On Setting the Reference Point for the Hypervolume Indicator in Multi-Objective Bayesian Optimisation

The Hypervolume (HV) indicator is widely used in Bayesian optimisation for expensive multi-objective optimisation problems, such as in Expected Hypervolume Improvement (EHVI). A key issue in HV-based methods is the specification of the reference point, which critically influences optimisation performance. In situations where an estimate of the problem's true Pareto front is unavailable (typically the case in real-world applications), a common strategy is to dynamically update the reference point according to the boundaries of the observed Pareto front. However, different regions of the Pareto front often exhibit different levels of reachability during the optimisation process. For example, certain boundary regions may be particularly difficult to discover. As such, selecting the reference point dynamically based on the currently observed Pareto front may bias the search and hinder the identification of the entire Pareto front, as solutions beyond the reference point contribute (approximately) zero to the EHVI/HV value. In this paper, we propose a simple method to address this issue. Specifically, we adaptively adjust the HV reference point to shift the search focus between the boundary and interior regions of the Pareto front, aiming to identify the boundaries first and then the interior regions. We validate the proposed method on a diverse set of 69 benchmark and real-world problems. Extensive experimental results demonstrate that it consistently outperforms various reference-point specification strategies.