## 2 Materials and Methods
### Outline of Computational Approach
#### 2.5.6 Multi-Population Based Differential Evolution (MDE) Algorithm  
MDE, developed by [PERSON] [50], is an iterative, population-based stochastic global optimization algorithm designed for both unimodal and multimodal problems. Unlike recursive methods, MDE is nonrecursive, enabling independent and parallel evolution of solution vectors. It does not require a parameter tuning phase and is capable of operating in both constrained and unconstrained search spaces. MDE employs a unique mutation operator composed of elitist, random, and noise components. The elitist component enhances optimization performance in unimodal problems, while the random component maintains solution diversity in multimodal problems. Meanwhile, the noise component prevents premature convergence by preserving numerical diversity within the population [50]. The working mechanism of MDE can be conceptually compared to individuals temporarily leaving an imaginary sub-swarm to forage for better resources. These individuals, denoted as \(B\), explore more efficient food sources and then return to \(Q\), sharing the discovered information with the rest of the population. During each iteration, the best vector is recalculated to reflect the newly acquired knowledge. The differences in foraging behavior among \(B\) individuals are simulated using a crossover control matrix (\(map\)) and a scaling factor (\(scale\)), ensuring effective adaptation and exploration [50]. The mathematical expressions of the MDE algorithm, including Equations (40)-(53), were adapted from [50; 51]. The population matrix of MDE, represented by \(Q\), is defined based on the elements of \(\Lambda\), as formulated in Equation (40). Here, \(\Lambda_{low,k}\) and \(\Lambda_{up,k}\) denote the lower and upper search space boundaries for the \(k\) th variable of the optimization problem \(G\). The objective function values for the \(Q_{i^{*}}\) vectors are computed using Equation (41). Here, \(i^{*}\) represents all valid \(i\) values (e.g., from 1 to 3 N) instead of a single \(i\), generalizing the notation to indicate that the algorithm processes all rows of the population matrix. Subsequently, the best global solution vector, denoted as \(Q^{best}\), along with its corresponding objective function value, is determined using Equation (42) [50]. \[Q_{i^{*},k}\sim U(\Lambda_{low,k},\Lambda_{up,k})\ \forall[i^{*},k]\leftarrow[1,2,3, \ldots,k,\ldots,3\cdot N],\{1:K\} \tag{40}\]  
\[fitQ_{i^{*}}=G(Q_{i^{*}}) \tag{41}\]  
\[[BestVal,Q^{best}]\leftarrow(\min(g(Q_{k})),Q_{k})\ (k\in i^{*}) \tag{42}\]  
At the beginning of MDE's iterative computation process, the \(b_{i}\) vectors are randomly selected from the set \(Q\) using Equation (43). Here, \(j_{0}\) is defined according to Equation (44). The permute \((x,y)\) function randomly shuffles the elements of the set \(\{1:x\}\) according to the Discrete Uniform Distribution and returns the first \(N\) elements of the reordered set as the output [50]. \[B=\left\{b_{i}\ |\ b_{i}=Q_{j_{0}(i)}\right\}\forall i=\{1:N\} \tag{43}\]  
\[j_{0}=permute(\tau\cdot N,N)\ \forall N\gets size(j_{0}) \tag{44}\]  
The general system equation of MDE is given in Equation (45). In this equation, \(T\), \(map\), \(temp\), and \(noise\) represent the trial vectors, crossover control matrix, temporary vectors, and noise values, respectively. The noise component prevents numerical entropy degradation in the trial vectors. The map matrix, which governs the crossover operation, is generated using Equation (46). The \(temp_{i}(k)\) vector, representing the \(k\) th variable of the \(i\) th temporary vector, is calculated using Equation (47). Finally, the scaling variable is obtained using Equation (48) [50]. \[T=B+map\odot(temp+noise-B) \tag{45}\]  
\[map=|\Lambda-\beta\odot\delta|<0.50\Rightarrow\left\{ \begin{array}{l}\alpha _{1:N}\sim U\{0,1\}\\ \beta_{(i=1:N,j=1:K)}\sim U(0,1)\\ \delta_{1:N,1:\xi}\sim U\{0,5\}\end{array} \right.