However, the sigmoid approximation can result in non-zero error. Recall from Section 3.3 that, given posterior samples, the time complexity on a single-threaded machine is For the purpose of stating our convergence results, we recall some concepts and notation from Balandat et al. We leave this to future work. We consider the setting from Balandat et al. As noted in Wilson et al.

Multi-objective Bayesian optimization (MOBO) provides a principled framework for navigating trade-offs in molecular design. However, its empirical advantages over scalarized alternatives remain underexplored. We benchmark a simple Pareto-based MOBO strategy -- Expected Hypervolume Improvement (EHVI) -- against a simple fixed-weight scalarized baseline using Expected Improvement (EI), under a tightly controlled setup with identical Gaussian Process surrogates and molecular representations. Across three molecular optimization tasks, EHVI consistently outperforms scalarized EI in terms of Pareto front coverage, convergence speed, and chemical diversity. While scalarization encompasses flexible variants -- including random or adaptive schemes -- our results show that even strong deterministic instantiations can underperform in low-data regimes. These findings offer concrete evidence for the practical advantages of Pareto-aware acquisition in de novo molecular optimization, especially when evaluation budgets are limited and trade-offs are nontrivial.

Summary and Contributions: Multi-objective optimization (MOO) problems involve more than one objective function that are to be minimized or maximized. For non-trivial instances of MOO problems, no unique solution exists that simultaneously optimizes all objectives. In that case, the aim to identify the set of Pareto optimal solutions of the problem. Bayesian Optimization (BO) approaches rely on acquisition functions (AF), to evaluate promising query points for function evaluations. BO approaches for MOO require to define AFs that are applicable to the notion of Pareto optimality.

Performing multi-objective Bayesian optimisation by scalarising the objectives avoids the computation of expensive multi-dimensional integral-based acquisition functions, instead of allowing one-dimensional standard acquisition functions\textemdash such as Expected Improvement\textemdash to be applied. Here, two infill criteria based on hypervolume improvement\textemdash one recently introduced and one novel\textemdash are compared with the multi-surrogate Expected Hypervolume Improvement. The reasons for the disparities in these methods' effectiveness in maximising the hypervolume of the acquired Pareto Front are investigated. In addition, the effect of the surrogate model mean function on exploration and exploitation is examined: careful choice of data normalisation is shown to be preferable to the exploration parameter commonly used with the Expected Improvement acquisition function. Finally, the effectiveness of all the methodological improvements defined here is demonstrated on a real-world problem: the optimisation of a wind turbine blade aerofoil for both aerodynamic performance and structural stiffness. With effective scalarisation, Bayesian optimisation finds a large number of new aerofoil shapes that strongly dominate standard designs.

In the field of multi-objective optimization algorithms, multi-objective Bayesian Global Optimization (MOBGO) is an important branch, in addition to evolutionary multi-objective optimization algorithms (EMOAs). MOBGO utilizes Gaussian Process Models learned from previous objective function evaluations to decide the next evaluation site by maximizing or minimizing an infill criterion. A common criterion in MOBGO is the Expected Hypervolume Improvement (EHVI), which shows a good performance on a wide range of problems, with respect to exploration and exploitation. However, so far it has been a challenge to calculate exact EHVI values efficiently. In this paper, an efficient algorithm for the computation of the exact EHVI for a generic case is proposed. This efficient algorithm is based on partitioning the integration volume into a set of axis-parallel slices. Theoretically, the upper bound time complexities are improved from previously $O (n^2)$ and $O(n^3)$, for two- and three-objective problems respectively, to $\Theta(n\log n)$, which is asymptotically optimal. This article generalizes the scheme in higher dimensional case by utilizing a new hyperbox decomposition technique, which was proposed by D{\"a}chert et al, EJOR, 2017. It also utilizes a generalization of the multilayered integration scheme that scales linearly in the number of hyperboxes of the decomposition. The speed comparison shows that the proposed algorithm in this paper significantly reduces computation time. Finally, this decomposition technique is applied in the calculation of the Probability of Improvement (PoI).

In multi-objective Bayesian optimization and surrogate-based evolutionary algorithms, Expected HyperVolume Improvement (EHVI) is widely used as the acquisition function to guide the search approaching the Pareto front. This paper focuses on the exact calculation of EHVI given a nondominated set, for which the existing exact algorithms are complex and can be inefficient for problems with more than three objectives. Integrating with different decomposition algorithms, we propose a new method for calculating the integral in each decomposed high-dimensional box in constant time. We develop three new exact EHVI calculation algorithms based on three region decomposition methods. The first grid-based algorithm has a complexity of $O(m\cdot n^m)$ with $n$ denoting the size of the nondominated set and $m$ the number of objectives. The Walking Fish Group (WFG)-based algorithm has a worst-case complexity of $O(m\cdot 2^n)$ but has a better average performance. These two can be applied for problems with any $m$. The third CLM-based algorithm is only for $m=3$ and asymptotically optimal with complexity $\Theta(n\log{n})$. Performance comparison results show that all our three algorithms are at least twice faster than the state-of-the-art algorithms with the same decomposition methods. When $m>3$, our WFG-based algorithm can be over $10^2$ faster than the corresponding existing algorithms. Our algorithm is demonstrated in an example involving efficient multi-objective material design with Bayesian optimization.