## 3 Technical Background
### Overview of Differential Evolution (DE)  
Differential Evolution (DE) [29] is known for its efficiency and effectiveness in the exploitation phase of optimization tasks. Grounded in evolutionary principles, the typical workflow of DE encompasses initializing a population of candidate solutions, performing mutation and crossover on each individual to create a trial vector, evaluating the fitness of these trial vectors, and applying a selection to establish the population for the next generation. This process is repeated until a predefined stopping criterion is met [30]. In addition, DE employs a straightforward yet potent set of operators (mutation, crossover, and selection) to maneuver through the solution space and refine solutions toward the global optimum. DE's approach is centered on a population-based methodology. It operates on a pool of candidate solutions, iteratively working through them to explore and exploit the solution space. Each candidate, also referred to as an individual or a vector, represents a possible solution to the optimization problem at hand. A critical aspect of DE is its mutation strategy, which plays a pivotal role in introducing variability and novel traits into the population [30]. The mutation strategy in differential evolution (DE) is encapsulated by an equation that outlines the process of mutated vector generation for each population member. This generation is accomplished by adding the weighted difference between two distinct, randomly chosen population members, \(X_{r2}\) and \(X_{r3}\), to a third member, \(X_{r1}\)[30]. The formula illustrating this mutation strategy is presented in Equation (3):  
\[V_{i}=X_{r1}+F\cdot(X_{r2}-X_{r3}) \tag{3}\]  
where \(V_{i}\) represents the mutated vector, \(X_{r1}\), \(X_{r2}\), and \(X_{r3}\) are the three distinct, randomly selected vectors from the population, and \(F\) is a scaling factor that controls the mutation extent. Following mutation, DE employs a crossover operation to further diversify the population. This operation integrates the mutated vector with an existing population member, creating a trial vector. It operates by interchanging components between both vectors, influenced by a crossover probability \(CR\), to enhance the search of the solution space [30]. The selection mechanism in DE hinges on the comparative fitness of the trial vector and the original individual. The one exhibiting superior fitness is retained for the next generation, fostering gradual enhancement in the overall quality of the population. The Role of Crossover in the DE Phase  
While the traditional role of the crossover operator in Differential Evolution (DE) is to introduce diversity by mixing components from the mutant and target vectors, it is important to address a subtle aspect of its function in our implementation. Specifically, the crossover operator can sometimes reverse parts of the mutation operation, which may give the impression that it is primarily a corrective mechanism rather than one that enhances diversity. Reversing Mutation:  
In our approach, the crossover operation decides whether to retain components from the target vector \(\mathbf{P}_{i}(j)\) or incorporate components from the mutant vector \(\mathbf{V}_{i}(j)\). Due to this decision-making process, particularly when the crossover rate \(C_{r}\) is low, the crossover may indeed undo some of the changes introduced by mutation. This occurs when more components from \(\mathbf{P}_{i}(j)\) are retained, effectively reversing parts of the mutation's effects on those components. Introducing Diversity:  
However, the crossover operation still plays a crucial role in maintaining diversity within the population. By selectively incorporating components from the mutant vector \(\mathbf{V}_{i}(j)\), especially when \(C_{r}\) is high, the crossover ensures that new information is introduced into the trial vector \(\mathbf{U}_{i}\).