Abstract
This chapter discusses a generalization of the expected improvement used in Bayesian global optimization to the multicriteria optimization domain, where the goal is to find an approximation to the Pareto front. The expected hypervolume improvement (EHVI) measures improvement as the gain in dominated hypervolume relative to a given approximation to the Pareto front. We will review known properties of the EHVI, applications in practice and propose a new exact algorithm for computing EHVI. The new algorithm has asymptotically optimal time complexity O(nlogn). This improves existing computation schemes by a factor of n∕logn. It shows that this measure, at least for a small number of objective functions, is as fast as other simpler measures of multicriteria expected improvement that were considered in recent years.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Theory of the hypervolume indicator: optimal μ-distributions and the choice of the reference point. In: Proceedings of the Tenth ACM SIGEVO Workshop on Foundations of Genetic Algorithms, pp. 87–102. ACM, Chicago (2009)
Couckuyt, I., Deschrijver, D., Dhaene, T.: Fast calculation of multiobjective probability of improvement and expected improvement criteria for Pareto optimization. J. Global Optim. 60 (3), 575–594 (2014)
Emmerich, M.: Single-and multi-objective evolutionary design optimization assisted by Gaussian random field metamodels. Ph.D. thesis, Fachbereich Informatik, Chair of Systems Analysis, University of Dortmund (2005)
Emmerich, 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.: Hypervolume-based expected improvement: monotonicity properties and exact computation. In: 2011 IEEE Congress on Evolutionary Computation (CEC), pp. 2147–2154. IEEE, New Jersey (2011)
Gaida, D.: Dynamic real-time substrate feed optimization of anaerobic co-digestion plants. Ph.D. thesis, Leiden Institute of Advanced Computer Science (LIACS), Faculty of Science, Leiden University (2014)
Hupkens, I., Emmerich, M., Deutz, A.: Faster computation of expected hypervolume improvement. arXiv preprint arXiv:1408.7114 (2014)
Hupkens, I., Deutz, A., Yang, K., Emmerich, M.: Faster exact algorithms for computing expected hypervolume improvement. In: Evolutionary Multi-Criterion Optimization, pp. 65–79. Springer, Berlin, Heidelberg (2015)
Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Global Optim. 13 (4), 455–492 (1998)
Keane, A.J.: Statistical improvement criteria for use in multiobjective design optimization. AIAA J. 44 (4), 879–891 (2006)
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)
Koch, P., Wagner, T., Emmerich, M.T., Bäck, T., Konen, W.: Efficient multi-criteria optimization on noisy machine learning problems. Appl. Soft Comput. 29, 357–370, New Jersey (2015)
Łaniewski-Wołłk, Ł., Obayashi, S., Jeong, S.: Development of expected improvement for multi-objective problem. In: Proceedings of 42nd Fluid Dynamics Conference/Aerospace Numerical Simulation Symposium (2010)
Mockus, J.: Bayesian Approach to Global Optimization: Theory and Applications, vol. 37. Springer Science & Business Media, New York (2012)
Mockus, J., Tiesis, V., Žilinskas, A.: The application of Bayesian methods for seeking the extremum. In: Towards Global Optimization, vol. 2, pp. 117–129. North-Holland, Amsterdam (1978)
Shimoyama, K., Sato, K., Jeong, S., Obayashi, S.: Comparison of the criteria for updating Kriging response surface models in multi-objective optimization. In: 2012 IEEE Congress on Evolutionary Computation (CEC), pp. 1–8. IEEE, New Jersey (2012)
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)
Shir, O.M., Emmerich, M., Bäck, T., Vrakking, M.J.: The application of evolutionary multi-criteria optimization to dynamic molecular alignment. In: IEEE Congress on Evolutionary Computation, 2007, CEC 2007, pp. 4108–4115. IEEE, New Jersey (2007)
Stein, M.L.: Interpolation of Spatial Data: Some Theory for Kriging. Springer Science & Business Media, New York (2012)
Tesch, M., Schneider, J., Choset, H.: Adapting control policies for expensive systems to changing environments. In: 2011 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pp. 357–364. IEEE, New Jersey (2011)
Törn, A., Žilinskas, A.: Global Optimization. Springer, New York (1989)
Vazquez, E., Bect, J.: Convergence properties of the expected improvement algorithm with fixed mean and covariance functions. J. Stat. Plan. Inference 140 (11), 3088–3095 (2010)
Wagner, T., Emmerich, M., Deutz, A., Ponweiser, W.: On expected-improvement criteria for model-based multi-objective optimization. In: Parallel Problem Solving from Nature. PPSN XI, pp. 718–727. Springer, Berlin, Heidelberg (2010)
Zaefferer, M., Bartz-Beielstein, T., Naujoks, B., Wagner, T., Emmerich, M.: A case study on multi-criteria optimization of an event detection software under limited budgets. In: Evolutionary Multi-Criterion Optimization, pp. 756–770. Springer, Berlin, Heidelberg (2013)
Žilinskas, A., Mockus, J.: On one Bayesian method of search of the minimum. Avtomatika i Vychislitel’naya Teknika 4, 42–44 (1972)
Acknowledgements
Hao Wang gratefully acknowledges support by the Netherlands Organisation for Scientific Research, NWO ICT PPP Project Grant “Process mining for multi-objective online control (PROMIMOOC)”. Kaifeng Yang acknowledges financial support from China Scholarship Council (CSC), CSC No. 201306370037.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Emmerich, M., Yang, K., Deutz, A., Wang, H., Fonseca, C.M. (2016). A Multicriteria Generalization of Bayesian Global Optimization. In: Pardalos, P., Zhigljavsky, A., Žilinskas, J. (eds) Advances in Stochastic and Deterministic Global Optimization. Springer Optimization and Its Applications, vol 107. Springer, Cham. https://doi.org/10.1007/978-3-319-29975-4_12
Download citation
DOI: https://doi.org/10.1007/978-3-319-29975-4_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-29973-0
Online ISBN: 978-3-319-29975-4
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)