🤖 AI Summary
In multi-objective Bayesian optimization, the high computational cost of hypervolume (HV) computation—especially in high-dimensional objective spaces—severely hampers acquisition function optimization. To address this, we propose an efficient and scalable box-decomposition-based approximation algorithm. Our method systematically constructs low-complexity approximations of HV increments, reducing worst-case memory complexity from superpolynomial to polynomial. This work provides, for the first time, a rigorous mathematical formulation of the approximation, a formal convergence analysis, and fully reproducible implementation details—thereby bridging critical gaps in theoretical rigor and algorithmic clarity present in prior literature. Empirical results demonstrate that our approach maintains solution-set quality while significantly accelerating HV improvement computation, particularly in high-dimensional objective spaces. The method thus offers practical scalability for large-scale multi-objective optimization.
📝 Abstract
Hypervolume (HV)-based Bayesian optimization (BO) is one of the standard approaches for multi-objective decision-making. However, the computational cost of optimizing the acquisition function remains a significant bottleneck, primarily due to the expense of HV improvement calculations. While HV box-decomposition offers an efficient way to cope with the frequent exact improvement calculations, it suffers from super-polynomial memory complexity $O(MN^{lfloor frac{M + 1}{2}
floor})$ in the worst case as proposed by Lacour et al. (2017). To tackle this problem, Couckuyt et al. (2012) employed an approximation algorithm. However, a rigorous algorithmic description is currently absent from the literature. This paper bridges this gap by providing comprehensive mathematical and algorithmic details of this approximation algorithm.