Faster Exact Algorithms for Computing Expected Hypervolume Improvement

Abstract

This paper is about computing the expected improvement of the hypervolume indicator given a Pareto front approximation and a predictive multivariate Gaussian distribution of a new candidate point. It is frequently used as an infill or prescreening criterion in multiobjective optimization with expensive function evaluations where predictions are provided by Kriging or Gaussian process surrogate models. The expected hypervolume improvement has good properties as an infill criterion, but exact algorithms for its computation have so far been very time consuming even for the two and three objective case. This paper introduces faster exact algorithms for computing the expected hypervolume improvement for independent Gaussian distributions. A new general computation scheme is introduced and a lower bound for the time complexity. By providing new algorithms, upper bounds for the time complexity for problems with two as well as three objectives are improved. For the 2-D case the time complexity bound is reduced from previously $O(n^3 \log n)$ to $O(n^2)$. For the 3-D case the new upper bound of $O(n^3)$ is established; previously $O(n^4 \log n)$. It is also shown how an efficient implementation of these new algorithms can lead to a further reduction of running time. Moreover it is shown how to process batches of multiple predictive distributions efficiently. The theoretical analysis is complemented by empirical speed comparisons of C++ implementations of the new algorithms to existing implementations of other exact algorithms.

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