## 5 An Intermediate Improvement of the Particle Positions  
The authors of [19] boost the convergence rate of the DE algorithm with a contour prediction strategy advanced in [30]. This method is integrated into each optimization iteration. Essentially, the approach aims at improving the position of selected particles from contours formed by surrounding particles. In [19; 30], this step forms contours around the best particles. In [19], the process examines the best particle of each niche. On the other hand, since the study in [30] concentrates on single optimum functions, there is no niche. The process considers only the first few best particles of the population. It thus has to sort them first. The approach developed in this paper does not involve any niche. Moreover, for the sake of efficiency, it is preferable to avoid sorting the particles at each iteration. Therefore, instead of considering the best particles, this step examines a fraction (_fp_) of the particles, regardless of their fitness value, their position or their type. It selects the first particles randomly, and then successively picks the following with respect to _fp_. For instance, if \(N\) is 1000 particles, with an _fp_ of 0.25, 250 particles will be visited respecting a step size of \(\frac{1}{fp}\). Here, since the particles are initially inserted into the swarm according to the generation sequence, to cover the swarm, the process randomly selects the first particle number among the first \(\frac{1}{fp}\) particles. The present section refers to point 4 of the introduction. It describes the procedure introduced in Part B-3, Figure 1. Figure 10 presents the corresponding pseudocode. Again, in this figure, numbers mark the beginning position of operation groups. For each visited particle, the procedure identifies its _cnb_ closest neighbours (Figure 10, Part E-2). These neighbours are those situated at the lowest Euclidian distances. After that, following the technique developed in [30], the position of a potentially better particle is extra/interpolated between the considered particle and each of its neighbours taken individually. The interpolation process considers a target fitness value _tFv_. Equation (25) describes the linear interpolation giving the new \(\left(x_{jd}\right)^{\prime}\) positions, while Equation (26) gives _tFv_. Equation (26) integrates a fraction \(\rho\) (\(\leq 1\)) to establish the target to improve the particle fitness value _Fv\({}_{l}\)_ for a minimization operation. In these expressions \(j\in\{1\) to _cnb_\(\}\), \(x_{id}\) and \(x_{jd}\) are respectively the position of the improved and the neighbour particles along  
Figure 10: Pseudocode of the particle position improvement. dimension \(d\), while \(Fv_{i}\) and \(Fv_{j}\) are respectively the fitness value of the improved and the neighbour particles. The final location of the improved particle \(\left(x_{id}\right)^{*}\) corresponds to the average position expressed by Equation (27) (Figure 10, Part E-3):  
\[\left(x_{jd}\right)^{\prime}=x_{id}+\frac{tFv-Fv_{i}}{Fv_{j}-Fv_{i}}\left(x_{ jd}-x_{id}\right) \tag{25}\]  
\[tFv=(1-sgn(Fv_{i})\rho)Fv_{i} \tag{26}\]  
\[\left(x_{id}\right)^{*}=\frac{1}{cnb}\sum_{j=1}^{cnb}\left(x_{jd}\right)^{\prime} \tag{27}\]  
If the real fitness \((Fv_{i})^{\prime}\) of the predicted improved position is better than \(Fv_{i}\), particle \(I\) is relocated to that position (Figure 10, Part E-4). Otherwise, it remains possible that \(\left(Fv_{i}\right)^{\prime}\) will outperform any of the considered \(cnb\) neighbours. In that case, in order to avoid leaving behind valuable knowledge, the new position may replace the closest less performant neighbour. However, it is worth mentioning here that this neighbour replacement may lead to premature convergence. Therefore, it is not recommended to integrate it systematically into the iteration. Since the whole operation does not target the best particles of the swarm, the proposed approach repeats it twice at each iteration (Figure 10, Part E-1).