Abstract
This paper addresses the problem of derivative-free multi-objective optimization of real-valued functions under multiple inequality constraints. Both the objective and constraint functions are assumed to be smooth, nonlinear, expensive-to-evaluate functions. As a consequence, the number of evaluations that can be used to carry out the optimization is very limited. The method we propose to overcome this difficulty has its roots in the Bayesian and multi-objective optimization literatures. More specifically, we make use of an extended domination rule taking both constraints and objectives into account under a unified multi-objective framework and propose a generalization of the expected improvement sampling criterion adapted to the problem. A proof of concept on a constrained multi-objective optimization test problem is given as an illustration of the effectiveness of the method.
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References
Au, S.K., Beck, J.L.: Estimation of small failure probabilities in high dimensions by subset simulation. Probab. Eng. Mech. 16(4), 263–277 (2001)
Bader, J., Zitzler, E.: Hype: An algorithm for fast hypervolume-based many-objective optimization. Evol. Comput. 19(1), 45–76 (2011)
Benassi, R., Bect, J., Vazquez, E.: Bayesian optimization using sequential Monte Carlo. In: Hamadi, Y., Schoenauer, M. (eds.) LION 2012. LNCS, vol. 7219, pp. 339–342. Springer, Heidelberg (2012)
Emmerich, M.T.M., Giannakoglou, K.C., Naujoks, B.: Single- and multi-objective evolutionary optimization assisted by Gaussian random field metamodels. IEEE Trans. Evol. Comput. 10(4), 421–439 (2006)
Emmerich, M., Klinkenberg, J.W.: The computation of the expected improvement in dominated hypervolume of Pareto front approximations. Leiden University, Rapport Technique (2008)
Fonseca, C.M., Fleming, P.J.: Multiobjective optimization and multiple constraint handling with evolutionary algorithms. I. A unified formulation. IEEE Trans. Syst. Man Cybern. B Cybern. Part A: Syst. Hum. 28(1), 26–37 (1998)
Gramacy, R.L., Lee, H.: Optimization under unknown constraints. In: Bayesian Statistics 9. In: Proceedings of the Ninth Valencia International Meeting, pp. 229–256. Oxford University Press (2011)
Hupkens, I., Emmerich, M., Deutz, A.: Faster computation of expected hypervolume improvement. arXiv preprint arXiv:1408.7114 (2014)
Jeong, S., Minemura, Y., Obayashi, S.: Optimization of combustion chamber for diesel engine using kriging model. J. Fluid Sci. Technol. 1(2), 138–146 (2006)
Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Global Optim. 13(4), 455–492 (1998)
Li, L., Bect, J., Vazquez, E.: Bayesian Subset Simulation: a kriging-based subset simulation algorithm for the estimation of small probabilities of failure. In: Proceedings of PSAM 2011 & ESREL 2012, 25–29 June 2012, Helsinki, Finland. IAPSAM (2012)
Liu, J.S.: Monte Carlo strategies in scientific computing. Springer, Heidelberg (2008)
Mockus, J.: Application of bayesian approach to numerical methods of global and stochastic optimization. J. Global Optim. 4(4), 347–365 (1994)
Osyczka, A., Kundu, S.: A new method to solve generalized multicriteria optimization problems using the simple genetic algorithm. Struct. Optim. 10(2), 94–99 (1995)
Oyama, A., Shimoyama, K., Fujii, K.: New constraint-handling method for multi-objective and multi-constraint evolutionary optimization. Trans. Jpn. Soc. Aeronaut. Space Sci. 50(167), 56–62 (2007)
Parr, J.M., Keane, A.J., Forrester, A.I.J., Holden, C.M.E.: Infill sampling criteria for surrogate-based optimization with constraint handling. Eng. Optim. 44(10), 1147–1166 (2012)
Picheny, V.: Multiobjective optimization using Gaussian process emulators via stepwise uncertainty reduction. Stat. Comput. 16 p. (2014). doi:10.1007/s11222-014-9477-x
Picheny, V.: A stepwise uncertainty reduction approach to constrained global optimization. In: Proceedings of the 17th International Conference on Artificial Intelligence and Statistics (AISTATS), Reykjavik, Iceland. vol. 33, pp. 787–795. JMLR: W&CP (2014)
Ray, T., Tai, K., Seow, K.C.: Multiobjective design optimization by an evolutionary algorithm. Eng. Optim. 33(4), 399–424 (2001)
Sasena, M.J., Papalambros, P., Goovaerts, P.: Exploration of metamodeling sampling criteria for constrained global optimization. Eng. Optim. 34(3), 263–278 (2002)
Schonlau, M., Welch, W.J., Jones, D.R.: Global versus local search in constrained optimization of computer models. In: New Developments and Applications in Experimental Design: Selected Proceedings of a 1997 Joint AMS-IMS-SIAM Summer Conference. IMS Lecture Notes-Monographs Series, vol. 34, pp. 11–25. Institute of Mathematical Statistics (1998)
Shimoyama, K., Sato, K., Jeong, S., Obayashi, S.: Updating kriging surrogate models based on the hypervolume indicator in multi-objective optimization. J. Mech. Des. 135(9), 094503 (2013)
Wagner, T., Emmerich, M., Deutz, A., Ponweiser, W.: On expected-improvement criteria for model-based multi-objective optimization. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds.) PPSN XI. LNCS, vol. 6238, pp. 718–727. Springer, Heidelberg (2010)
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This research work has been carried out in the frame of the Technological Research Institute SystemX, and therefore granted with public funds within the scope of the French Program Investissements d’Avenir.
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Feliot, P., Bect, J., Vazquez, E. (2015). A Bayesian Approach to Constrained Multi-objective Optimization. In: Dhaenens, C., Jourdan, L., Marmion, ME. (eds) Learning and Intelligent Optimization. LION 2015. Lecture Notes in Computer Science(), vol 8994. Springer, Cham. https://doi.org/10.1007/978-3-319-19084-6_24
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