Cone-Based Hypervolume Indicators: Construction, Properties, and Efficient Computation
In this paper we discuss cone-based hypervolume indicators (CHI) that generalize the classical hypervolume indicator (HI) in Pareto optimization. A family of polyhedral cones with scalable opening angle γ is studied. These γ-cones can be efficiently constructed and have a number of favorable properties. It is shown that for γ-cones dominance can be checked efficiently and the CHI computation can be reduced to the computation of the HI in linear time with respect to the number of points μ in an approximation set. Besides, individual contributions to these can be computed using a similar transformation to the case of Pareto dominance cones.
Furthermore, we present first results on theoretical properties of optimal μ-distributions of this indicator. It is shown that in two dimensions and for linear Pareto fronts the optimal μ-distribution has uniform gap. For general Pareto curves and γ approaching zero, it is proven that the optimal μ-distribution becomes equidistant in the Manhattan distance. An important implication of this theoretical result is that by replacing the classical hypervolume indicator by CHI with γ-cones in hypervolume-based algorithms, such as the SMS-EMOA, the distribution can be shifted from a distribution that is focussed more on the knee point region to a distribution that is uniformly distributed. This is illustrated by numerical examples in 2-D. Moreover, in 3-D a similar dependency on γ is observed.
KeywordsHypervolume Indicator Cone-based Hypervolume Indicator Optimal μ-distribution Complexity Cone-orders SMS-EMOA
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- 1.Auger, A., Bader, J., Brockhoff, D., Zitzler, E.: Theory of the hypervolume indicator: optimal μ-distributions and the choice of the reference point. In: FOGA 2009, pp. 87–102. ACM, NY (2009)Google Scholar
- 3.Batista, L.S., Campelo, F., Guimarães, F.G., Ramírez, J.A.: Pareto cone -dominance: Improving convergence and diversity in multiobjective evolutionary algorithms. In: Takahashi, et al. (eds.) , pp. 76–90Google Scholar
- 4.Billingsley, P.: Probability and Measure, 3rd edn. Wiley (1995)Google Scholar
- 6.Ehrgott, M.: Multicriteria Optimization, 2nd edn. Springer (2005)Google Scholar
- 9.Emmerich, M.T.M., Fonseca, C.M.: Computing hypervolume contributions in low dimensions: Asymptotically optimal algorithm and complexity results. In: Takahashi, et al. (eds.) , pp. 121–135.Google Scholar
- 10.Guerreiro, A.P., Fonseca, C.M., Emmerich, M.T.M.: A Fast Dimension-Sweep Algorithm for the Hypervolume Indicator in Four Dimensions. In: CCCG 2012, pp. 77–82 (2012)Google Scholar
- 11.Fischer, G.: Lineare Algebra, 11th edn. Vieweg Studium (1997)Google Scholar
- 16.Sawaragi, Y., Nakayama, H., Tanino, T.: Theory of multiobjective optimization. Academic Press Inc. (1985)Google Scholar
- 17.Shukla, P.K., Hirsch, C., Schmeck, H.: Towards a Deeper Understanding of Trade-offs Using Multi-objective Evolutionary Algorithms. In: Di Chio, C., Agapitos, A., Cagnoni, S., Cotta, C., de Vega, F.F., Di Caro, G.A., Drechsler, R., Ekárt, A., Esparcia-Alcázar, A.I., Farooq, M., Langdon, W.B., Merelo-Guervós, J.J., Preuss, M., Richter, H., Silva, S., Simões, A., Squillero, G., Tarantino, E., Tettamanzi, A.G.B., Togelius, J., Urquhart, N., Uyar, A.Ş., Yannakakis, G.N. (eds.) EvoApplications 2012. LNCS, vol. 7248, pp. 396–405. Springer, Heidelberg (2012)CrossRefGoogle Scholar
- 20.Yıldız, H., Suri, S.: On Klee’s measure problem on grounded boxes. In: Proceedings of the 28th Annual Symposium on Computational Geometry (SoCG), pp. 111–120 (June 2012)Google Scholar