# Faster Hypervolume-Based Search Using Monte Carlo Sampling

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Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 634))

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## Abstract

In recent years, the hypervolume indicator – a set quality measure considering the dominated portion of the objective space – has gained increasing attention in the context of multiobjective search. This is mainly due to the following feature: whenever one Pareto set approximation completely dominates another approximation, the hypervolume of the former will be greater than the hypervolume of the latter. Unfortunately, the calculation of the hypervolume measure is computationally highly demanding, and current algorithms are exponential in the number of objectives. This paper proposes a methodology based on Monte Carlo sampling to estimate the hypervolume contribution of single solutions regarding a specific Pareto set approximation. It is therefore designed to be used in the environmental selection process of an evolutionary algorithm, and allows substantial speedups in hypervolume-based search as the experimental results demonstrate.

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## Notes

1. 1.

A solution x weakly dominates another solution y, denoted by xy, if and only if $$\forall \;1 \leq i \leq n : {f}_{i}(x) \leq {f}_{i}(y)$$ referred to as f(x) ≤ f(y).

2. 2.

All parameters, such as recombination and mutation probabilities, were set according to Deb et al. (2001).

3. 3.

α was set to 0.9 for m max ≤ 100, 0.99 for m max ∈ (100, 1000), 0.999 for m max ∈ (1000, 104), 0.9999 for m max ∈ (104, 105), 0.99999 for m max ∈ (105, 106) and 0.999999 for m max > 106.

4. 4.

Around 5,000 and 20,000 samples per removal for the three and five dimensional case respectively.

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## Acknowledgements

Johannes Bader has been supported by the Indo-Swiss Joint Research Program IT14.

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### Cite this paper

Bader, J., Deb, K., Zitzler, E. (2010). Faster Hypervolume-Based Search Using Monte Carlo Sampling. In: Ehrgott, M., Naujoks, B., Stewart, T., Wallenius, J. (eds) Multiple Criteria Decision Making for Sustainable Energy and Transportation Systems. Lecture Notes in Economics and Mathematical Systems, vol 634. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04045-0_27

• DOI: https://doi.org/10.1007/978-3-642-04045-0_27

• Published:

• Publisher Name: Springer, Berlin, Heidelberg

• Print ISBN: 978-3-642-04044-3

• Online ISBN: 978-3-642-04045-0