## 2 Materials and Methods
### Optimization Techniques  
Surrogate optimization is suitable for those problems with computationally expensive function evaluations. Using the surrogate model instead of the actual simulation model can reduce the computation time, save computational efforts, and preserve acceptable approximation accuracy. The general flowchart of a surrogate optimization program is plotted in Figure 5. The surrogate model algorithms consist of the steps described next. First, an initial experimental design is generated to determine some sample points at which the function evaluation is to be done. Then, the program will spend time obtaining the objective function values at all these points. After that, there is going to be a surrogate model (response surface) built through the data fitting for the initial design points and adopted to predict the objective function values at unsampled points in the variable domain to decide at which point(s) to conduct the subsequent expensive function evaluation. the function evaluation will be conducted at that (those) selected in the previous step. The surrogate model can be updated using the new data set. All of the above steps will be repeated until the user-defined stop criterion is met.  
Surrogate model algorithms in the reported literature differ mainly concerning the generation of the initial experimental design, the chosen surrogate model, and the strategy for selecting the sample point in each iteration. In this paper, we used the symmetric Latin hypercube design (\"SLHD\") [49] to generate 30 initial sample points in the design spaces. The return value of the objective function is calculated for these sample points based on this output analysis of the UBET. Next, a surrogate model is built to find a globally good approximation at an unobserved location for this optimization problem. A mixture surrogate model (\"MIX_RcKg\") containing cubic radial basis function interpolation [50], the Kriging model with a Gaussian correlation function (first order regression polynomial) [19], and a multivariate adaptive regression spline [51], are employed to prevent accidentally selecting the worst surrogate model, especially in the case where nothing about the behavior of the black-box objective function is known. For this problem with only continuous variables, we used the candidate point approach (\"CAND\") [52] for selecting the sample point in each iteration. In this approach, two groups of candidates for the next sample point are created. Those in the first group are generated by perturbing the best point found so far. Those in the second group are combined by uniformly selecting sample points from the whole box-constrained variable domain. Then, the objective function values at the candidate points are predicted using the established surrogate model. The distance from each candidate sample point to the set of already sampled points is computed. A weighted sum of these two results is used to determine the best candidate point. That avoids selecting the one that is locally optimal rather than globally optimal. Computational experiments also show that the candidate point approach leads to the best results. In the current problem, the iteration generally starts outside the feasible domain. The sample point that violates the constraint receives the corresponding 'penalty' and is excluded from the candidate points. As the number of iterations increases, the increasing penalty forces the iteration point to approach the feasible domain. A surrogate model optimization toolbox [53] and self-developed UBET program were used to solve the optimization problem, and the flowchart of the optimization program is shown in Figure 6.  
Figure 5: General flowchart of a surrogate optimization program.