Abstract
We present a new analysis of the LebMeasure algorithm for calculating hypervolume. We prove that although it is polynomial in the number of points, LebMeasure is exponential in the number of objectives in the worst case, not polynomial as has been claimed previously. This result has important implications for anyone planning to use hypervolume, either as a metric to compare optimisation algorithms, or as part of a diversity mechanism in an evolutionary algorithm.
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Deb, K., Pratap, A., Agarwal, S., Meyarivan, T.: A fast and elitist multi-objective genetic algorithm: NSGA-II. IEEE TEC 6(2), 182–197 (2002)
Fleischer, M.: The measure of Pareto optima: Applications to multi-objective metaheuristics. Technical Report ISR TR 2002-32, University of Maryland (2002)
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)
Huband, S., Hingston, P., While, L., Barone, L.: An evolution strategy with probabilistic mutation for multi-objective optimization. In: CEC 2003, vol. 4, pp. 2284–2291. IEEE, Los Alamitos (2003)
Knowles, J., Corne, D.: M-PAES: A memetic algorithm for multi-objective optimization. In: CEC 2000, vol. 1, pp. 325–332. IEEE, Los Alamitos (2000)
Knowles, J., Corne, D., Fleischer, M.: Bounded archiving using the Lebesgue measure. In: CEC 2003, vol. 4, pp. 2490–2497. IEEE, Los Alamitos (2003)
Okabe, T., Jin, Y., Sendhoff, B.: A critical survey of performance indices for multi-objective optimisation. In: CEC 2003, vol. 2, pp. 878–885. IEEE, Los Alamitos (2003)
Purshouse, R.: On the evolutionary optimisation of many objectives. PhD thesis, The University of Sheffield (2003)
Wu, J., Azarm, S.: Metrics for quality assessment of a multi-objective design optimization solution set. Journal of Mechanical Design 123, 18–25 (2001)
Zitzler, E.: Evolutionary algorithms for multi-objective optimization: Methods and applications. PhD thesis, Swiss Federal Inst of Technology (ETH) Zurich (1999)
Zitzler, E., Laumanns, M., Thiele, L.: SPEA2: Improving the strength Pareto evolutionary algorithm for multi-objective optimization. In: EUROGEN 2001, Int Center for Numerical Methods in Engineering, Barcelona, pp. 95–100 (2001)
Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C.M., da Fonseca, V.G.: Performance assessment of multi-objective optimizers: An analysis and review. IEEE TEC 7(2), 117–132 (2003)
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While, L. (2005). A New Analysis of the LebMeasure Algorithm for Calculating Hypervolume. In: Coello Coello, C.A., Hernández Aguirre, A., Zitzler, E. (eds) Evolutionary Multi-Criterion Optimization. EMO 2005. Lecture Notes in Computer Science, vol 3410. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31880-4_23
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DOI: https://doi.org/10.1007/978-3-540-31880-4_23
Publisher Name: Springer, Berlin, Heidelberg
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