Abstract
Surrogate model assisted evolutionary algorithms (SAEAs) are metamodel-based strategies usually employed on the optimization of problems that demand a high computational cost to be evaluated. SAEAs employ metamodels, like Kriging and radial basis function (RBF), to speed up convergence towards good quality solutions and to reduce the number of function evaluations. However, investigations concerning the influence of metamodels in SAEAs performance have not been developed yet. In this context, this paper performs an investigative study on commonly adopted metamodels to compare the ordinary Kriging (OK), first-order universal Kriging (UK1), second-order universal Kriging (UK2), blind Kriging (BK) and RBF metamodels performance when embedded into a single-objective SAEA Framework (SAEA/F). The results obtained suggest that the OK metamodel presents a slightly better improvement than the others, although it does not present statistically significant difference in relation to UK1, UK2, and BK. The RBF showed the lowest computational cost, but the worst performance. However, this worse performance is around 2% in relation to the other metamodels. Furthermore, the results show that BK presents the highest computational cost without any significant improvement in solution quality when compared to OK, UK1, and UK2.
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References
Coello, C.C., Lamont, G.B., van Veldhuizen, D.A.: Evolutionary Algorithms for Solving Multi-objective Problems. Genetic and Evolutionary Computation, 2nd edn. Springer, Berlin (2007)
Collette, Y., Siarry, P.: Multiobjective Optimization: Principles and Case Studies. Springer, Berlin (2004)
Deb, K.: Multi-objective Optimization using Evolutionary Algorithms, 1st edn. Wiley, Hoboken (2001)
Jin, Y.: A comprehensive survey of fitness approximation in evolutionary computation. Soft. Comput. 9(1), 3–12 (2005)
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)
Liu, B., Zhang, Q., Gielen, G.G.E.: A Gaussian process surrogate model assisted evolutionary algorithm for medium scale expensive optimization problems. IEEE Trans. Evol. Comput. 18(2), 180–192 (2014)
Sun, C., Jin, Y., Cheng, R., Ding, J., Zeng, J.: Surrogate-assisted cooperative swarm optimization of high-dimensional expensive problems. IEEE Trans. Evol. Comput. 21(4), 644–660 (2017)
Büche, D., Scharaudolph, N.N., Koumountsakos, P.: Accelerating evolutionary algorithms with Gaussian process fitness function models. IEEE Trans. Syst. Man Cybern. C (Appl. Rev) 35(2), 183–194 (2005)
Zhou, Z., Ong, Y.S., Nair, P.B., Keane, A.J., Lum, K.Y.: Combining global and local surrogate models to accelerate evolutionary optimization. IEEE Trans. Syst. Man Cybern. C (Appl. Rev) 37(1), 66–76 (2007)
Lim, D., Jin, Y., Ong, Y.S., Sendhoff, B.: Generalizing surrogate-assisted evolutionary computation. IEEE Trans. Evol. Comput. 14, 329–355 (2010)
Hao, W., Shaoping, W., Tomovic, M.M.: Modified sequential kriging optimization for multidisciplinary complex product simulation. Chin. J. Aeronaut. 23(5), 616–622 (2010)
Schonlau, M.: Computer experiments and global optimization. Ph.D. thesis, University of Waterloo (1997)
Forrester, A.I.J., Sóbester, A., Keane, A.J.: Engineering Design via Surrogate Modelling: A Practical Guide. Wiley, Hoboken (2008)
Jin, Y., Olhofer, M., Sendhoff, B.: On evolutioary optimization with approximate fitness function. In: Genetic and Evolutionary Computation Conference, pp. 786–793 (2000)
Zhao, L., Choi, K.K., Lee, I.: Metamodeling method using dynamic kriging for design optimization. AIAA J. 49(9), 2034–2046 (2011)
Xia, B., Baatar, N., Ren, Z., Koh, C.S.: A numerically efficient muli-objective optimization algorithm: combination of dynamic Taylor Kriging and differential evolution. IEEE Trans. Magn. 51(3), 1–4 (2015)
Xia, B., Ren, Z., seop Koh, C.: Comparative study on Kriging surrogate models for metaheuristic optimization of multidimensional eletromagnetic problems. IEEE Trans. Magn. 51(3), 1–4 (2015)
Palar, P.S., Shimoyama, K.: On efficient global optimization via universal Kriging surrogate models. Struct. Multidiscip. Optim. 57(6), 2377–2397 (2017)
Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Global Optim. 13(4), 455–492 (1998)
Giunta, A.A., Watson, L.T.: A comparison of approximation modeling techniques-polynomial versus interpolating models. In: 7th AIAA/USAF/NASA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, p. 4758 (1998)
Jin, R., Chen, W., Simpson, T.W.: Comparative studies of metamodelling techniques under multiple modelling criteria. Struct. Multidiscip. Optim. 23(1), 1–13 (2001)
Daberkow, D.D., Mavris, D.N.: New approaches to conceptual and preliminary aircraft design: a comparative assessment of a neural network formulation and a response surface methodology. In: 1998 World Aviation Conference, 985509, pp. 1–13. American Institute of Aeronautics and Astronautics (AIAA), Anaheim, CA (1998)
Trosset, M.W., Torczon, V.: Numerical optimization using computer experiments. Technical Report 97-38, Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton VA (1997)
Torczon, V., Trosset, M.W.: Using approximation to accelerate engineering design optimization. Technical Report 98-33, Institute for Computer Applications in Science and Engineering, NASA Langley Research Center (1998)
Ma, H., Fei, M., Simon, D., Mo, H.: Update-based evolution control: a new fitness approximation method for evolutionary algorithms. Eng. Optim. 47(9), 1177–1190 (2015)
Regis, R.G., Shoemaker, C.A.: Local function approximation in evolutionary algorithms for the optimization of costly functions. IEEE Trans. Evol. Comput. 8(5), 490–505 (2004)
Sacks, J., Welch, W.J., Mitchell, T.J., Wynn, H.P.: Design and analysis of computer experiments. Stat. Sci. 4(4), 409–423 (1989)
Joseph, V.R., Hung, Y., Sudjianto, A.: Blind Kriging: a new method for developing metamodels. J. Mech. Des. 130, 031102(1–7) (2008)
Mackay, D.J.C.: Introduction to Gaussian Process. Cambridge University (1998). http://www.inference.org.uk/mackay/gpB.pdf
Lophanev, S.N., Nielsen, H.B., Søndergaard, J.: DACE—a MATLAB kriging toolbox. Technical Report IMM-TR-2002-12, Technical University of Denmark (2002)
Roustant, O., Ginsbourger, D., Deville, Y.: DiceKriging, DiceOptim: two R packages for the analysis of computer experiments by Kriging-based metamodeling and optimization. J. Stat. Softw. 51(1), 1–55 (2012)
Ginsbourger, D., Riche, R.L., Carraro, L.: Kriging is well-suited to parallelize optimization. In: Tenne, Y., Goh, C.K. (eds.) Computational Intelligence in Expensive Optimization Problems, Adaptation, Learning and Optimization, vol. 2, pp. 131–162. Springer, Berlin (2010). Chap. 6
Couckuyt, I., Forrester, A., Gorissen, D., Turck, F.D., Dhaene, T.: Blind Kriging: Implementation and performance analysis. Adv. Eng. Softw. 49, 1–13 (2012)
Martin, J.D., Simpson, T.W.: Use of Kriging models to approximate deterministic computer models. AIAA J. 43(4), 853–863 (2005)
Forrester, A.I.J., Keane, A.J.: Recent advances in surrogate-based optimization. Prog. Aerosp. Sci. 45(1–3), 50–79 (2009)
Jin, R., Chen, W., Sudjianto, A.: On sequential sampling for global metamodeling in engineering design. In: International Design Engineering Technical Conferences and Computers and Information in Engineering Conference, p. 10. American Society of Mechanical Engineers (ASME), Montreal, Canada (2002)
Rocha, H.: On the selection of the most adequate radial basis function. Appl. Math. Model. 33(3), 1573–1583 (2009)
Laguna, M., Martí, R.: Experimental testing of advanced scatter search designs for global optimization of multimodal functions. J. Global Optim. 33(2), 235–255 (2005)
Surjanovic, S., Bingham, D.: Virtual library of simulation experiments: test functions and datasets. http://www.sfu.ca/~ssurjano (2018). Retrieved 27 April 2018
Price, K.V., Storn, R.M.: Differential Evolution: A Practical Approach to Global Optimization. Springer, Berlin (2005)
Couckuyt, I., Dhaene, T., Demeester, P.: ooDACE toolbox a Matlab Kriging toolbox: getting started, 3rd June edn (2013). http://sumo.intec.ugent.be/ooDACE
Viana, F.A.C.: SURROGATES toolbox user’s guide, version 2.1 edn (2010). http://sites.google.com/site/felipeacviana/surrogatestoolbox
Jēkabsons, G.: RBF: radial basis function interpolation for MATLAB/OCTAVE, version 1.1 edn (2009). http://www.cs.rtu.lv/jekabsons/regression.html
Dean, A., Voss, D.: Design and Analysis of Experiments. Springer Texts in Statistic. Springer, New York (1999)
Acknowledgements
This work was partially supported by PRPq/UFMG and by the Brazilian agencies CNPq, FAPEMIG and CAPES. The authors would also like to thank the comments and invaluable suggestions to improve the quality of this manuscript offered by our colleagues André L. Maravilha and Fillipe Goulart (ORCS Lab. / UFMG).
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Valadão, M.A.C., Batista, L.S. A comparative study on surrogate models for SAEAs. Optim Lett 14, 2595–2614 (2020). https://doi.org/10.1007/s11590-020-01575-2
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DOI: https://doi.org/10.1007/s11590-020-01575-2