## 2 Materials and Methods
### Methodology
#### 2.6.2 The Global and Local Models
13: Mutate parent \(p_{i+NP/2}\) slightly to give an offspring \(c_{i+NP/2}\);  
14:endif  
15: Assign skill factors by parent candidates to offspring through vertical cultural transmission;  
16:if\(k>\theta\) and \(mod\) (\(k\), \(\theta\)) = 0,then  
17: Calculate the translated direction of each task;  
18:endif  
19: Update the position of each offspring according to the translated direction of its corresponding task;  
20:endfor  
21: Predict the fitness of each offspring with the RBF surrogates and the ensemble surrogate and select better individuals as next parents;  
22:\(k=k+1\);  
23:endwhileAfter the decision space is divided, the samples are selected from the DB to build RBF models (denoted as \(M_{g}=\left\{M_{g}^{1},M_{g}^{2},\cdots,M_{g}^{n}\right\}\)) in each subregion separately. Given the \(i\)th region, if the number of accurately assessed samples in this region is greater than \(5\times D\), all accurately assessed samples are chosen for modeling. Otherwise, the \(5\times D\) neighboring accurately assessed samples to this region are selected for modeling. As a dynamic ensemble of RBF, PRS, and SVR, the DES model has been used for high-dimensional optimization problems [28]. However, its accuracy and robustness need to be improved. On the one hand, the DES requires two parameters \(\left(R^{2}\right)_{lb}\) and \(\left(R^{2}\right)_{ub}\) to select appropriate component surrogates, which may make it not well suited for various optimization problems. For another, the DES makes the first selection of component surrogates by \(R^{2}\) and then makes the second selection by \(R^{2}\) or \(NRMSE\) according to the number of remaining component surrogates in the first selection. Therefore, the DES may use only one error metric when selecting component surrogates, which may lead to deviations in the accuracy of the final ensemble surrogate. To enhance the accuracy of the ensemble surrogate, a novel dynamic selection for the ensemble surrogate (DSES) is presented in this paper. Algorithm 2 provides the DSES technique in detail. The main distinctions between the DSES and the DES include the following two points. First, the DSES uses \(R^{2}\) and \(NRMSE\) error metrics simultaneously to select component surrogates, ensuring that the final components used to formulate the ensemble surrogate have better fitting performance. Second, the DSES selects component surrogates from two perspectives at the same time. Therefore, it does not need to limit surrogates by setting thresholds, avoiding the trouble of adjusting parameters. The final DSES (denoted as \(M_{i}\)) can be a single surrogate or an ensemble of two or three surrogates. To demonstrate that the DSES has better performance than the DES, Figure 2 gives the \(NRMSE\) values of both on the 20-dimensional F1 and F3 in the search process. From Figure 2, it was observed that the \(NRMSE\) values of DSES and DES fluctuated greatly in the early period, but both gradually stabilized as the number of real function evaluations increased. Obviously, after stabilization, the DSES had smaller \(NRMSE\) values than the DES. Therefore, one can consider that for different optimization problems, the DSES is more advantageous in building surrogate models with higher reliability than the DES. 6.3 Multi-Tasking Optimization Based on Decision Space Partition and a Dynamic Ensemble of Surrogates  
It has been demonstrated that multi-tasking optimization may effectively handle multiple optimization tasks simultaneously [30; 31]. In this study, the generalized multi-factorial evolutionary algorithm (G-MFEA) [35] is utilized to identify the best solution. The G-MFEA is a generalization of the MFEA that adds two strategies, decision variable translation and decision variable shuffling, to promote knowledge transfer between optimization problems with different optimal solutions. The \(n\) RBF models and one dynamic ensemble of surrogates are optimized simultaneously as \(n+1\) tasks in the G-MFEA. The pseudocode for multi-tasking optimization is given in Algorithm 3.