# Differential Evolution: A Survey and Analysis
## 3 Parameter Control
### Hybridization of DE with Other Evolution Algorithms  
DE has been frequently hybridized with PSO because both algorithms implement simple difference processes to perturb the current population [82]. The variation between the current and the best individual is utilized both in the refresh population method of PSO and in the DE/current-to-best/1 mutation strategy. The particle swarm optimization (PSO) method was proposed by [PERSON] and [PERSON] [83; 84]. The technique shows perfect action compared with that of other evolutionary algorithms or metaheuristics. This approach mimics human cognition and has been applied to optimization problems. The goal is to apply a group of individuals called a swarm of particles [85]. The same notation used for DE is used for PSO; a vector is used as a solution for an optimization task t. At each loop t, a particle alteration index, affected by its velocity \(v_{i}(t)\) via the equation \(x_{i}(t)=x_{i}(t-1)+v_{i}(t)\). However, two equations control the updating of the velocity \(v_{i}(t)\). \[v_{i}(t)=v_{i}(t-1)+\rho_{1}*(p_{i}-\ x_{i}(t-1))\]  
\(g_{best}\) represent the whole population and lbest describes the subpopulation encompassing the particle. The \(g_{best}\) is practical for best results. Let \(p_{g}\) be the best results of the population; thus, social influence is mathematically expressed as \(\rho_{2}*(p_{g}-\ x_{i}(t-1))\). Therefore, updating the particles at each loop as follows:  
\[v_{i}(t)=v_{i}(t-1)+\rho_{1}*(p_{i}-\ x_{i}(t-1))+\rho_{2}*(p_{g}-\ x_{i}(t-1))\]  
\[x_{i}(t)=x_{i}(t-1)+v_{i}(t)\]  
where \(\rho_{1}\) and \(\rho_{2}\) are the control parameters. PSO has several disadvantages, the most significant of which is its premature convergence. PSO consists of three components: previous velocities \(v_{i}(t-1)\), present behavior \(\rho_{1}*(p_{i}-x_{i}(t-1))\), and social behavior \(\rho_{2}*(p_{g}-x_{i}(t-1))\). Because PSO is built on these three components, it will not operate if any of those components has any issue; for example, a vector consisting of a bad solution will retard the optimal solution. However, DE does not carry the initial two features of PSO. The individual construct is based on a random walk algorithm in the search space, which then selects the optimal position index [86]. In PSO, the next position is based on the present optimal position \(p_{i}\) and by the particle's velocity \(v_{i}\). In addition, the third feature of PSO could be inferred in DE as the RAND/BEST strategy. PSO refresh the velocity of a particle applying three expressions. In the proposed strategy, the particle velocities are updated by carrying the subtract of the index vectors of any two dissimilar particles arbitrarily selected from the swarm. [PERSON] et al. proposed PSO-DV (particle swarm with differentially perturbed velocity) [87]. In the proposed scheme, particle velocities are perturbed by a new term containing the weighted difference of the position vectors of any two dissimilar particles randomly selected from the swarm. This differential velocity term mimics the DE mutation [88]. PSO-DV applies the DE differential operator to update the velocity of PSO. Two vectors are chosen randomly from the population. Then, unlike in PSO, a particle is moved to a new position only if the new position produces a better fitness value. In PSO-DV, for each particle \(i\) in the swarm, two other separate particles \(j\) and \(k\) (\(i\
eq j\
eq k\)) are chosen randomly. The difference between their locations is calculated as a difference vector:  
\[v_{id}(t+1) =\omega*v_{id}(t)+\beta*\delta_{d}+C_{2}\ \rho_{2}*(p_{gd}-\ jx_{id}(t))\] \[T_{ri} =x_{i}(t)+\beta*\delta_{d}+C_{2}\rho_{2}*(p_{gd}-jx_{id}(t))\]  
where \(CR\) is the crossover rate, \(\delta_{d}\) is the component of the subtract vector and \(\beta\) is a factor rate in the range [0, 1]. [PERSON] proposed the first combination of DE and PSO and called it SDEA as the individuals comply swarm principles [60]. DE is used to transfer the individuals to the promised region in a random fashion. [PERSON] et al.