Abstract
The attainment function provides a description of the location of the distribution of a random non-dominated point set. This function can be estimated from experimental data via its empirical counterpart, the empirical attainment function (EAF). However, computation of the EAF in more than two dimensions is a non-trivial task. In this article, the problem of computing the empirical attainment function is formalised, and upper and lower bounds on the corresponding number of output points are presented. In addition, efficient algorithms for the two and three-dimensional cases are proposed, and their time complexities are related to lower bounds derived for each case.
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References
Bleuler, S., Laumanns, M., Thiele, L., Zitzler, E.: PISA – A platform and programming language independent interface for search algorithms. In: Fonseca, C.M., et al. (eds.) EMO 2003. LNCS, vol. 2632, pp. 494–508. Springer, Heidelberg (2003)
Fonseca, C.M., Fleming, P.J.: On the performance assessment and comparison of stochastic multiobjective optimizers. In: Voigt, H.M., et al. (eds.) PPSN 1996. LNCS, vol. 1141, pp. 584–593. Springer, Heidelberg (1996)
Fonseca, C.M., Grunert da Fonseca, V., Paquete, L.: Exploring the performance of stochastic multiobjective optimisers with the second-order attainment function. In: Coello, C.C., et al. (eds.) EMO 2005. LNCS, vol. 3410, pp. 250–264. Springer, Heidelberg (2005)
Grunert da Fonseca, V., Fonseca, C.M.: The attainment-function approach to stochastic multiobjective optimizer assessment and comparison. In: Bartz-Beielstein, T., et al. (eds.) Experimental Methods for the Analysis of Optimization Algorithms, pp. 103–130. Springer, Berlin (2010)
Grunert da Fonseca, V., Fonseca, C.M., Hall, A.O.: Inferential performance assessment of stochastic optimisers and the attainment function. In: Zitzler, E., et al. (eds.) EMO 2001. LNCS, vol. 1993, pp. 213–225. Springer, Heidelberg (2001)
Knowles, J.D.: A summary-attainment-surface plotting method for visualizing the performance of stochastic multiobjective optimizers. In: Proceedings of the 5th International Conference on Intelligent Systems Design and Applications, pp. 552–557 (2005)
Kung, H.T., Luccio, F., Preparata, F.P.: On finding the maxima of a set of vectors. Journal of the ACM 22(4), 469–476 (1975)
López-Ibáñez, M., Paquete, L., Stützle, T.: Exploratory analysis of stochastic local search algorithms in biobjective optimization. In: Bartz-Beielstein, T., et al. (eds.) Experimental Methods for the Analysis of Optimization Algorithms, pp. 209–222. Springer, Berlin (2010)
Preparata, F.P., Shamos, M.I.: Computational Geometry. An Introduction, 2nd edn. Springer, Berlin (1988)
Zitzler, E., Thiele, L.: Multiobjective optimization using evolutionary algorithms – A comparative case study. In: Eiben, A.E., et al. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 292–301. Springer, Heidelberg (1998)
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Fonseca, C.M., Guerreiro, A.P., López-Ibáñez, M., Paquete, L. (2011). On the Computation of the Empirical Attainment Function. In: Takahashi, R.H.C., Deb, K., Wanner, E.F., Greco, S. (eds) Evolutionary Multi-Criterion Optimization. EMO 2011. Lecture Notes in Computer Science, vol 6576. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-19893-9_8
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DOI: https://doi.org/10.1007/978-3-642-19893-9_8
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