Abstract
Several indicator-based evolutionary multiobjective optimization algorithms have been proposed in the literature. The notion of optimal μ-distributions formalizes the optimization goal of such algorithms: find a set of μ solutions that maximizes the underlying indicator among all sets with μ solutions. In particular for the often used hypervolume indicator, optimal μ-distributions have been theoretically analyzed recently. All those results, however, cope with bi-objective problems only. It is the main goal of this paper to extend some of the results to the 3-objective case. This generalization is shown to be not straight-forward as a solution’s hypervolume contribution has not a simple geometric shape anymore in opposition to the bi-objective case where it is always rectangular. In addition, we investigate the influence of the reference point on optimal μ-distributions and prove that also in the 3-objective case situations exist for which the Pareto front’s extreme points cannot be guaranteed in optimal μ-distributions.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Investigating and Exploiting the Bias of the Weighted Hypervolume to Articulate User Preferences. In: Genetic and Evolutionary Computation Conference (GECCO 2009), pp. 563–570. ACM, New York (2009)
Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Theory of the Hypervolume Indicator: Optimal μ-Distributions and the Choice of the Reference Point. In: Foundations of Genetic Algorithms (FOGA 2009), pp. 87–102. ACM, New York (2009)
Bader, J.: Hypervolume-Based Search for Multiobjective Optimization: Theory and Methods. PhD thesis, ETH Zurich, Switzerland (2010)
Bartle, R.G.: The Elements of Integration and Lebesgue Measure. Wiley, Chichester (1995)
Beume, N., Fonseca, C.M., Lopez-Ibanez, M., Paquete, L., Vahrenhold, J.: On the Complexity of Computing the Hypervolume Indicator. IEEE T. Evolut. Comput. 13(5), 1075–1082 (2009)
Beume, N., Naujoks, B., Emmerich, M.: SMS-EMOA: Multiobjective Selection Based on Dominated Hypervolume. Eur. J. Oper. Res. 181, 1653–1669 (2007)
Bringmann, K., Friedrich, T.: The Maximum Hypervolume Set Yields Near-optimal Approximation. In: Genetic and Evolutionary Computation Conference, GECCO 2010, pp. 511–518. ACM, New York (2010)
Deb, K., Thiele, L., Laumanns, M., Zitzler, E.: Scalable Test Problems for Evolutionary Multi-Objective Optimization. In: Evolutionary Multiobjective Optimization: Theoretical Advances and Applications, ch. 6, pp. 105–145. Springer, Heidelberg (2005)
Emmerich, M., Deutz, A., Beume, N.: Gradient-Based/Evolutionary Relay Hybrid for Computing Pareto Front Approximations Maximizing the S-Metric. In: Bartz-Beielstein, T., Blesa Aguilera, M.J., Blum, C., Naujoks, B., Roli, A., Rudolph, G., Sampels, M. (eds.) HCI/ICCV 2007. LNCS, vol. 4771, pp. 140–156. Springer, Heidelberg (2007)
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)
Friedrich, T., Horoba, C., Neumann, F.: Multiplicative Approximations and the Hypervolume Indicator. In: Genetic and Evolutionary Computation Conference (GECCO 2009), pp. 571–578. ACM, New York (2009)
Huband, S., Hingston, P., Barone, L., While, L.: A Review of Multiobjective Test Problems and a Scalable Test Problem Toolkit. IEEE T. Evolut. Comput. 10(5), 477–506 (2006)
Igel, C., Hansen, N., Roth, S.: Covariance Matrix Adaptation for Multi-objective Optimization. Evolutionary Computation 15(1), 1–28 (2007)
Rudin, W.: Principles of Mathematical Analysis, 3rd edn. McGraw-Hill, New York (1976)
Van Veldhuizen, D.A., Lamont, G.B.: Multiobjective evolutionary algorithm test suites. In: Symposium on Applied Computing, pp. 351–357. ACM, New York (1999)
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., Künzli, S.: Indicator-Based Selection in Multiobjective Search. In: Yao, X., Burke, E.K., Lozano, J.A., Smith, J., Merelo-Guervós, J.J., Bullinaria, J.A., Rowe, J.E., Tiňo, P., Kabán, A., Schwefel, H.-P. (eds.) PPSN 2004. LNCS, vol. 3242, pp. 832–842. Springer, Heidelberg (2004)
Zitzler, E., Thiele, L.: Multiobjective Evolutionary Algorithms: A Comparative Case Study and the Strength Pareto Approach. IEEE T. Evolut. Comput. 3(4), 257–271 (1999)
Zitzler, E., Thiele, L., Bader, J.: On Set-Based Multiobjective Optimization. IEEE T. Evolut. Comput. 14(1), 58–79 (2009)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Auger, A., Bader, J., Brockhoff, D. (2010). Theoretically Investigating Optimal μ-Distributions for the Hypervolume Indicator: First Results for Three Objectives. In: Schaefer, R., Cotta, C., Kołodziej, J., Rudolph, G. (eds) Parallel Problem Solving from Nature, PPSN XI. PPSN 2010. Lecture Notes in Computer Science, vol 6238. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-15844-5_59
Download citation
DOI: https://doi.org/10.1007/978-3-642-15844-5_59
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-15843-8
Online ISBN: 978-3-642-15844-5
eBook Packages: Computer ScienceComputer Science (R0)