Abstract
Bilevel problems (BLPs) involve solving two nested levels of optimization, namely the upper (leader) and the lower (follower) level, usually motivated by real-world situations involving a hierarchical structure. BLPs are known to be hard and computationally expensive. When the computation of the objective functions and/or constraints require an expensive computer simulation, as in “black-box” optimization, evolutionary algorithms (EAs) are often used. As EAs may become very expensive by requiring a large number of function evaluations, surrogate models can help overcome this drawback, either by replacing expensive evaluations or allowing for increased exploration of the search space. Here we apply different types of local surrogate models at the upper level optimization, in order to enhance the overall performance of the proposed method, which is studied by means of computational experiments conducted on the well-known SMD benchmark problems.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
References
Angelo, J.S., Barbosa, H.J.C.: A study on the use of heuristics to solve a bilevel programming problem. Int. Trans. Oper. Res. 22(5), 861–882 (2015)
Angelo, J.S., Krempser, E., Barbosa, H.J.C.: Differential evolution for bilevel programming. In: IEEE Congress on Evolutionary Computation, pp. 470–477 (2013)
Angelo, J.S., Krempser, E., Barbosa, H.J.C.: Differential evolution assisted by a surrogate model for bilevel programming problems. In: IEEE Congress on Evolutionary Computation, pp. 1784–1791 (2014)
Bard, J.F.: Practical Bilevel Optimization. Kluwer Academic Publisher, Boston (1998)
Barjhoux, P.J., Diouane, Y., Grihon, S., Bettebghor, D., Morlier, J.: A bilevel methodology for solving a structural optimization problem with both continuous and categorical variables. In: 2018 Multidisciplinary Analysis and Optimization Conference, pp. 1–16 (2018)
Broomhead, D., Lowe, D.: Multivariable functional interpolation and adaptive networks. Complex Syst. 2, 321–355 (1988)
Ciccazzo, A., Latorre, V., Liuzzi, G., Lucidi, S., Rinaldi, F.: Derivative-free robust optimization for circuit design. J. Optim. Theory Appl. 164(3), 842–861 (2015)
Colson, B., Marcotte, P., Savard, G.: A trust-region method for nonlinear bilevel programming: algorithm and computational experience. Comput. Optim. Appl. 30(3), 211–227 (2005)
Conn, A.R., Vicente, L.N.: Bilevel derivative-free optimization and its application to robust optimization. Optim. Methods Softw. 27(3), 561–577 (2012)
Deb, K., Sinha, A.: Solving bilevel multi-objective optimization problems using evolutionary algorithms. In: Ehrgott, M., Fonseca, C.M., Gandibleux, X., Hao, J.-K., Sevaux, M. (eds.) EMO 2009. LNCS, vol. 5467, pp. 110–124. Springer, Heidelberg (2009). https://doi.org/10.1007/978-3-642-01020-0_13
Dempe, S.: Foundations of Bilivel Programming. Kluwer Academic Publisher, Dordrecht (2002)
Dempe, S.: Bilevel optimization: theory, algorithms and applications (2018). https://tu-freiberg.de/sites/default/files/media/fakultaet-fuer-mathematik-und-informatik-fakultaet-1-9277/prep/preprint_2018_11_dempe.pdf
Hansen, P., Jaumard, B., Savard, G.: New branch-and-bound rules for linear bilevel programming. SIAM J. Sci. Stat. Comput. 13(5), 1194–1217 (1992)
Islam, M.M., Singh, H.K., Ray, T.: A surrogate assisted approach for single-objective bilevel optimization. IEEE Trans. Evol. Comput. 21(5), 681–696 (2017)
Islam, M.M., Singh, H.K., Ray, T.: Efficient global optimization for solving computationally expensive bilevel optimization problems. In: 2018 IEEE Congress on Evolutionary Computation, pp. 1–8, July 2018
Krempser, E., Bernardino, H.S., Barbosa, H.J., Lemonge, A.C.: Performance evaluation of local surrogate models in differential evolution-based optimum design of truss structures. Eng. Comput. 34(2), 499–547 (2017)
Shepard, D.: A two-dimensional interpolation function for irregularly-spaced data. In: Proceedings of the 23rd ACM National Conference, New York, NY, USA, pp. 517–524 (1968)
da Silva, E.K., Barbosa, H.J.C., Lemonge, A.C.C.: An adaptive constraint handling technique for differential evolution with dynamic use of variants in engineering optimization. Optim. Eng. 12, 31–54 (2011)
Sinha, A., Malo, P., Deb, K.: An improved bilevel evolutionary algorithm based on quadratic approximations. In: 2014 IEEE Congress on Evolutionary Computation, pp. 1870–1877 (2014)
Sinha, A., Malo, P., Deb, K.: A review on bilevel optimization: from classical to evolutionary approaches and applications. IEEE Trans. Evol. Comput. 22(2), 276–295 (2018)
Sinha, A., Lu, Z., Deb, K., Malo, P.: Bilevel optimization based on iterative approximation of multiple mappings. arXiv preprint arXiv:1702.03394 (2017)
Sinha, A., Malo, P., Deb, K.: Efficient evolutionary algorithm for single-objective bilevel optimization. CoRR abs/1303.3901 (2013)
Sinha, A., Malo, P., Deb, K.: Test problem construction for single-objective bilevel optimization. Evol. Comput. 22(3), 439–477 (2014). pMID: 24364674
Stackelberg, H.V.: Marktform und Gleichgewicht. Springer, Berlin (1934). English translation: The Theory of the Market Economy. Oxford University Press, Oxford (1952)
Storn, R., Price, K.V.: Differential evolution: a simple and efficient adaptive scheme for global optimization over continuous spaces (1995). iCSI, USA, Technical report, TR-95-012 (1995). http://icsi.berkeley.edu/~storn/litera.html
Storn, R., Price, K.V.: Differential evolution - a simple and efficient heuristic for global optimization over continuous spaces. J. Global Optim. 11, 341–359 (1997)
Vicente, L.N., Savard, G., Júdice, J.J.: Descent approaches for quadratic bilevel programming. J. Optim. Theory Appl. 81(2), 379–399 (1994)
Yan, X., Su, X.G.: Linear Regression Analysis: Theory and Computing. World Scientific Publishing Company, Singapore (2009)
Zhang, D., Lin, G.H.: Bilevel direct search method for leader-follower problems and application in health insurance. Comput. Oper. Res. 41, 359–373 (2014)
Acknowledgement
The authors thank the reviewers for their comments and observations, and PNPD/CAPES and CNPq (grant 312337/2017-5), for the financial support.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Angelo, J.S., Krempser, E., Barbosa, H.J.C. (2019). Performance Evaluation of Local Surrogate Models in Bilevel Optimization. In: Nicosia, G., Pardalos, P., Umeton, R., Giuffrida, G., Sciacca, V. (eds) Machine Learning, Optimization, and Data Science. LOD 2019. Lecture Notes in Computer Science(), vol 11943. Springer, Cham. https://doi.org/10.1007/978-3-030-37599-7_29
Download citation
DOI: https://doi.org/10.1007/978-3-030-37599-7_29
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-37598-0
Online ISBN: 978-3-030-37599-7
eBook Packages: Computer ScienceComputer Science (R0)