Computing 3-D Expected Hypervolume Improvement and Related Integrals in Asymptotically Optimal Time

  • Kaifeng Yang
  • Michael Emmerich
  • André Deutz
  • Carlos M. Fonseca
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10173)


The Expected Hypervolume Improvement (EHVI) is a frequently used infill criterion in surrogate-assisted multi-criterion optimization. It needs to be frequently called during the execution of such algorithms. Despite recent advances in improving computational efficiency, its running time for three or more objectives has remained in \(O(n^d)\) for \(d\ge 3\), where d is the number of objective functions and n is the size of the incumbent Pareto-front approximation. This paper proposes a new integration scheme, which makes it possible to compute the EHVI in \(\varTheta (n \log n)\) optimal time for the important three-objective case (\(d=3\)). The new scheme allows for a generalization to higher dimensions and for computing the Probability of Improvement (PoI) integral efficiently. It is shown, both theoretically and empirically, that the hidden constant in the asymptotic notation is small. Empirical speed comparisons were designed between the C++ implementations of the new algorithm (which will be in the public domain) and those recently published by competitors, on randomly-generated non-dominated fronts of size 10, 100, and 1000. The experiments include the analysis of batch computations, in which only the parameters of the probability distribution change but the incumbent Pareto-front approximation stays the same. Experimental results show that the new algorithm is always faster than the other algorithms, sometimes over \(10^4\) times faster.


Expected hypervolume improvement Time complexity Surrogate-assisted multi-criterion optimization Efficient global optimization Probability of improvement 



Kaifeng Yang acknowledges financial support from the China Scholarship Council (CSC), CSC No. 201306370037. Carlos M. Fonseca was supported by national funds through the Portuguese Foundation for Science and Technology (FCT), and by the European Regional Development Fund (FEDER) through COMPETE 2020 – Operational Program for Competitiveness and Internationalization (POCI).


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  • Kaifeng Yang
    • 1
  • Michael Emmerich
    • 1
  • André Deutz
    • 1
  • Carlos M. Fonseca
    • 2
  1. 1.LIACS, Leiden UniversityLeidenThe Netherlands
  2. 2.CISUC, Department of Informatics EngineeringUniversity of CoimbraCoimbraPortugal

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