# A Hybrid JADE-Sine Cosine Approach for Advanced Metaheuristic Optimization
## 2 Literature Review
### Overview of Jade  
JADE (adaptive differential evolution with an optional external archive) is an advanced evolutionary algorithm designed to enhance the traditional differential evolution (DE) framework. It introduces adaptive mechanisms and an optional external archive to improve performance in terms of convergence speed, robustness, and solution quality. A key innovation in JADE is its dynamic adjustment of mutation and crossover parameters, which balances exploration and exploitation throughout the optimization process. Unlike traditional DE, JADE continuously adapts these parameters based on the current state of the search, ensuring effective performance across different optimization stages.  
Another significant feature is the external archive, which stores successful solutions from previous generations. This archive serves as a memory, guiding future searches and helping to avoid premature convergence to local optima. JADE employs a modified DE/current-to-p-best/1 mutation strategy, generating difference vectors based on the current population and a subset of the best solutions. This enhances the algorithm's ability to exploit promising search space regions while maintaining diversity. Additionally, JADE uses a binomial crossover mechanism with a dynamically adjusted crossover rate, promoting diversity and preventing stagnation by ensuring offspring inherit a mix of traits from parent solutions.  
The mutation strategy in JADE is given as shown in Equation (1):  
\[\mathbf{v}_{i,g}=\mathbf{x}_{i,g}+F\cdot\Big{(}\mathbf{x}_{pbest,g}-\mathbf{ x}_{i,g}\Big{)}+F\cdot\big{(}\mathbf{x}_{r1,g}-\mathbf{x}_{r2,g}\big{)}, \tag{1}\]  
where \(\mathbf{x}_{i,g}\) is the current vector, \(\mathbf{x}_{pbest,g}\) is the p-best individual, and \(\mathbf{x}_{r1,g}\) and \(\mathbf{x}_{r2,g}\) are two randomly selected vectors from the current population.  
The crossover operation is defined as follows:  
\[\mathbf{u}_{i,j,g}=\begin{cases}\mathbf{v}_{i,j,g}&\text{if }\text{rand}_{j}[0,1 )\leq CR\text{ or }j=j_{rand}\\ \mathbf{x}_{i,j,g}&\text{otherwise}\end{cases},\]  
where \(\mathbf{u}_{i,j,g}\) is the trial vector, \(CR\) is the crossover rate, and \(j_{rand}\) is a randomly chosen index to ensure that \(\mathbf{u}_{i,g}\) gets at least one component from \(\mathbf{v}_{i,g}\).  
The adaptive parameter control and use of the external archive contribute to JADE's improved convergence speed, allowing it to reach high-quality solutions more quickly than traditional DE and other optimization algorithms. The dynamic adjustment of search parameters and utilization of historical information also enhance JADE's robustness, making it effective for a wide range of optimization problems, including those with complex landscapes and multiple local optima. JADE's versatility extends to various types of optimization problems, such as continuous, discrete, and mixed-variable problems, making it suitable for real-world engineering design challenges and other complex tasks. JADE has been successfully applied in numerous domains, including engineering design, machine learning, and energy systems. In engineering design, JADE optimizes structural designs and control systems, handling highly complex and nonlinear solution spaces. In machine learning, it optimizes hyperparameters and feature selection, improving model performance and generalization. In energy systems, JADE optimizes the operation and control of renewable energy systems, enhancing their efficiency and stability.