## 2 Material and Methods
### Parallel Bayesian Optimization
#### 2.2.2 Acquisition Process for Optimization
If more candidates are required, a partial model update is necessary (line 11) using the predicted value (\(y_{PV}\)) as suggested in the KB heuristic. The loop continues until \(q\) candidate points are selected. * BSP-EGO [13] is an algorithm developed by the authors with the objective of tackling time-consuming black-box objective functions for which the execution time does not dominate the acquisition time. It uses a dynamic binary partition of the search space to optimize the AF inside sub-regions. Thereby, a local AP based on a global model is conducted simultaneously in each sub-domain. Candidate points coming from sub-regions are aggregated and sorted according to their infill value in order to choose the \(n_{batch}\) most promising ones. By the end of a cycle, the partition evolves in accordance with the performance of each sub-region: the one containing the best candidate in terms of the AF will be split further while the supposedly less valuable sub-regions are merged. At any time, the partition covers the entire search space and involves the same number of sub-regions. The number of sub-regions is chosen to be a multiple of \(n_{batch}\) to balance the load between parallel workers. Having \(n_{cand}>n_{batch}\) avoids sampling into clearly low potential sub-regions by letting the choice of \(n_{batch}\) among \(n_{cand}\). Typically in this work, \(n_{cand}=2\times n_{batch}\). As the supposed best sub-region is split at each cycle, diversification is imposed at the beginning, while intensification is favored as the budget fades. This considerably reduces the acquisition time compared to \(q\)-EGO since it can be conducted in parallel thanks to the spatial decomposition, which is valuable since the optimization must be completed in very restricted time. * EI in this study. The latter allows to approximate the AF value and time-efficiently optimize the batch of candidates. However, optimizing the multi-points AF still requires optimizing a \(n_{batch}\times d\) objective function which can become tricky and, depending on the inner optimization algorithm, time-consuming when increasing \(n_{batch}\). * TuRBO [14] uses trust regions to progressively reduce the size of the search space with the objective of compensating for the overemphasized exploration resulting from global AP. The trust region takes the form of a hyper-rectangle centered at the best solution found so far. The side length for each dimension of the hyper-rectangle is scaled according to the length scale \(\lambda_{i}\) from the GP model while maintaining a total volume of \(L^{d}\). Then, an AP is performed in the trust region that yields a batch of candidates. One or several trust regions can be maintained simultaneously. In this work, one trust region is used with multi-point EI (\(q\)-EI) evaluated through MC approximation, similarly to MC-based \(q\)-EGO. Therefore, the main characteristic of TuRBO is that the search space is re-scaled according to the length scale of the \(\mathcal{GP}\) model and evolves according to the following rule: if the current cycle improves the best solution, the trust region is extended, or else it is shrunk. As mentioned in [PERSON] et al., running with only one trust region including the whole domain \(\Omega\), TuRBO is equivalent to running the standard \(q\)-EGO algorithm. Both KB-\(q\)-EGO and MC-based \(q\)-EGO algorithms are common parallel BO approaches that use a global model fitted on the entire data set. KB-\(q\)-EGO relies on a sequential heuristic to select a batch of candidates while MC-based \(q\)-EGO uses Monte Carlo sampling to evaluate the \(q\)-points AF and create batches of candidates. The main difference is that MC-based \(q\)-EGO considers the combined utility of the set of candidates while KB-\(q\)-EGO only considers one candidate at a time, sequentially. mic-\(q\)-EGO uses a global model and exploits different AFs to limit the number of surrogate updates and add a diversity in candidates.