## 4 A Strategy Based on a Segmentation of the Search Domain's
### Evaluation of Landscapes Jaggedness  
The Jaggedness (_J_) considered here also refers to the frequency spectrum representing the landscape. While a precise representation of the frequency spectrum of a D-dimensional domain is not easy, a 2D landscape provides a simple illustration of the concept, which then becomes equivalent to surface roughness. Hence, high energy levels at higher frequencies should reflect a higher jaggedness of surfaces. However, while a Fast Fourier Transform (FFT) would normally provide the required information, in the present context involving D-dimensional domains, obtaining a low computation cost evaluation remains challenging. The proposed strategy is close to the quantitative method put forward in 1970 by [PERSON] for EEG activity evaluation [26]. The [PERSON] method aimed at predicting properties of time signals in the frequency domain from time-based measures of physical parameters. Figure 6: Landscapes (**a**) 2D Vincent function and (**b**) 2D Himmelblau function. The author proposed three parameters: Activity, Complexity and Mobility. These parameters are also described as normalized slope descriptors [27]. The first reflects the squared standard deviation of the signal amplitude. It thus evaluates the mean power. Complexity provides an evaluation of the signal bandwidth. Finally, Mobility is designed to provide an estimate of the mean frequency of the signal [27]. Mobility thus offers information closely related to the needs of the present study. If a time signal \(T(t)\) is known by \(n\) sample values \(\tau_{u}\), Activity (\(A\)) is expressed as \(A^{2}=\frac{1}{n}\sum_{u=1}^{n}\tau_{u}^{u}\), and when defining \(\delta_{u}=\tau_{u+1}-\tau_{u}\), Mobility (\(M\)) is expressed as \(M=\frac{1}{2\pi}\sqrt{\frac{1}{(A\Omega)^{T}n}\sum_{u=1}^{n}\delta_{u}^{2}}\). \(M\) is measured in Hz units [28]. Refs. [27; 28] demonstrate that \(M\) provides accurate predictions of the main signal frequency. In the present context, a good evaluation of the domain jaggedness or mean frequency is sufficient. However, since \(M\) is formulated for a time signal or a 1D domain, it is not directly transferable to D-dimensional spaces. Basically, \(M\) evaluates the average slope of the signal from regularly spaced samples. In the proposed scheme, the segmentation of the D-dimensional domains or multidimensional spaces results in multidimensional zones. The jaggedness of each of them is assumed to be correlated with its equivalent mean frequency. We may then postulate that the particle distribution established before the optimization should respect the jaggedness distribution since domains with oscillations at higher frequencies are more susceptible to containing numerous optima. Instead of the mean frequency, the calculation scheme assumes that the maximum slope amplitude calculated between consecutive evaluation points best describes the \(J\) amplitudes of the multidimensional zones or their potential to contain optima. The algorithm lines between positions 2 and 3 in Figure 5, Part D integrate the formulation developed below. Equation (12) formulates \(J\) between the consecutive points \(u\) and \(u\pm 1\) separated by the Euclidian distance \(l_{D}\). In fact, since distances \(l_{D}\) are calculated in D-dimensional subdomains between individuals uniformly distributed, each position potentially possesses \(3^{D}-1\) adjacent neighbours with higher \(l_{D}\) values along the space diagonals. Therefore, to restrict the calculations to the closest neighbours, they exclude the diagonal directions. \[J_{u}=Max\bigg{[}\frac{|Fv_{u}-Fv_{u\pm 1}|}{l_{D}}\bigg{]} \tag{12}\]  
To obtain a representative description of the jaggedness for a subdomain \(k\), Equation (13) establishes an averaged evaluation \(\overline{J}_{uk}\) of the \(J_{u}\) values calculated among the \(nk\) particles of the subdomain \(k\). Finally, Equation (14) writes the normalized jaggedness \(J_{k}\) of subdomain \(k\) determined when considering the \(S^{D}\) subdomains.