Empirical Performance of the Approximation of the Least Hypervolume Contributor

  • Krzysztof Nowak
  • Marcus Märtens
  • Dario Izzo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8672)


A fast computation of the hypervolume has become a crucial component for the quality assessment and the performance of modern multi-objective evolutionary optimization algorithms. Albeit recent improvements, exact computation becomes quickly infeasible if the optimization problems scale in their number of objectives or size. To overcome this issue, we investigate the potential of using approximation instead of exact computation by benchmarking the state of the art hypervolume algorithms for different geometries, dimensionality and number of points. Our experiments outline the threshold at which exact computation starts to become infeasible, but approximation still applies, highlighting the major factors that influence its performance.


Hypervolume indicator performance indicators multi- objective optimization many-objective optimization approximation algorithms 


  1. 1.
    Fleischer, M.: The measure of Pareto optima applications to multi-objective metaheuristics. In: Fonseca, C.M., Fleming, P.J., Zitzler, E., Deb, K., Thiele, L. (eds.) EMO 2003. LNCS, vol. 2632, pp. 519–533. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  2. 2.
    Zitzler, E.: Evolutionary algorithms for multiobjective optimization: Methods and applications, vol. 63. Shaker Ithaca (1999)Google Scholar
  3. 3.
    Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., Da Fonseca, V.G.: Performance assessment of multiobjective optimizers: An analysis and review. IEEE Transactions on Evolutionary Computation 7(2), 117–132 (2003)CrossRefGoogle Scholar
  4. 4.
    Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithms - A comparative case study. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 292–301. Springer, Heidelberg (1998)CrossRefGoogle Scholar
  5. 5.
    Zitzler, E., Brockhoff, D., Thiele, L.: The hypervolume indicator revisited: On the design of Pareto-compliant indicators via weighted integration. In: Obayashi, S., Deb, K., Poloni, C., Hiroyasu, T., Murata, T. (eds.) EMO 2007. LNCS, vol. 4403, pp. 862–876. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Transactions on Evolutionary Computation 6(2), 182–197 (2002)CrossRefGoogle Scholar
  7. 7.
    Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the strength Pareto evolutionary algorithm (2001)Google Scholar
  8. 8.
    Zitzler, E., Künzli, S.: Indicator-based selection in multiobjective search. In: Yao, X., et al. (eds.) PPSN 2004. LNCS, vol. 3242, pp. 832–842. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  9. 9.
    Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: Multiobjective selection based on dominated hypervolume. European Journal of Operational Research 181(3), 1653–1669 (2007)CrossRefzbMATHGoogle Scholar
  10. 10.
    Bader, J., Zitzler, E.: HypE: An algorithm for fast hypervolume-based many-objective optimization. Evolutionary Computation 19(1), 45–76 (2011)CrossRefGoogle Scholar
  11. 11.
    Igel, C., Hansen, N., Roth, S.: Covariance matrix adaptation for multi-objective optimization. Evolutionary Computation 15(1), 1–28 (2007)CrossRefGoogle Scholar
  12. 12.
    Bringmann, K., Friedrich, T.: Approximating the least hypervolume contributor: Np-hard in general, but fast in practice. Theoretical Computer Science 425, 104–116 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Bringmann, K., Friedrich, T.: Parameterized average-case complexity of the hypervolume indicator. In: Proceeding of the Fifteenth Annual Conference on Genetic and Evolutionary Computation Conference, GECCO 2013, pp. 575–582. ACM, New York (2013)CrossRefGoogle Scholar
  14. 14.
    Beume, N., Fonseca, C.M., López-Ibáñez, M., Paquete, L., Vahrenhold, J.: On the complexity of computing the hypervolume indicator. IEEE Transactions on Evolutionary Computation 13(5), 1075–1082 (2009)CrossRefGoogle Scholar
  15. 15.
    Guerreiro, A.P., Fonseca, C.M., Emmerich, M.T.: A fast dimension-sweep algorithm for the hypervolume indicator in four dimensions. In: CCCG, pp. 77–82 (2012)Google Scholar
  16. 16.
    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)CrossRefGoogle Scholar
  17. 17.
    Bringmann, K.: An improved algorithm for Klee’s measure problem on fat boxes. Computational Geometry 45(5), 225–233 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Yildiz, H., Suri, S.: On Klee’s measure problem for grounded boxes. In: Proceedings of the 2012 Symposuim on Computational Geometry, pp. 111–120. ACM (2012)Google Scholar
  19. 19.
    Beume, N.: S-metric calculation by considering dominated hypervolume as Klee’s measure problem. Evolutionary Computation 17(4), 477–492 (2009)CrossRefGoogle Scholar
  20. 20.
    Fonseca, C.M., Paquete, L., López-Ibánez, M.: An improved dimension-sweep algorithm for the hypervolume indicator. In: IEEE Congress on Evolutionary Computation, CEC 2006, pp. 1157–1163. IEEE (2006)Google Scholar
  21. 21.
    While, L., Bradstreet, L., Barone, L.: A fast way of calculating exact hypervolumes. IEEE Transactions on Evolutionary Computation 16(1), 86–95 (2012)CrossRefGoogle Scholar
  22. 22.
    Priester, C., Narukawa, K., Rodemann, T.: A comparison of different algorithms for the calculation of dominated hypervolumes. In: Proceeding of the Fifteenth Annual Conference on Genetic and Evolutionary Computation Conference, GECCO 2013, pp. 655–662. ACM, New York (2013)CrossRefGoogle Scholar
  23. 23.
    Bringmann, K., Friedrich, T.: Approximating the volume of unions and intersections of high-dimensional geometric objects. Computational Geometry 43(6), 601–610 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Ishibuchi, H., Tsukamoto, N., Sakane, Y., Nojima, Y.: Indicator-based evolutionary algorithm with hypervolume approximation by achievement scalarizing functions. In: Proceedings of the 12th Annual Conference on Genetic and Evolutionary Computation, pp. 527–534. ACM (2010)Google Scholar
  25. 25.
    Bringmann, K., Friedrich, T., Igel, C., Voß, T.: Speeding up many-objective optimization by Monte Carlo approximations. Artificial Intelligence 204, 22–29 (2013)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Knowles, J.: ParEGO: A hybrid algorithm with on-line landscape approximation for expensive multiobjective optimization problems. IEEE Transactions on Evolutionary Computation 10(1), 50–66 (2006)CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Krzysztof Nowak
    • 1
  • Marcus Märtens
    • 2
  • Dario Izzo
    • 1
  1. 1.European Space AgencyNoordwijkThe Netherlands
  2. 2.TU DelftDelftThe Netherlands

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