Abstract
Multidimensional stochastic objective functions can be optimized by finding values in decision space for which the expected output is optimal and the uncertainty is minimal. We investigate the optimization of expensive stochastic black box functions \(f: \mathbb {R}^a \times \mathbb {R}^b \rightarrow \mathbb {R}\) with controllable parameter \(x \in \mathbb {R}^a\) and a b-dimensional random variable C. To estimate the expectation \(\mathrm {E}(f(x, C))\) and the standard deviation \(\mathrm {S}(f(x, C))\) as a measure of uncertainty with few evaluations of f, we use a metamodel of f. We compare an integration and a sampling approach for the estimation of \(\mathrm {E}(f(x, C))\) and \(\mathrm {S}(f(x, C))\). With the sampling approach, the runtime is much lower at almost no loss of quality.
The source code for the experiments and analyses of this paper is publicly available at https://github.com/mariusbommert/LOD2020.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arrow, K.J., Harris, T., Marschak, J.: Optimal inventory policy. Econometrica 19(3), 250–272 (1951). https://www.jstor.org/stable/1906813
Beachkofski, B.K., Grandhi, R.V.: Improved distributed hypercube sampling. In: 43rd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, p. 1274 (2002). https://doi.org/10.2514/6.2002-1274
Bommert, M., Rudolph, G.: Reliable biobjective solution of stochastic problems using metamodels. In: Deb, K., Goodman, E., Coello Coello, C.A., Klamroth, K., Miettinen, K., Mostaghim, S., Reed, P. (eds.) Evolutionary Multi-Criterion Optimization, pp. 581–592. Springer International Publishing, Cham (2019). https://doi.org/10.1007/978-3-030-12598-1_46
Bossek, J.: ecr 2.0: A modular framework for evolutionary computation in R. In: Proceedings of the Genetic and Evolutionary Computation Conference Companion, pp. 1187–1193. ACM (2017). https://doi.org/10.1145/3067695.3082470
Branke, J., Deb, K., Miettinen, K., Słowiński, R. (eds.): Multiobjective Optimization. LNCS, vol. 5252. Springer, Heidelberg (2008). https://doi.org/10.1007/978-3-540-88908-3
Carnell, R.: lhs: Latin Hypercube Samples (2019). https://CRAN.R-project.org/package=lhs, R package version 1.0.1
Fonseca, C.M., Guerreiro, A.P., López-Ibáñez, M., Paquete, L.: On the computation of the empirical attainment function. In: Takahashi, R.H.C., Deb, K., Wanner, E.F., Greco, S. (eds.) Evolutionary Multi-Criterion Optimization, pp. 106–120. Springer, Berlin, Heidelberg (2011). https://doi.org/10.1007/978-3-642-19893-9_8
Grunert da Fonseca, V., Fonseca, C.M., Hall, A.O.: Inferential performance assessment of stochastic optimisers and the attainment function. In: Zitzler, E., Deb, K., Thiele, L., Coello Coello, C.A., Corne, D. (eds.) Evolutionary Multi-Criterion Optimization, pp. 213–225. Springer, Berlin, Heidelberg (2001). https://doi.org/10.1007/3-540-44719-9_15
Gutjahr, W.J., Pichler, A.: Stochastic multi-objective optimization: a survey on non-scalarizing methods. Ann. Oper. Res. 236(2), 475–499 (2013). https://doi.org/10.1007/s10479-013-1369-5
Huang, D., Allen, T.T., Notz, W.I., Zeng, N.: Global optimization of stochastic black-box systems via sequential kriging meta-models. J. Global Optim. 34(3), 441–466 (2006). https://doi.org/10.1007/s10898-005-2454-3
Kleywegt, A.J., Shapiro, A., Homem-de-Mello, T.: The sample average approximation method for stochastic discrete optimization. SIAM J. Optim. 12(2), 479–502 (2002). https://doi.org/10.1137/S1052623499363220
Lang, M., Bischl, B., Surmann, D.: batchtools: Tools for R to work on batch systems. J. Open Source Softw. 2(10) (2017). https://doi.org/10.21105/joss.00135
López-Ibáñez, M., Paquete, L., Stützle, T.: Exploratory analysis of stochastic local search algorithms in biobjective optimization. In: Bartz-Beielstein, T., Chiarandini, M., Paquete, L., Preuss, M. (eds.) Experimental Methods for the Analysis of Optimization Algorithms, pp. 209–222. Springer, Heidelberg (2010). https://doi.org/10.1007/978-3-642-02538-9_9
Markowitz, H.: Portfolio selection. J. Finan. 7(1), 77–91 (1952). https://doi.org/10.1111/j.1540-6261.1952.tb01525.x
Ponsich, A., Jaimes, A.L., Coello Coello, C.A.: A survey on multiobjective evolutionary algorithms for the solution of the portfolio optimization problem and other finance and economics applications. IEEE Trans. Evolution. Comput. 17(3), 321–344 (2013). https://doi.org/10.1109/TEVC.2012.2196800
R Core Team: R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria (2019). https://www.R-project.org/
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). https://doi.org/10.18637/jss.v051.i01
Swamy, C.: Risk-averse stochastic optimization: probabilistically-constrained models and algorithms for black-box distributions. In: Proceedings of the twenty-second annual ACM-SIAM symposium on Discrete Algorithms, pp. 1627–1646. SIAM (2011). https://doi.org/10.1137/1.9781611973082.126
Acknowledgments
The authors gratefully acknowledge the computing time provided on the Linux HPC cluster at TU Dortmund University (LiDO3), partially funded in the course of the Large-Scale Equipment Initiative by the German Research Foundation (DFG) as project 271512359.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2020 Springer Nature Switzerland AG
About this paper
Cite this paper
Bommert, M., Rudolph, G. (2020). Reliable Solution of Multidimensional Stochastic Problems Using Metamodels. In: Nicosia, G., et al. Machine Learning, Optimization, and Data Science. LOD 2020. Lecture Notes in Computer Science(), vol 12565. Springer, Cham. https://doi.org/10.1007/978-3-030-64583-0_20
Download citation
DOI: https://doi.org/10.1007/978-3-030-64583-0_20
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-64582-3
Online ISBN: 978-3-030-64583-0
eBook Packages: Computer ScienceComputer Science (R0)