Abstract
The problem of finding sets of bounded cardinality maximizing dominated hypervolume is considered for explicitly parameterized Pareto fronts of multi-objective optimization problems. A parameterization of the Pareto front it often known (by construction) for synthetic benchmark functions. For the widely used ZDT and DTLZ families of benchmarks close-to-optimal sets have been obtained only for two objectives, although the three-objective variants of the DTLZ problems are frequently applied. Knowledge of the dominated hypervolume theoretically achievable with an approximation set of fixed cardinality facilitates judgment of (differences in) optimization results and the choice of stopping criteria, two important design decisions of empirical studies. The present paper aims to close this gap. An efficient optimization strategy is presented for two and three objectives. Optimized sets are provided for standard benchmarks.
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References
ZDT and DTLZ test problems, http://people.ee.ethz.ch/~sop/download/supplementary/testproblems/ (accessed: March 17, 2014)
Auger, A., Bader, J., Brockhoff, D.: Theoretically Investigating Optimal μ-Distributions for the Hypervolume Indicator: First Results for Three Objectives. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds.) PPSN XI, Part I. LNCS, vol. 6238, pp. 586–596. Springer, Heidelberg (2010)
Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Theory of the Hypervolume Indicator: Optimal μ-Distributions and the Choice of the Reference Point. In: Proceedings of the Tenth ACM SIGEVO Workshop on Foundations of Genetic Algorithms, pp. 87–102. ACM (2009)
Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Hypervolume-based Multiobjective Optimization: Theoretical Foundations and Practical Implications. Theoretical Computer Science 425, 75–103 (2012)
Bringmann, K.: Klee’s measure problem on fat boxes in time \(\mathcal{O}(n(d+2)/3)\). In: Proceedings of the Twenty-Sixth Annual Symposium on Computational Geometry, pp. 222–229. ACM (2010)
Bringmann, K., Friedrich, T.: Approximating the least hypervolume contributor: NP-hard in general, but fast in practice. Theoretical Computer Science 425, 104–116 (2012)
Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable Multi-Objective Optimization Test Problems. In: Congress on Evolutionary Computation, CEC 2002, pp. 825–830. IEEE Press (2002)
Emmerich, M., Deutz, A.: Time complexity and zeros of the hypervolume indicator gradient field. In: Schuetze, O., Coello Coello, C.A., Tantar, A.-A., Tantar, E., Bouvry, P., Del Moral, P., Legrand, P. (eds.) EVOLVE - A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation III. SCI, vol. 500, pp. 169–193. Springer, Heidelberg (2014)
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)
Fonseca, C.M., Paquete, L., Lopez-Ibanez, M.: An improved dimension-sweep algorithm for the hypervolume indicator. In: IEEE Congress on Evolutionary Computation, CEC 2006, pp. 1157–1163 (2006)
Friedrich, T., Neumann, F., Thyssen, C.: Multiplicative Approximations, Optimal Hypervolume Distributions, and the Choice of the Reference Point. Technical Report arXiv:1309.3816, arXiv.org (2013)
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)
Zitzler, E., Deb, K., Thiele, L.: Comparison of Multiobjective Evolutionary Algorithms: Empirical Results. Evolutionary Computation 8(2), 173–195 (2000)
Zitzler, E., Thiele, L., Laumanns, M., Fonseca, C., Da Fonseca, V.: Performance assessment of multiobjective optimizers: An analysis and review. IEEE Transactions on Evolutionary Computation 7(2), 117–132 (2003)
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Glasmachers, T. (2014). Optimized Approximation Sets for Low-Dimensional Benchmark Pareto Fronts. In: Bartz-Beielstein, T., Branke, J., Filipič, B., Smith, J. (eds) Parallel Problem Solving from Nature – PPSN XIII. PPSN 2014. Lecture Notes in Computer Science, vol 8672. Springer, Cham. https://doi.org/10.1007/978-3-319-10762-2_56
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DOI: https://doi.org/10.1007/978-3-319-10762-2_56
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