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Logarithmic-Time Updates in SMS-EMOA and Hypervolume-Based Archiving

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Part of the Advances in Intelligent Systems and Computing book series (AISC,volume 227)


The hypervolume indicator is frequently used in selection procedures of evolutionary multi-criterion optimization algorithms (EMOA) and in bounded size archivers for Pareto non-dominated points. We propose and study an algorithm that updates all hypervolume contributions and identifies a minimal hypervolume contributor after the removal or insertion of a single point in ℝ2 in amortized time complexity O(logn). This algorithm will be tested for the efficient update of bounded-size archives and for a fast implementation of the steady state selection in the bi-criterion SMS-EMOA. To achieve an amortized time complexity of O(logn) for SMS-EMOA iterations a constant-time update method for establishing a ranking among dominated solutions is suggested as an alternative to non-dominated sorting. Besides the asymptotical analysis, we discuss empirical results on several test problems and discuss the impact of the overhead caused by maintaining additional AVL tree data structures, including scalability studies with very large population size that will yield high resolution approximations.


  • Pareto Front
  • Objective Vector
  • Binary Search Tree
  • Sorting Order
  • True Pareto Front

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Correspondence to Iris Hupkens .

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Hupkens, I., Emmerich, M. (2013). Logarithmic-Time Updates in SMS-EMOA and Hypervolume-Based Archiving. In: Emmerich, M., et al. EVOLVE - A Bridge between Probability, Set Oriented Numerics, and Evolutionary Computation IV. Advances in Intelligent Systems and Computing, vol 227. Springer, Heidelberg.

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  • Publisher Name: Springer, Heidelberg

  • Print ISBN: 978-3-319-01127-1

  • Online ISBN: 978-3-319-01128-8

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