Bayesian Optimization specifically aims to increase sample efficiency for hard optimization algorithms, and consequently can help achieve better solutions without incurring large societal costs. For instance, as demonstrated in this work, automotive design problems may be solved much faster, reducing the amount of computationally costly simulations and thus the energy footprint during development. At the same time, improved solutions mean that high crash safety can be achieved with lighter cars, resulting in fewer resources required for their production and, importantly, improving fuel economy of the whole vehicle fleet. Increased robustness to noisy observations further helps reduce the resources spent on evaluating regions of the search space that are not promising. Improvements to the optimization performance and practicality of multi-objective Bayesian optimization have the potential to allow decision makers to better understand and make more informed decisions across multiple trade-offs. We expect these directions to be particularly important as Bayesian optimization is increasingly used for applications such as recommender systems [35], where auxiliary goals such as fairness must be accounted for. Of course, at the end of the day, exactly what objectives decision makers choose to optimize, and how they balance those trade-offs (and whether that is done in equitable fashion) is up to the individuals themselves. Such a partitioning allows for efficient piece-wise computation of the hypervolume improvement from a new point f(xi) by computing the volume of the intersection of the region dominated exclusively by the new point with ({f(xi),P,r) (and not dominated by the P) with each hyperrectangle Sk.

Bayesian Optimization specifically aims toincrease sample efficiencyfor hard optimization algorithms, and consequently can help achieve better solutions without incurring large societal costs. In the 2-objective case, instead of padding the box decomposition, the Pareto frontier under each posterior sample can be padded instead by repeating a point on the Pareto Frontier such that the padded Pareto frontier under every posterior sample has exactlymaxt|Pt| points. Since the sequentialNEHVIis equivalent to theqNEHVIwith q = 1, we prove Theorem 1 for the generalq > 1 case. Recall from Section C.2, that using the method of common random numbers tofixthebasesamples, theIEPandCBDformulations areequivalent. Note that the box decomposition of the non-dominated space{S1,...,SKt} and the number of rectangles in the box decomposition depend onζt.