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Cone-Based Hypervolume Indicators: Construction, Properties, and Efficient Computation

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Part of the Lecture Notes in Computer Science book series (LNTCS,volume 7811)


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.


  • Hypervolume Indicator
  • Cone-based Hypervolume Indicator
  • Optimal μ-distribution
  • Complexity
  • Cone-orders

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Emmerich, M., Deutz, A., Kruisselbrink, J., Shukla, P.K. (2013). Cone-Based Hypervolume Indicators: Construction, Properties, and Efficient Computation. In: Purshouse, R.C., Fleming, P.J., Fonseca, C.M., Greco, S., Shaw, J. (eds) Evolutionary Multi-Criterion Optimization. EMO 2013. Lecture Notes in Computer Science, vol 7811. Springer, Berlin, Heidelberg.

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