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

  • Michael Emmerich
  • André Deutz
  • Johannes Kruisselbrink
  • Pradyumn Kumar Shukla
Part of the Lecture Notes in Computer Science book series (LNCS, 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 SMS-EMOA 


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© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Michael Emmerich
    • 1
  • André Deutz
    • 1
  • Johannes Kruisselbrink
    • 1
  • Pradyumn Kumar Shukla
    • 2
  1. 1.LIACS. Leiden UniversityLeidenNetherland
  2. 2.Karlsruhe Institute of TechnologyKarlsruheGermany

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