## 4 A Strategy Based on a Segmentation of the Search Domain's
### Reduction of the Domain Dimensionality  
The PCA approach integrated in [19] can efficiently identify the principal dimensions among correlated variables and thus allows reducing the dimensionality of real-life problems described by subjectively selected variables. On the other hand, in functions such as those of the Ref. [17] benchmark involving independent variables, PCA is not an efficient option. Of note, the dimensionality reduction of problem formulations based on independent variables should refer to the influence of the variable on the final function value.  
The present section refers to point 3 of the introduction. It also corresponds to Part C of the pseudocode in Figure 5.  
The dominant variable identification examined here rests on a sensitivity analysis. The authors of [29] proposed a statistical approach based on the variance decomposition identity of ANOVA to classify the variables of a problem in order of their influence on the final output. Equation (22) shows the standard form of the sensitivity index \(S_{ed}\) considered in [29]. This formulation is adapted to the present variable definition. Thus, \(x_{d}\) denotes the input positions along dimension \(d\), while \(F_{v}\) represents the fitness or the function value:  
\[S_{ed}=\frac{\text{Var}(\text{E}(F_{v}|x_{d}))}{\text{Var}(F_{v})} \tag{22}\]  
When considering each dimension \(d\) individually, the particles of the swarm may be classified based on the \(v_{d}\) distinct position \(x_{d}\) they take along the considered dimension (or the variable). The process thus generates \(v_{d}\) subsets. Each subset \(k\) encompasses \(n_{dk}\) particles. Equation (23) writes \(\overline{F}_{vdk}\), the average objective function of subset \(k\) of dimension \(d\):  
\[\overline{F}_{vdk}=\frac{1}{n_{dk}}\sum_{k=1}^{n_{dk}}F_{vk} \tag{23}\]  
After some manipulation of Equation (22), the authors of [29] rewrite the sensitivity index as \(\hat{S}_{ed}\). Equation (24) presents this descriptor adapted to the present nomenclature:  
\[\hat{S}_{ed}=\frac{\sum_{k=1}^{n_{dk}}n_{dk}(\overline{F}_{vdk}-\mu_{D})^{2}} {\sum_{u=1}^{N}(F_{vu}-\mu_{D})^{2}} \tag{24}\]  
where \(F_{vu}\) denotes the fitness function of particle \(u\), while the average fitness value \(\mu_{D}\) is given by Equation (17b).  
Thus, ranking the dimension in descending order of their \(\hat{S}_{ed}\) evaluation presents them from the most significant dimension of the domain to the least. In addition, since it just requires keeping track of the values of the particles that are already known, this sensitivity index formulation only involves simple calculations.  
The proposed procedure first divides the search domain into \(S^{D}\) subdomains, uniformly distributes the particles over each of them, and then evaluates the fitness values of the particles (Figure 1, Part A). To improve the precision of this first series of preparatory evaluations, the number of distributed particles may be selected larger than the number of particles employed during the optimization process itself. Thereafter, the procedure applies Equation (24) and identifies the \(D^{*}\) dominant dimensions. It then eliminates the divisions generated along the remaining \(D-D^{*}\) dimensions, and forms larger subdomains. The process finally relinks the particles to the right subdomains.  
After the above preparation steps, the proposed method allows distributing the optimization particles over the \(S^{D^{*}}\) subdomains based on biased decisions. As indicated, the process uniformly distributes the particles along the \(D^{*}\) dominant dimensions, whereas along the remaining dimensions, it simply establishes their positions randomly.