Computing 3-D Expected Hypervolume Improvement and Related Integrals in Asymptotically Optimal Time

  • Kaifeng Yang
  • Michael Emmerich
  • André Deutz
  • Carlos M. Fonseca
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10173)

Abstract

The Expected Hypervolume Improvement (EHVI) is a frequently used infill criterion in surrogate-assisted multi-criterion optimization. It needs to be frequently called during the execution of such algorithms. Despite recent advances in improving computational efficiency, its running time for three or more objectives has remained in \(O(n^d)\) for \(d\ge 3\), where d is the number of objective functions and n is the size of the incumbent Pareto-front approximation. This paper proposes a new integration scheme, which makes it possible to compute the EHVI in \(\varTheta (n \log n)\) optimal time for the important three-objective case (\(d=3\)). The new scheme allows for a generalization to higher dimensions and for computing the Probability of Improvement (PoI) integral efficiently. It is shown, both theoretically and empirically, that the hidden constant in the asymptotic notation is small. Empirical speed comparisons were designed between the C++ implementations of the new algorithm (which will be in the public domain) and those recently published by competitors, on randomly-generated non-dominated fronts of size 10, 100, and 1000. The experiments include the analysis of batch computations, in which only the parameters of the probability distribution change but the incumbent Pareto-front approximation stays the same. Experimental results show that the new algorithm is always faster than the other algorithms, sometimes over \(10^4\) times faster.

Keywords

Expected hypervolume improvement Time complexity Surrogate-assisted multi-criterion optimization Efficient global optimization Probability of improvement 

Notes

Acknowledgements

Kaifeng Yang acknowledges financial support from the China Scholarship Council (CSC), CSC No. 201306370037. Carlos M. Fonseca was supported by national funds through the Portuguese Foundation for Science and Technology (FCT), and by the European Regional Development Fund (FEDER) through COMPETE 2020 – Operational Program for Competitiveness and Internationalization (POCI).

References

  1. 1.
    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: Purshouse, R.C., Fleming, P.J., Fonseca, C.M., Greco, S., Shaw, J. (eds.) EMO 2013. LNCS, vol. 7811, pp. 756–770. Springer, Heidelberg (2013). doi: 10.1007/978-3-642-37140-0_56 CrossRefGoogle Scholar
  2. 2.
    Yang, K., Deutz, A., Yang, Z., Bäck, T., Emmerich, M.: Truncated expected hypervolume improvement: exact computation and application. In: IEEE Congress on Evolutionary Computation (CEC). IEEE (2016)Google Scholar
  3. 3.
    Yang K, Gaida D, Bäck T, Emmerich M.: Expected hypervolume improvement algorithm for PID controller tuning and the multiobjective dynamical control of a biogas plant. In: 2015 IEEE Congress on Evolutionary Computation (CEC), pp. 1934–1942, May 2015Google Scholar
  4. 4.
    Michael, 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)CrossRefGoogle Scholar
  5. 5.
    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 (2015)CrossRefGoogle Scholar
  6. 6.
    Shimoyama, K., Jeong, S., Obayashi, S.: Kriging-surrogate-based optimization considering expected hypervolume improvement in non-constrained many-objective test problems. In: IEEE Congress on Evolutionary Computation (CEC), pp. 658–665. IEEE (2013)Google Scholar
  7. 7.
    Shimoyama, K., Sato, K., Jeong, S., Obayashi, S.: Comparison of the criteria for updating kriging response surface models in multi-objective optimization. In: IEEE Congress on Evolutionary Computation, pp. 1–8. IEEE (2012)Google Scholar
  8. 8.
    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)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Jones, D.R., Schonlau, M., Welch, W.J.: Efficient global optimization of expensive black-box functions. J. Global Optim. 13(4), 455–492 (1998)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Mockus, J., Tiešis, V., Žilinskas, A.: The application of Bayesian methods for seeking the extremum. In: Towards Global Optimization, vol. 2, pp. 117–131. North-Holland, Amsterdam (1978)Google Scholar
  11. 11.
    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 2010. LNCS, vol. 6238, pp. 718–727. Springer, Heidelberg (2010). doi: 10.1007/978-3-642-15844-5_72 Google Scholar
  12. 12.
    Emmerich, M.T., Deutz, A.H., Klinkenberg, J.W.: Hypervolume-based expected improvement: monotonicity properties and exact computation. In: IEEE Congress on Evolutionary Computation (CEC), pp. 2147–2154. IEEE (2011)Google Scholar
  13. 13.
    Emmerich, M., Yang, K., Deutz, A., Wang, H., Fonseca, C.M.: A multicriteria generalization of bayesian global optimization. In: Pardalos, P.M., Zhigljavsky, A., Žilinskas, J. (eds.) Advances in Stochastic and Deterministic Global Optimization. SOIA, vol. 107, pp. 229–242. Springer, Cham (2016). doi: 10.1007/978-3-319-29975-4_12 CrossRefGoogle Scholar
  14. 14.
    Hupkens, I., Deutz, A., Yang, K., Emmerich, M.: Faster exact algorithms for computing expected hypervolume improvement. In: Gaspar-Cunha, A., Henggeler Antunes, C., Coello, C.C. (eds.) EMO 2015. LNCS, vol. 9019, pp. 65–79. Springer, Cham (2015). doi: 10.1007/978-3-319-15892-1_5 Google Scholar
  15. 15.
    Yang, K., Li, L., Deutz, A., Bäck, T., Emmerich, M.: Preference-based multiobjective optimization using truncated expected hypervolume improvement. In: 12th International Conference on Natural Computation, Fuzzy Systems and Knowledge Discovery. IEEE (2016)Google Scholar
  16. 16.
    Vazquez, E., Bect, J.: Convergence properties of the expected improvement algorithm with fixed mean and covariance functions. J. Stat. Plan. Infer. 140(11), 3088–3095 (2010)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Knowles, J., Hughes, E.J.: Multiobjective optimization on a budget of 250 evaluations. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds.) EMO 2005. LNCS, vol. 3410, pp. 176–190. Springer, Heidelberg (2005). doi: 10.1007/978-3-540-31880-4_13 CrossRefGoogle Scholar
  18. 18.
    Keane, A.J.: Statistical improvement criteria for use in multiobjective design optimization. AIAA J. 44(4), 879–891 (2006)CrossRefGoogle Scholar
  19. 19.
    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), 094503–094503-7 (2013)CrossRefGoogle Scholar
  20. 20.
    Svenson, J., Santner, T.: Multiobjective optimization of expensive-to-evaluate deterministic computer simulator models. Comput. Stat. Data Anal. 94, 250–264 (2016)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Emmerich, M.T.M.: Single-and multi-objective evolutionary design optimization assisted by Gaussian random field metamodels. Ph.D. thesis, FB Informatik, University of Dortmund, ELDORADO, Dortmund, 10 (2005)Google Scholar
  22. 22.
    Shir, O.M., Emmerich, M., Bäckck, T., Vrakking, M.J.: The application of evolutionary multi-criteria optimization to dynamic molecular alignment. In: IEEE Congress on Evolutionary Computation, pp. 4108–4115. IEEE (2007)Google Scholar
  23. 23.
    Łaniewski-Wołłk, P, Obayashi S, Jeong S.: Development of expected improvement for multi-objective problems. In: Proceedings of 42nd Fluid Dynamics Conference/Aerospace Numerical, Simulation Symposium (CD ROM), Varna, Bulgaria (2010)Google Scholar
  24. 24.
    Luo, C., Shimoyama, K., Obayashi, S.: Kriging model based many-objective optimization with efficient calculation of expected hypervolume improvement. In: IEEE Congress on Evolutionary Computation (CEC), pp. 1187–1194. IEEE (2014)Google Scholar
  25. 25.
    Zitzler, E., Thiele, L.: Multiobjective evolutionary algorithms: a comparative case study and the strength pareto approach. IEEE Trans. Evol. Comput. 3(4), 257–271 (1999)CrossRefGoogle Scholar
  26. 26.
    Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Fonseca, V.G.D.: Performance assessment of multiobjective optimizers: an analysis and review. IEEE Trans. Evol. Comput. 7(2), 117–132 (2003)CrossRefGoogle Scholar
  27. 27.
    Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Hypervolume-based multiobjective optimization: theoretical foundations and practical implications. Theor. Comput. Sci. 425, 75–103 (2012)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Emmerich, M.T.M., Fonseca, C.M.: Computing hypervolume contributions in low dimensions: asymptotically optimal algorithm and complexity results. In: Takahashi, R.H.C., Deb, K., Wanner, E.F., Greco, S. (eds.) EMO 2011. LNCS, vol. 6576, pp. 121–135. Springer, Heidelberg (2011). doi: 10.1007/978-3-642-19893-9_9 CrossRefGoogle Scholar
  29. 29.
    Lacour, R., Klamroth, K., Fonseca, C.M.: A box decomposition algorithm to compute the hypervolume indicator. Comput. Oper. Res. 79, 347–360 (2016)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Kaifeng Yang
    • 1
  • Michael Emmerich
    • 1
  • André Deutz
    • 1
  • Carlos M. Fonseca
    • 2
  1. 1.LIACS, Leiden UniversityLeidenThe Netherlands
  2. 2.CISUC, Department of Informatics EngineeringUniversity of CoimbraCoimbraPortugal

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