# Differential Evolution: A Survey and Analysis
## 2 Classic Differential Evolution
### Differential Evolution Strategies  
The various equations underpinning DE possess certain aspects in common when applied to continuous optimization. All consist of an original point, sometimes referred to as the base point. The original algorithm carries out the search operation such that it finds the optimum as soon as possible. We can generalize the DE formula to the form \(\alpha=\beta+F\times\delta\), where \(\beta\) represents the base vectors and \(\delta\) the difference between vectors. Thus, the main goal of all DE equations is to provide the optimal direction based on the differential \(\beta\) and base vector \(\delta\) (Figure 2). Establishing \(\beta\) and \(\delta\) is crucial to creating an efficient strategy that can be applied to the chosen individuals from the population. However, all possible combinations of \(\beta\) and \(\delta\) can be classified into  
Figure 2: The differential \(\beta\) and base vector \(\delta\) provide the optimal direction. the following strategies: local, random, directed, and hybrid. In random strategies, abbreviated as \"Rand\", all individuals are formed randomly, and there is no prior information about the objective function. In directed strategies, abbreviated as \"DIR\", a suitable value for the base vector is chosen according to the objective function to ensure a suitable direction. Hybrid strategies include the combination of \"Rand\" and \"DIR\", labeled RAND/DIR. In another approach, the best overall vector is used, not only the best among the selected individuals; this approach is referred to as the \"BEST\". Combining the \"Rand\" and \"BEST\" yields the hybrid RAND/BEST strategy. In addition, the combination of more than two approaches, e.g., RAND/BEST/DIR, can yield favorable results by exploiting the advantages of each approach. Table 1 shows that all DE strategies employed are formed based on the DE/rand/x variation, which applies pairs of difference vectors:  
\[u_{i}=x_{r_{1}}+F_{1}(x_{r_{1}}-x_{r_{1}})+\cdots\ldots\ldots+F_{k}(x_{r_{2k}} -x_{r_{2k}})\]  
whereas the scaling factors are frequently presumed to be the same \(F_{1}=F_{2}=\ldots=F_{k}=F\). Substituting an arbitrary base vector \(\mathbf{x}_{1}\) as \(\mathbf{v}_{\text{best}}\), \"the best vector\" from the population, provides a different DE approach, indicated as DE/best/1:  
\[\mathbf{v}_{\text{y}}=\mathbf{v}_{\text{best}}+F*(\mathbf{v}_{1}-\mathbf{v}_{ 2})\]  
Most mutation strategies can be formed by a general formula based on the sum of \(k\) scaled difference vectors and a weighted average among the best vector and arbitrary ones:  
\[\mathbf{v}_{\text{y}}=\lambda\mathbf{v}_{\text{best}}+(1-\lambda)\mathbf{v}_ {\text{x}}+\sum F_{i}*(\mathbf{v}_{r2_{i}}-\mathbf{v}_{r2_{i+1}})\]  
One aspect common to all the mutation strategy methods is the base vector, which controls the search direction. The difference vector provides a mutation rate term, such as a self-adaptive term, that is added to an arbitrary or guided base vector to construct a trial individual. Over generations, the individuals of a population reside in increasingly better positions and reform themselves. The various combinations of these vectors can be categorized into four groups based on information pertaining to the values gathered from the objective function: random, directed, local and hybrid. The RAND approach consists of strategies in which the trial individual is produced without knowledge of the value of the objective function. Similarly, the RAND/DIR approach includes strategies that use the values of the objective function to determine a promising direction. Likewise, the RAND/BEST approach applies the best individual approach to proceed with a trial. Additionally, the RAND/BEST/DIR approach combines the last two groups into one that includes all of their