Abstract
Bilevel optimization involves searching for the optimum of an upper level problem subject to optimality of a nested lower level problem. These are also referred to as the leader and follower problems, since the lower level problem is formulated based on the decision variables at the upper level. Most evolutionary algorithms designed to deal with such problems operate in a nested mode, which makes them computationally prohibitive in terms of the number of function evaluations. In the classical literature, one of the common ways of solving the problem has been to re-formulate it as a single-level problem using optimality measures (such as Karush-Kuhn-Tucker conditions) for lower level problem as complementary constraint(s). However, the mathematical properties such as linearity/convexity limits their application to more complex or black-box functions. In this study, we explore a non-nested strategy in the context of evolutionary algorithm. The constraints of the upper and lower level problems are considered together at a single-level while optimizing the upper level objective function. An additional constraint is formulated based on local exploration around the lower level decision vector, which reflects an estimate of its optimality. The approach is further enhanced through the use of periodic local search and selective “re-evaluation” of promising solutions. The proposed approach is implemented in a commonly used evolutionary algorithm framework and empirical results are shown for the SMD suite of test problems. A comparison is done with other established algorithms in the field such as BLEAQ, NBLEA, and BLMA to demonstrate the potential of the proposed approach.
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Please note that the current version of the code has certain modifications from the one used for the study in [20].
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References
Aiyoshi, E., Shimizu, K.: A solution method for the static constrained Stackelberg problem via penalty method. IEEE Trans. Autom. Control 29(12), 1111–1114 (1984)
Angelo, J.S., Krempser, E., Barbosa, H.J.: Differential evolution for bilevel programming. In: IEEE Congress on Evolutionary Computation, pp. 470–477 (2013)
Angelo, J.S., Krempser, E., Barbosa, H.J.: Differential evolution assisted by a surrogate model for bilevel programming problems. In: IEEE Congress on Evolutionary Computation, pp. 1784–1791 (2014)
Barbosa, H.J., Bernardino, H.S., Barreto, A.: Using performance profiles to analyze the results of the 2006 CEC constrained optimization competition. In: IEEE Congress on Evolutionary Computation, pp. 1–8 (2010)
Bard, J.F., Falk, J.E.: An explicit solution to the multi-level programming problem. Comput. Oper. Res. 9(1), 77–100 (1982)
Bhattacharjee, K.S., Singh, H.K., Ray, T.: Multi-objective optimization with multiple spatially distributed surrogates. J. Mech. Des. 138(9), 091401 (2016)
Byrd, R.H., Hribar, M.E., Nocedal, J.: An interior point algorithm for large-scale nonlinear programming. SIAM J. Optim. 9(4), 877–900 (1999)
Colson, B., Marcotte, P., Savard, G.: An overview of bilevel optimization. Ann. Oper. Res. 153(1), 235–256 (2007)
Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)
Islam, M.M., Singh, H.K., Ray, T.: A surrogate assisted approach for single-objective bilevel optimization. IEEE Trans. Evol. Comput. (2017)
Islam, M.M., Singh, H.K., Ray, T.: A memetic algorithm for solving single objective bilevel optimization problems. In: 2015 IEEE Congress on Evolutionary Computation (CEC), pp. 1643–1650. IEEE (2015)
Islam, M.M., Singh, H.K., Ray, T., Sinha, A.: An enhanced memetic algorithm for single-objective bilevel optimization problems. Evol. Comput. (2016). doi:10.1162/EVCO_a_00198
Jin, Y.: A comprehensive survey of fitness approximation in evolutionary computation. Soft. Comput. 9(1), 3–12 (2005)
Kirjner-Neto, C., Polak, E., Der Kiureghian, A.: An outer approximations approach to reliability-based optimal design of structures. J. Optim. Theory Appl. 98(1), 1–16 (1998)
Koh, A.: A metaheuristic framework for bi-level programming problems with multi-disciplinary applications. In: Talbi, E.-G. (ed.) Metaheuristics for Bi-level Optimization, vol. 482, pp. 153–187. Springer, Heidelberg (2013)
Migdalas, A.: Bilevel programming in traffic planning: models, methods and challenge. J. Global Optim. 7(4), 381–405 (1995)
Sinha, A., Malo, P., Deb, K.: An improved bilevel evolutionary algorithm based on quadratic approximations. In: IEEE Congress on Evolutionary Computation, pp. 1870–1877 (2014)
Sinha, A., Malo, P., Deb, K.: Nested bilevel evolutionary algorithm (N-BLEA) code and user guide. http://www.bilevel.org
Sinha, A., Malo, P., Deb, K.: Efficient evolutionary algorithm for single-objective bilevel optimization. arXiv preprint (2013). arXiv:1303.3901
Sinha, A., Malo, P., Deb, K.: Test problem construction for single-objective bilevel optimization. Evol. Comput. 22(3), 439–477 (2014)
Sinha, A., Malo, P., Frantsev, A., Deb, K.: Multi-objective Stackelberg game between a regulating authority and a mining company: a case study in environmental economics. In: IEEE Congress on Evolutionary Computation, pp. 478–485 (2013)
Sun, H., Gao, Z., Wu, J.: A bi-level programming model and solution algorithm for the location of logistics distribution centers. Appl. Math. Model. 32(4), 610–616 (2008)
Vicente, L.N., Calamai, P.H.: Bilevel and multilevel programming: a bibliography review. J. Global Optim. 5(3), 291–306 (1994)
Zhu, X., Yu, Q., Wang, X.: A hybrid differential evolution algorithm for solving nonlinear bilevel programming with linear constraints. IEEE International Conference on Cognitive Informatics, pp. 126–131 (2006)
Acknowledgments
The second author would like to acknowledge the Early Career Researcher (ECR) grant from the University of New South Wales, Australia.
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Islam, M.M., Singh, H.K., Ray, T. (2017). Use of a Non-nested Formulation to Improve Search for Bilevel Optimization. In: Peng, W., Alahakoon, D., Li, X. (eds) AI 2017: Advances in Artificial Intelligence. AI 2017. Lecture Notes in Computer Science(), vol 10400. Springer, Cham. https://doi.org/10.1007/978-3-319-63004-5_9
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