## 2 Material and Methods
### Parallel Bayesian Optimization
#### 2.2.2 Acquisition Process for Optimization  
The BO framework is illustrated in Algorithm 1. The mentioned parallel algorithms follow the same scheme but differ in the candidate selection phase which is represented by the optimization of the \(\alpha\) function in Algorithm 1. Line 4 of Algorithm 1 states that the algorithm is searching for the best (batch of) candidate(s) in terms of \(\alpha\) to be added. The candidate selection, referred to as the AP, is also an optimization problem sometimes called inner optimization. ```
1:Initial DoE: \(\{\mathbf{X},\mathbf{y}\}\)
2:while Budget available do
3:\(\mathcal{M}=\mathcal{GP}(\{\mathbf{X},\mathbf{y}\})\)
4:\(\mathbf{x}_{new}=argmax_{\Omega}(\alpha(\mathbf{x}))\)
5:\(y_{new}=f(\mathbf{x}_{new})\)
6:\((\mathbf{X},\mathbf{y})=(\mathbf{X},\mathbf{y})\cup(\mathbf{x}_{new},y_{new})\)
7:endwhile
```  
**Algorithm 1** Bayesian Optimization  
Even though BO and GP seems a legitimate choice in the context of optimization with a relatively small budget and strong time constraint the choice of the AP is difficult and must be done in accordance with the time constraint. This is why several different APs presenting different behaviors are investigated in this study. * The KB-\(q\)-EGO algorithm refers to the work of [PERSON] et al. [9; 36] where they present heuristics to approximate multi-point criteria which are difficult to exploit when the required number of candidates exceeds \(q=2\) (\(q\) being the batch size, also denoted as \(n_{batch}\) throughout this paper). The idea is to replace the time-consuming simulation with a fast-to-obtain temporary/fantasy value in order to update the surrogate model \(\mathcal{M}\), which allows the suggestion of a new distinct candidate. In the KB heuristic, it is decided to trust the surrogate model (hence, Kriging Believer) and use its output as fantasy value. The operation can be repeated sequentially until \(q\) design points are selected. Then, it is possible to exactly evaluate them in parallel and replace the temporary values with the real costs. However, this strategy has the major drawback of requiring \(q\) sequential updates of \(\mathcal{M}\) per cycle. In [PERSON] et al. [9], inside the heuristic loop, the Kriging model is used without hyperparameter re-estimation. To preserve this aspect and in order to alleviate the fitting cost, the budget allocated to the intermediate fitting of the surrogate model is reduced compared to a full update performed at the beginning of a cycle. * We propose Multi-infill-criteria \(q\)-EGO (mic-\(q\)-EGO), an alternative AP that uses multiple IC in the same cycle in order to limit the number of surrogate fitting operations. This approach is a variation of \(q\)-EGO using a multi-infill strategy such as in [PERSON] et al. [10] or [PERSON] et al. [11]. Using multiple AFs allows to select different candidate points without updating the surrogate model, assuming that their optimization yields distinct candidate points. Moreover, it has been observed that resorting to different AFs can favorably impact the objective value, especially when the batch size is large. Indeed, repeatedly updating \(\mathcal{M}\) using non-simulated data may degrade the relevance of proposed candidates [8]. Since standard EI, used as the primary criterion, is known to sometimes over-explore the search space, it is decided to rely on the Upper Confidence Bound (UCB) criterion to improve the exploitation. In this paper, two ICs are used with the same number of candidates from each criterion. Therefore, albeit generalizable to more criteria not necessarily evenly distributed, the description of Algorithm 2 considers the previous statement for sake of simplicity. We denote \(\alpha_{1}\) and \(\alpha_{2}\) the two AFs, then for the same model, a candidate can be chosen according to each AF (line 6 and 7). Despite not being implemented yet, the optimizations of \(\alpha_{1,2}\) could be conducted in parallel.