## 2 Literature Review
### Overview of Adaptive Differential Evolution with an Optional External Archive (JADE)  
Adaptive differential evolution with an optional external archive, usually referred to as JADE, is an improved variant of differential evolution (DE). Its design focuses on increasing convergence speed and maintaining solution diversity. JADE accomplishes these goals by adapting two central parameters--the scale factor and crossover probability--and by keeping an external archive of solutions. This archive retains historical information that can help the algorithm explore the search space more effectively. JADE employs a special mutation strategy referred to as _current-to-pbest_. Equation (24) shows how the mutant vector \(\mathbf{v}_{i,G}\) is formed from the target vector \(\mathbf{x}_{i,G}\) at generation \(G\). \[\mathbf{v}_{i,G}=\mathbf{x}_{i,G}+F_{i}\big{(}\mathbf{x}_{\text{pbest},G}- \mathbf{x}_{i,G}\big{)}+F_{i}\big{(}\mathbf{x}_  
After producing the mutant vector, JADE relies on binomial crossover to generate a trial vector \(\mathbf{u}_{i,G}\) by mixing the components of \(\mathbf{v}_{i,G}\) and \(\mathbf{x}_{i,G}\). The crossover step is illustrated in Equation (16). \[u_{i,G}^{d}= \begin{cases}v_{i,G}^{d},&\text{if rand}_{d}\leq CR_{i}\text{ or }d=\text{rand}\_\text{index},\\ x_{i,G}^{d},&\text{otherwise},\end{cases}  \tag{2}\]  
In Equation (16), \(u_{i,G}^{d}\) is the \(d\)th component of the trial vector \(\mathbf{u}_{i,G}\). The term \(v_{i,G}^{d}\) represents the \(d\)th component of the mutant vector \(\mathbf{v}_{i,G}\), while \(x_{i,G}^{d}\) is the \(d\)th component of the target vector \(\mathbf{x}_{i,G}\). The quantity \(CR_{i}\) is the crossover probability for the \(i\)th individual, and rand\({}_{d}\) is a uniformly distributed random number in \([0,1]\). The index rand_index ensures that at least one component of \(\mathbf{u}_{i,G}\) is inherited from the mutant vector. The selection step compares \(\mathbf{u}_{i,G}\) against \(\mathbf{x}_{i,G}\) and keeps the superior one according to Equation (3). \[\mathbf{x}_{i,G+1}= \begin{cases}\mathbf{u}_{i,G},&\text{if }f(\mathbf{u}_{i,G}) \leq f(\mathbf{x}_{i,G}),\\ \mathbf{x}_{i,G},&\text{otherwise}.\end{cases}  \tag{3}\]  
Equation (3) indicates that \(\mathbf{x}_{i,G+1}\) is the solution that exhibits the better (or equal) objective function value among \(\mathbf{x}_{i,G}\) and \(\mathbf{u}_{i,G}\). If \(\mathbf{u}_{i,G}\) replaces \(\mathbf{x}_{i,G}\), the replaced vector \(\mathbf{x}_{i,G}\) may be stored in the archive \(\mathcal{A}\). Thus, the archive collects solutions that have been displaced during the evolutionary process, which can help retain search diversity in subsequent generations. One of JADE's core strengths is the adaptation of the scale factor \(F_{i}\) and crossover probability \(CR_{i}\). The means of these parameters, \(\mu_{F}\) and \(\mu_{CR}\), are updated dynamically as indicated in Equations (4) and (5). \[\mu_{F}\leftarrow(1-c)\,\mu_{F}\ +\ c\,\text{mean}_{\text{L}}(S_{F}) \tag{4}\]  
\[\mu_{CR}\leftarrow(1-c)\,\mu_{CR}\ +\ c\,\text{mean}(S_{CR}) \tag{5}\]  
In Equations (4) and (5), \(c\) is a small constant (for instance, 0.1) that controls how much emphasis is placed on newly gathered information. The sets \(S_{F}\) and \(S_{CR}\) contain the scale factors and crossover probabilities that produced improvements in the current generation. The function mean\({}_{\text{L}}\) typically denotes the Lehmer mean, used for \(F\), while mean represents the arithmetic mean, used for \(CR\). By sampling \(F_{i}\) and \(CR_{i}\) from probability distributions centered around \(\mu_{F}\) and \(\mu_{CR}\) and then adaptively updating those means based on successful parameter values, JADE steers its parameters toward beneficial settings over time. In summary, JADE proceeds by generating new mutant vectors according to Equation (24), performing crossover as shown in Equation (16), selecting superior solutions using Equation (3), and adapting its key parameters with Equations (4) and (5).