## 4 Discussion  
Both benchmark analysis and UPHES management application highlight conceptual differences between algorithms, especially regarding their behavior when increasing the batch size. Using PBO in a time-restricted setting necessitates to take into account the quickly increasing time needed to fit a surrogate model and to optimize the acquisition function. Our experiments have indeed shown that there is a \"breaking point\" beyond which an increase in the batch size deteriorates performance instead of adding value to the optimization process. Generally speaking, the breaking point in our context appears between \(n_{batch}=4\) and \(n_{batch}=16\), depending on the algorithm and the objective function. Indeed, the landscape of the optimized function impacts the model learning time and the acquisition time. This becomes clear when looking at the number of simulations achieved in a fixed amount of time for the three benchmark functions (Figure 2a-c). The Schwefel function is highly multi-modal with mode values of the same amplitude, hence, the greater time cost of AP compared to the Ackley function, where the minimum value area can be identified clearly. The breaking point is reached faster given that the learning time and acquisition time are significant. The addition of a complementary criterion (i.e., mic-\(q\)-EGO using UCB and EI) is efficient to obtain better outcomes in this situation in comparison with KB-\(q\)-EGO. However, the main bottleneck being the global model fitting, the breaking point is only slightly delayed thanks to the smaller number of model updates compared to KB-\(q\)-EGO. The mic-\(q\)-EGO approach is not suited for large batch sizes either. Regarding the number of performed simulations in Figure 9, we can conclude similarly: despite a faster AP of mic-\(q\)-EGO, the global model remains restricting. The same conclusion can be drawn for any approach using global models, fitted on the full data set. Another characteristic of the investigated APs is the search space decomposition. On the one side, TuRBO focuses on a trust region and we observed a significant impact on AP execution speed, as well as a clear advantage in convergence towards better outcomes, especially for benchmark functions. On the other side, BSP-EGO exploits space decomposition in the service of parallel computing. The gain in execution time allows multiplying the number of sub-regions so that each computing core is in charge of two sub-regions, yielding \(2\times n_{batch}\) candidates before selecting \(n_{batch}\) for evaluation. In both cases, the total number of evaluations is increased compared to other approaches thanks to space partitioning. For the UPHES management problem, the final expected profit is considerably improved compared to the initial sampling thanks to PBO. Even considering a large random sample of almost 12,000 objective function evaluations, the best-observed profit is around EUR \(-1200\). All investigated BO algorithms allow to achieve much better profits with significantly fewer simulations. This demonstrates the need for efficient optimization algorithms for this application. The latter observations allow extracting some general recommendations regarding BO within restricted time. Firstly, the surrogate model should remain fast to train even with a large data set. If it becomes excessively time-consuming, one can use subsets of data, or even rely on different models faster to fit: Bayesian Neural Networks [47], Sparse GP and variational inference [48], Ensemble Methods using additive structures of GPs [49] or low rank approximations of the Gram matrix [50]. Secondly, AP must also remain fast enough. In some cases AP time can become larger than learning and simulation time (for instance in MC-based \(q\)-EGO) and considerably reduce the number of possible cycles in a given time. Using multiple criteria has proven its efficiency in this study, and might be further experimented with more AFs as in [PERSON] et al.