Abstract
This paper proposes a novel infill sampling criterion for constraint handling in multi-objective optimization of computationally expensive black-box functions. To reduce the computational burden, Kriging models are used to emulate the objective and constraint functions. The challenge of this multi-objective optimization problem arises from the fact that the epistemic uncertainty of the Kriging models should be taken into account to find Pareto-optimal solutions in the feasible domain. This is done by the proposed sampling criterion combining the Expected HyperVolume Improvement of the front of nondominated solutions and the Probability of Feasibility of new candidates. The proposed criterion is non-intrusive and derivative-free, and it is oriented to: (1) problems in which the computational cost is mainly from the function evaluation rather than optimization, and (2) problems that use complex in-house or commercial software that cannot be modified. The results using the proposed sampling criterion are compared with the results using Multi-Objective Evolutionary Algorithms. These results show that the proposed sampling criterion permits to identify both the feasible domain and an approximation of the Pareto front using a reduced number of computationally expensive simulations.
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References
Arora, J.S.: Introduction to Optimum Design. McGraw-Hill, New York (1989)
Bautista, D.C.: A Sequential Design for Approximating the Pareto Front Using the Expected Pareto Improvement Function. Ohio State University, PhD thesis (2009)
Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: multiobjective selection based on dominated hypervolume. Eur. J. Oper. Res. 181(3), 1653–1669 (2007)
Coello Coello, C.A.: Handling preferences in evolutionary multiobjective optimization: a survey. In: Proceedings of IEEE Evolutionary Computation, La Jolla, CA, pp. 30–37 (2000)
Coello Coello, C.A., Lamont, G.B., Van-Veldhuizen, D.A.: Evolutionary algorithms for solving multi-objective problems. In: Goldberg, D.E., Koza, J.R. (eds.) Genetic Algorithms and Evolutionary Computation. Springer, New York (2007)
Collette, Y., Siarry, P.: Multiobjective Optimization—Principles and Case Studies. Springer, Berlin (2003)
Couckuyt, I., Deschrijver, D., Dhaene, T.: Fast calculation of multiobjective probability of improvement and expected improvement criteria for Pareto optimization. J. Glob. Optim. 60(3), 575–594 (2014)
Deb, K.: Multi-objective Optimization Using Evolutionary Algorithms. Wiley, New York (2001)
Deb, K., Pratap, A., Agarwal, S., Meyarivan, : A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput. 6(2), 182–197 (2002)
Ehrgott, M.: Multiple Criteria Optimization—Classification and Methodology. Shaker Verlag, Aachen (1997)
Emmerich, M.T.M., Giannakoglou, K.C., Naujoks, B.: Single- and multiobjective evolutionary optimization assisted by Gaussian random field metamodels. IEEE Trans. Evol. Comput. 10(4), 421–439 (2006)
Emmerich, M., Deutz, A.H., Klinkenberg, J.W.: The computation of the expected improvement in dominated hypervolume of Pareto front approximations, Technical report, Leiden University (2008)
Feng, Z., Zhang, Q., Zhang, Q., Tang, Q., Yang, T., Ma, Y.: A multiobjective optimization based framework to balance the global exploration and local exploitation in expensive optimization. J. Glob. Optim. 61(4), 677–694 (2015)
Finkel, D.E.: DIRECT Optimization Algorithm User Guide. North Carolina State University. http://www4.ncsu.edu/definkel/research/index.html (2003). Accessed 28 Sep 2014
Forrester, A.I.J., Sobester, A., Keane, A.J.: Engineering Design via Surrogate Modelling—A Practical Guide. Wiley, New York (2008)
Forrester, A.I.J., Keane, A.J.: Recent advances in surrogate-based optimization. Prog. Aerosp. Sci. 45(1–3), 50–79 (2009)
Gaspar-Cunha, A., Vieira, A.: A multi-objective evolutionary algorithm using neural networks to approximate fitness evaluations. Int. J. Comput. Syst. Signals 6(1), 18–36 (2005)
Hawe, G.I., Sykulski, J.K.: An enhanced probability of improvement utility function for locating Pareto-optimal solutions. In: Proceedings of the Conference on Computation of Electromagnetic Fields, Aachen, Germany, pp. 965–966 (2007)
Hawe, G.I., Sykulski, J.K.: Scalarizing cost-effective multi-objective optimization algorithms made possible with kriging. COMPEL 27(4), 836–844 (2008)
Henkenjohann, N., Kunert, J.: An efficient sequential optimization approach based on the multivariate expected improvement criterion. Qual. Eng. 19(4), 267–280 (2007)
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., Perttunen, C.D., Stuckmann, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory App. 79(1), 157–181 (1993)
Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13(4), 455–492 (1998)
Jones, D.R.: A taxonomy of global optimization methods based on response surfaces. J. Glob. Optim. 21(4), 345–383 (2001)
Keane, A.J.: Statistical improvement criteria for use in multiobjective design optimization. AIAA J. 44(4), 879–891 (2006)
Kleijnen, J.P.C.: Kriging metamodeling in simulation: a review. Eur. J. Oper. Res. 192(3), 707–716 (2009)
Knowles, J.: ParEGO: a hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE Trans. Evol. Comput. 10(1), 50–66 (2006)
Krige, D.G.: A Statistical approach to some basic mine valuation problems on the Witwatersrand. J. South. Afr. Inst. Min. Metall. 52(6), 119–139 (1951)
Li, M., Li, G., Azarm, S.: A Kriging metamodel assisted multi-objective genetic algorithm for design optimization. J. Mech. Des. 130(3), 1–10 (2008)
Li, G., Li, M., Azarm, S., Al Hashimi, S., Al Ameri, T., Al Qasas, N.: Improving multi-objective genetic algorithms with adaptive design of experiments and online metamodeling. Struct. Multidiscip. Optim. 37(5), 447–461 (2009)
Liu, Q., Cheng, W.: A modified DIRECT algorithm with bilevel partition. J. Glob. Optim. 60(3), 483–499 (2014)
Marler, R.T., Arora, J.S.: Survey of multi-objective optimization methods for engineering. Struct. Multidiscip. Optim. 26(6), 369–395 (2004)
Meckesheimer, M., Booker, A.J., Barton, R.R., Simpson, T.W.: Computationally inexpensive metamodel assessment strategies. AIAA J. 40(10), 2053–2060 (2002)
Mockus, J., Tiesis, V., Zilinskas, A.: The application of Bayesian methods for seeking the extremum. In: Dixon, L.C.W., Szego, G.P. (eds.) Toward Global Optimisation 2, pp. 117–130. Elsevier, Amsterdam (1978)
Nielsen, H.B., Lophaven, S.N., Søndergaard, J.: DACE—A Matlab Kriging Toolbox. Department of Informatics and Mathematical Modeling, Technical University of Denmark (2002)
Obayashi, S.: Multi-objective design exploration using efficient global optimization. In: Proceedings of the European Conference on Computational Fluid Dynamics (ECCOMAS CFD), TU Delft, pp. 1–8 (2006)
Pareto, V.: Manuale di Economia Politica. Societa Editrice, Milan (1906)
Ponweiser, W., Wagner, T., Biermann, D., Vincze, M.: Multiobjective optimization on a limited budget of evaluations using model-assisted S-Metric selection. In: Rudolph, G., Jansen, T., Lucas, S., Poloni, C., Beume, N. (eds.) Parallel Problem Solving from Nature—PPSN X, pp. 784–794. Springer, Heidelberg (2008)
Regis, R.G., Shoemaker, C.A.: Constrained global optimization of expensive black box functions using radial basis functions. J. Glob. Optim. 31(1), 153–171 (2005)
Regis, R.G.: Constrained optimization by radial basis function interpolation for high-dimensional expensive black-box problems with infeasible initial points. Eng. Optim. 46(2), 218–243 (2014)
Regis, R.G.: Evolutionary programming for high-dimensional constrained expensive black-box optimization using radial basis functions. IEEE Trans. Evolut. Comput. 18(3), 326–347 (2014)
Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4(4), 409–435 (1989)
Santner, T., Williams, B., Notz, W.: The Design and Analysis of Computer Experiments. Springer Series in Statistics. Springer, Berlin (2003)
Sasena, M.J., Papalambros, P.Y., Goovaerts, P.: Exploration of metamodeling sampling criteria for constrained global optimization. Eng. Optim. 34(3), 263–278 (2002)
Singh, P., Couckuyt, I., Ferranti, F., Dhaene, T.: A constrained multi-objective surrogate-based optimization algorithm. In: Proceedings of World Congress on Computational Intelligence, Beijing, China (2014)
Shimoyama, K., Jeong, S., Obayashi, S.: Kriging-surrogate-based optimization considering expected hypervolume improvement in non-constrained many-objective test problems. In: Proceedings of IEEE Congress on Evolutionary Computation (CEC), Cancun, Mexico, pp. 658–665 (2013)
Ulmer, H., Streichert, F., Zell, A.: Evolution strategies assisted by Gaussian processes with improved pre-selection criterion. In: Proceedings of IEEE Congress on Evolutionary Computation (CEC), Canberra, Australia, pp. 692–699 (2003)
Voutchkov, I., Keane, A.: Multiobjective optimization using surrogates. In: Tenne, Y., Goh, C.-K. (eds.) Computational Intelligence in Optimization—Applications and Implementations, pp. 155–175. Springer, Berlin (2010)
Wang, G., Shan, S.: An efficient Pareto set identification approach for multiobjective optimization on black-box functions. J. Mech. Des. 127(5), 866–874 (2005)
Wolfe, M.A.: An interval algorithm for bound constrained global optimization. Optim. Methods Softw. 6(2), 145–159 (1995)
Zhang, Y., Leithead, W.: Exploiting Hessian matrix and trust-region algorithm in hyperparameters estimation of Gaussian process. Appl. Math. Comput. 171(2), 1264–1281 (2005)
Zhang, Q., Liu, W., Tsang, E., Virginas, B.: Expensive multiobjective optimization by MOEA/D with Gaussian process model. IEEE Trans. Evol. Comp. 14(3), 456–474 (2010)
Zilinskas, A.: One-step Bayesian method of the search for extremum of an one-dimensional function. Cybernetics 1, 139–144 (1975)
Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: improving the strength Pareto evolutionary algorithm for multiobjective optimization. In: Giannakoglou, K.C., et al. (eds.) Evolutionary Methods for Design, Optimisation and Control with Application to Industrial Problems, pp. 95–100. CIMNE, Barcelona (2002)
Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comp. 7(2), 117–132 (2003)
Zuluaga, M., Krause, A., Sergent, G., Markus, P.: Active learning for multi-objective optimization. In: Proceedings of International Conference on Machine Learning, Atlanta, USA (2013)
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This work was supported through the research support programme from “Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia” under the contract 19274/PI/14.
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Martínez-Frutos, J., Herrero-Pérez, D. Kriging-based infill sampling criterion for constraint handling in multi-objective optimization. J Glob Optim 64, 97–115 (2016). https://doi.org/10.1007/s10898-015-0370-8
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DOI: https://doi.org/10.1007/s10898-015-0370-8