## 4 A Strategy Based on a Segmentation of the Search Domain's
### Evaluation of Subdomains Mean-Average Separation and Weight Factor Formulation  
The jaggedness approach works well with landscapes presenting shapes equivalent to the unidimensional sine profile displayed in Figure 8(a). Conversely, the process would concentrate the particles around the central peak of the theoretical domain shown in Figure 8(b). This result would obviously be unsuitable for a minimization problem. The proposed technique would also prompt an equivalent particle aggregation around the central minimum shown in Figure 8(c). In this case, however, this result would be beneficial to the search operation. The goal is therefore to find a proper correction of the jaggedness technique to include the complementary information illustrated in Figure 8.  
Figure 7: Uniform and pre-mapping distributions established based on the jaggedness distributions (**a**) 2D Vincent function and (**b**) 2D Himmelblau function. The three profiles in Figure 8 describe distributions with different levels of asymmetry. Their average and median positions thus deviate from one another. More precisely, the jaggedness approach is well adapted to landscapes having no substantial differences between their median and average positions, whereas domains demonstrating a greater separation between these two measures require an adaptation of the procedure. A comparison with the profile standard deviation represents an easy option to quantify the importance of the average-median difference. Equation (15) defines the difference evaluation factor (\(D_{f}\)), where \(\mu_{D}\), \(M_{D}\) and \(\sigma_{D}\) are the average and median positions, and the standard deviation defining the landscape, respectively. \(D_{f}\) may be interpreted as a measure of the influence of the outermost positions. The potential to contain optima of domains demonstrating low \(D_{f}\) values will be better described by the jaggedness, while domains with more important \(D_{f}\) evaluations require a tempered pre-probing strategy, with a reduced contribution of the jaggedness (Figure 5, Part D-1). \[D_{f}=\frac{|\mu_{D}-M_{D}|}{\sigma_{D}} \tag{15}\]  
The mean fitness value \(\overline{F}\overline{v}_{u,k}\) calculated over a subdomain \(k\) compared to the subdomain average \(\mu_{k}\) offers the required balance. The lines between positions 3 and 4 in Figure 5, Part D integrate the formulation developed below. Equations (16) and (17a) give the \(\overline{F}\overline{v}_{u,k}\) and \(\mu_{k}\) formulas, respectively. In addition, since the subdomain average \(\mu_{k}\) and the domain average \(\mu_{D}\) express different evaluations describing the same particle population, to prevent any ambiguous interpretation, Equation (17b) gives \(\mu_{D}\):  
\[\overline{F}\overline{v}_{u,k}=\frac{1}{n_{k}}\sum_{u=1}^{n_{k}}Fv_{u} \tag{16}\]  
\[\mu_{k}=\frac{1}{S^{D}}\sum_{k=1}^{S^{D}}\overline{F}\overline{v}_{u,k} \tag{17a}\]  
Figure 8: Idealized 2D landscape profiles: (a) Symmetric, (b) Prominent maximum, (c) Deep minimum. \[\mu_{D}=\frac{1}{N}\sum_{u=1}^{N}Fv_{u} \tag{17b}\]  
The difference between \(\overline{Fv}_{u,k}\) and \(\mu_{k}\) represents the separation of a subdomain \(k\) from the average position. The definition of an initial weighted distribution of the particles over the subdomains requires a normalized formulation, based on a balanced combination of the \(I_{k}\) and the influence of the separation between the subdomain means and the average. Equation (18) presents the proposed formulation for the mean-average separation or interval (\(I_{uk}\)) of subdomain \(k\). In this formula, \(\sigma_{k}\) denotes the standard deviation of the \(\overline{Fv}_{u,k}\). This description is similar to \(\sigma_{D}\), that is calculated over the entire particle population. Again, for the sake of clarity, Equation (19) gives both \(\sigma_{k}\) and \(\sigma_{D}\) definitions. In Equation (18), \(c_{4}\) is a constant arbitrary set at 4 in the present study.