# Hybrid Artificial Protozoa-Based JADE for Attack Detection
## 2 Literature Review
### Overview of JADE: Adaptive Differential Evolution with Optional External Archive  
JADE (Adaptive Differential Evolution with Optional External Archive) is an advanced variant of the Differential Evolution (DE) algorithm, which is designed to enhance the optimization performance by introducing adaptive control parameters and an external archive for better exploration and exploitation. Proposed by [PERSON] and [PERSON] in 2009 [12], JADE addresses some of the limitations of traditional DE by adapting mutation and crossover rates dynamically and utilizing historical information to guide the search process more effectively.  
JADE retains the core structure of DE, including population initialization, mutation, crossover, and selection, but it introduces several innovative features. The key improvement in JADE is the adaptation of the mutation factor \(F\) and the crossover rate \(C_{r}\) during the optimization process. These parameters are crucial for balancing exploration and exploitation. Instead of using fixed values, JADE adjusts \(F\) and \(C_{r}\) based on the success of previous generations, allowing the algorithm to adapt to different stages of the search process. This adaptability helps in maintaining diversity in the population and avoiding premature convergence to local optima.  
The mutation operation in JADE is enhanced with an external archive, which stores inferior solutions that were replaced in previous generations. The archive is used to provide additional information for generating mutant vectors, thus enriching the diversity of potential solutions. The mutation strategy in JADE can be expressed as  
\[\mathbf{v}_{i}=\mathbf{x}_{i}+F\cdot(\mathbf{x}_{p}-\mathbf{x}_{i})+F\cdot( \mathbf{x}_{r}1-\mathbf{x}_{r}2), \tag{4}\]  
where \(\mathbf{v}_{i}\) is the mutant vector, \(\mathbf{x}_{i}\) is the target vector, \(\mathbf{x}_{p}\) is the best solution in the current population, \(\mathbf{x}_{r}1\) and \(\mathbf{x}_{r}2\) are randomly selected vectors, and \(F\) is the mutation factor. The use of the best solution \(\mathbf{x}_{p}\) guides the search towards promising regions, while the random vectors \(\mathbf{x}_{r}1\) and \(\mathbf{x}_{r}2\) introduce variability.  
Crossover in JADE is performed similarly to traditional DE, but with an adaptive crossover rate \(C_{r}\), which is adjusted based on the performance of the previous generations. The crossover operation combines the mutant vector and the target vector to generate a trial vector \(\mathbf{u}_{i}\):  
\[u_{i,j}=\begin{cases}v_{i,j}&\text{if }\text{rand}(0,1)\leq C_{r}\text{ or }j=j_{rand},\\ x_{i,j}&\text{otherwise},\end{cases} \tag{5}\]  
where \(u_{i,j}\) is the trial vector, \(v_{i,j}\) is the mutant vector, \(x_{i,j}\) is the target vector, \(C_{r}\) is the crossover probability, and \(j_{rand}\) is a randomly chosen index to ensure that at least one parameter is taken from the mutant vector.  
Selection in JADE follows the DE principle, where the trial vector \(\mathbf{u}_{i}\) competes with the target vector \(\mathbf{x}_{i}\). The vector with the better fitness value is selected for the next generation:  
\[\mathbf{x}_{i}^{(t+1)}=\begin{cases}\mathbf{u}_{i}&\text{if }f(\mathbf{u}_{i})\leq f( \mathbf{x}_{i}),\\ \mathbf{x}_{i}&\text{otherwise},\end{cases} \tag{6}\]  
where \(\mathbf{x}_{i}^{(t+1)}\) is the individual in the next generation, \(\mathbf{u}_{i}\) is the trial vector, and \(f(\cdot)\) is the objective function.  
JADE's adaptive mechanism and the use of an external archive significantly enhance its performance on various optimization problems. The adaptability of \(F\) and \(C_{r}\) allows JADE to fine-tune its search strategy dynamically, improving its ability to find global optima and avoid stagnation. The external archive contributes to maintaining diversity and preventing the loss of valuable information throughout the optimization process.