Skip to main content
SpringerLink
Log in
Book cover

Australasian Joint Conference on Artificial Intelligence

AI 2016: AI 2016: Advances in Artificial Intelligence pp 243–254Cite as

Improving and Extending the HV4D Algorithm for Calculating Hypervolume Exactly

  • Conference paper
  • First Online:

Part of the book series: Lecture Notes in Computer Science ((LNAI,volume 9992))

Abstract

We describe extensions to the 4D hypervolume algorithm HV4D that greatly improve its performance in 4D, and that enable an extension of the algorithm to 5D. We add a facility to cope with dominated points, reducing the number of contribution calculations required; and a new representation of the front between slices, eliminating significant repeated work. The former also allows the algorithm to work efficiently with 5D data. The new algorithms can process sets containing 1,000 points in around 1 ms in 4D, and around 5–10 ms in 5D. They make a significant contribution to the state-of-the-art.

This is a preview of subscription content, log in via an institution.

References

  1. Purshouse, R.: On the Evolutionary Optimisation of Many Objectives. The University of Sheffield, UK (2003)

    Google Scholar 

  2. Fleischer, M.: The measure of Pareto optima applications to multi-objective metaheuristics. In: Fonseca, C.M., Fleming, P.J., Zitzler, E., Thiele, L., Deb, K. (eds.) EMO 2003. LNCS, vol. 2632, pp. 519–533. Springer, Heidelberg (2003). doi:10.1007/3-540-36970-8_37

    Chapter  Google Scholar 

  3. Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Theory of the hypervolume indicator: optimal \(\mu \)-distributions and the choice of the reference point. In: FOGA, pp. 87–102. ACM (2009)

    Google Scholar 

  4. While, L., Bradstreet, L., Barone, L.: A fast way of calculating exact hypervolumes. IEEE TEVC 16(1), 86–95 (2012)

    Google Scholar 

  5. Russo, L., Francisco, A.P.: Quick hypervolume. IEEE TEVC 18(4), 481–502 (2014)

    Google Scholar 

  6. Beume, N., Fonseca, C.M., López-Ibáñez, M., Paquete, L., Vahrenhold, J.: On the complexity of computing the hypervolume indicator. IEEE TEVC 13(5), 1075–1082 (2009)

    Google Scholar 

  7. Guerreiro, A.P., Fonseca, C.M., Emmerich, M.T.: A fast dimension-sweep algorithm for the hypervolume indicator in four dimensions. In: CCCG (2012)

    Google Scholar 

  8. Back, T., Fogel, D.B., Michalewicz, Z.: Handbook of Evolutionary Computation. IOP Publishing Ltd., Bristol (1997)

    Book  MATH  Google Scholar 

  9. Cox, W., While, L.: Improving the IWFG algorithm for calculating incremental hypervolume. In: IEEE CEC (2016)

    Google Scholar 

  10. Zitzler, E.: Hypervolume metric calculation (2001). ftp://ftp.tik.ee.ethz.ch/pub/people/zitzler/hypervol.c

  11. Knowles, J.: Local-Tearch and Hybrid Evolutionary Algorithms for Pareto Optimisation. The University of Reading, United Kingdom (2002)

    Google Scholar 

  12. While, L., Hingston, P., Barone, L., Huband, S.: A faster algorithm for calculating hypervolume. IEEE TEVC 10(1), 29–38 (2006)

    Google Scholar 

  13. While, L., Bradstreet, L., Barone, L., Hingston, P.: Heuristics for optimising the calculation of hypervolume for multi-objective optimisation problems. In: IEEE CEC, pp. 2225–2232 (2005)

    Google Scholar 

  14. Fonseca, C.M., Paquete, L., López-Ibáñez, M.: An improved dimension-sweep algorithm for the hypervolume indicator. In: IEEE CEC, pp. 3973–3979 (2006)

    Google Scholar 

  15. Bringmann, K., Friedrich, T.: Approximating the least hypervolume contributor: NP-hard in general, but fast in practice. In: Ehrgott, M., Fonseca, C.M., Gandibleux, X., Hao, J.-K., Sevaux, M. (eds.) EMO 2009. LNCS, vol. 5467, pp. 6–20. Springer, Heidelberg (2009). doi:10.1007/978-3-642-01020-0_6

    Chapter  Google Scholar 

  16. Everson, R.M., Fieldsend, J.E., Singh, S.: Full elite sets for multi-objective optimisation. In: Parmee, I.C. (ed.) Adaptive Computing in Design and Manufacture V, pp. 343–354. Springer, London (2002)

    Chapter  Google Scholar 

  17. Bader, J., Deb, K., Zitzler, E.: Faster hypervolume-based search using Monte Carlo sampling. In: Ehrgott, M., Naujoks, B., Stewart, T.J., Wallenius, J. (eds.) Multiple Criteria Decision Making for Sustainable Energy and Transportation Systems. Lecture Notes in Economics and Mathematical Systems, vol. 634, pp. 313–326. Springer, Heidelberg (2010). doi:10.1007/978-3-642-04045-0_27

    Chapter  Google Scholar 

  18. Bringmann, K., Friedrich, T.: Don’t be greedy when calculating hypervolume contributions. In: FOGA, pp. 103–112. ACM (2009)

    Google Scholar 

  19. Friedrich, T., Horoba, C., Neumann, F.: Multiplicative approximations and the hypervolume indicator. In: FOGA, pp. 103–112. ACM (2009)

    Google Scholar 

  20. Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable multi-objective optimization test problems. In: IEEE CEC, pp. 825–830 (2002)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Computer Science and Software Engineering, The University of Western Australia, Perth, Australia

    Wesley Cox & Lyndon While

Authors
  1. Wesley Cox
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lyndon While
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Lyndon While .

Editor information

Editors and Affiliations

  1. University of Tasmania, Hobart, Australia

    Byeong Ho Kang

  2. Auckland University of Technology, Auckland, New Zealand

    Quan Bai

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing AG

About this paper

Cite this paper

Cox, W., While, L. (2016). Improving and Extending the HV4D Algorithm for Calculating Hypervolume Exactly. In: Kang, B.H., Bai, Q. (eds) AI 2016: Advances in Artificial Intelligence. AI 2016. Lecture Notes in Computer Science(), vol 9992. Springer, Cham. https://doi.org/10.1007/978-3-319-50127-7_20

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-50127-7_20

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-50126-0

  • Online ISBN: 978-3-319-50127-7

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation