. Res. 153(1), 235–256 (2007)
Dempe, S.: Annotated bibliography on bilevel programming and mathematical programs with equilibrium constraints. Optimization 52(3), 333–359 (2003)
Dempe, S., Mordukhovich, B.S., Zemkoho, A.B.: Sensitivity analysis for two-level value functions with applications to bilevel programming. SIAM J. Optim. 22(4), 1309–1343 (2012)
Dempe, S., Zemkoho, A.B.: The bilevel programming problem: reformulations, constraint qualifications and optimality conditions. Math. Program. 138(1–2), 447–473 (2013)
Dempe, S., Zemkoho, A.B.: KKT reformulation and necessary conditions for optimality in nonsmooth bilevel optimization. SIAM J. Optim. 24(4), 1639–1669 (2014)
Facchinei, F., Jiang, H., Qi, L.: A smoothing method for mathematical programs with equilibrium constraints. Math. Program. 85(1), 107–134 (1999)
Fletcher, R., Leyffer, S., Ralph, D., Scholtes, S.: Local convergence of SQP methods for mathematical programs with equilibrium constraints. SIAM J. Optim. 17(1), 259–286 (2006)
Guo, L., Lin, G.H., Ye, J.J.: Solving mathematical programs with equilibrium constraints. J. Optim. Theory Appl. 166(1), 234–256 (2015)
Guo, P.: One-shot decision approach and its application to duopoly market. Int. J. Inf. Decis. Sci. 2(3), 213–232 (2010)
Guo, P.: Private real estate investment analysis within one-shot decision framework. Int. Real Estate Rev. 13(3), 238–260 (2010)
Guo, P.: One-shot decision theory. IEEE Trans. Syst. Man Cybern. A Syst. Hum. 41(5), 917–926 (2011)
Guo, P.: Focus theory of choice and its application to resolving the St. Petersburg, Allais, and Ellsberg paradoxes and other anomalies. Eur. J. Oper. Res. 276(3), 1034–1043 (2019)
Guo, P., Yan, R., Wang, J.: Duopoly market analysis within one-shot decision framework with asymmetric possibilistic information. Int. J. Comput. Int. Syst. 3(6), 786–796 (2010)
Guo, P., Li, Y.: Approaches to multistage one-shot decision making. Eur. J. Oper. Res. 236(2), 612–623 (2014)
Guo, P., Ma, X.: Newsvendor models for innovative products with one-shot decision theory. Eur. J. Oper. Res. 239(2), 523–536 (2014)
Guo, P., Zhu, X.: Focus programming: a fundamental alternative for stochastic optimization problems (2019). Available at SSRN, https://ssrn.com/abstract=3334211 or https://doi.org/10.2139/ssrn.3334211
Hoheisel, T., Kanzow, C., Schwartz, A.: Theoretical and numerical comparison of relaxation methods for mathematical programs with complementarity constraints. Math. Program. 137(1–2), 257–288 (2013)
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)
Luo, Z.Q., Pang, J.S., Ralph, D.: Mathematical Programs with Equilibrium Constraints. Cambridge University Press, Cambridge (1996)
Outrata, J.V.: On the numerical solution of a class of Stackelberg problems. Math. Methods Oper. Res. 34(4), 255–277 (1990)
Scheel, H., Scholtes, S.: Mathematical programs with complementarity constraints: stationarity, optimality, and sensitivity. Math. Oper. Res. 25(1), 1–22 (2000)
Vicente, L.N., Calamai, P.H.: Bilevel and multilevel programming: a bibliography review. J. Global Optim. 5(3), 291–306 (1994)
von Stackelberg, H.: The Theory of the Market Economy. Oxford University Press, Oxford (1952)
Wang, C., Guo, P.: Behavioral models for first-price sealed-bid auctions with the one-shot decision theory. Eur. J. Oper. Res. 261(3), 994–1000 (2017)
Ye, J.J.: Necessary and sufficient optimality conditions for mathematical programs with equilibrium constraints. J. Math. Anal. Appl. 307(1), 350–369 (2005)
Ye, J.J., Zhu, D.L., Zhu, Q.J.: Exact penalization and necessary optimality conditions for generalized bilevel programming problems. SIAM J. Optim. 7(2), 481–507 (1997)
Zhu, X., Guo, P.: Approaches to four types of bilevel programming problems with nonconvex nonsmooth lower level programs and their applications to newsvendor problems. Math. Methods Oper. Res. 86(2), 255–275 (2017)
Zhu, X., Guo, P.: Bilevel programming approaches to production planning for multiple products with short life cycles. 4OR-Q. J. Oper. Res. (2019). https://doi.org/10.1007/s10288-019-00407-z
Acknowledgements
This research was supported by JSPS KAKENHI Grant No. 15K03599.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflicts of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Zhu, X., Guo, P. Single-level reformulations of a specific non-smooth bilevel programming problem and their applications. Optim Lett 14, 1393–1406 (2020). https://doi.org/10.1007/s11590-019-01444-7
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11590-019-01444-7