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Maximising hypervolume and minimising ϵ-indicators using Bayesian optimisation over sets

Bayesian optimisation methods have been widely used to solve problems with computationally expensive objective functions. In the multi-objective case, these methods have been successfully applied to maximise the expected hypervolume improvement of individual solutions. However, the hypervolume, and other unary quality indicators such as multiplicative ϵ-indicator, measure the quality of an approximation set and the overall goal is to find the set with the best indicator value. Unfortunately, the literature on Bayesian optimisation over sets is scarce. This work uses a recent set-based kernel in Gaussian processes and applies it to maximise hypervolume and minimise ϵ-indicators in Bayesian optimisation over sets. The results on benchmark problems show that maximising hypervolume using Bayesian optimisation over sets gives a similar performance than non-set based methods. The performance of using ϵ indicator in Bayesian optimisation over sets needs to be investigated further. The set-based method is computationally more expensive than the non-set-based ones, but the overall time may be still negligible in practice compared to the expensive objective functions.