# An Analysis of Differential Evolution Population Size
###### Abstract
Accordingly, a plethora of research has emerged to address the challenge of improving DE performance by incorporating adaptive mechanisms to adjust the population size throughout the search process [19; 20; 21; 22; 23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41; 42; 43; 44; 45; 46; 47; 48; 49; 50]. Numerous adaptive strategies have focused on reducing the population size throughout generations to enhance the DE search abilities and to achieve better convergence while reducing the computational burden [19; 20; 21; 22; 23; 24; 25; 26; 27]. Another perspective is the adoption of variable population size strategies, wherein the population size is dynamically varied based on certain information gathered during the optimization process, such as the achieved performance level, convergence speed, and problem-specific information [28; 29; 30; 31; 32; 33; 34; 35; 36; 37; 38; 39; 40; 41]. In addition to the aforementioned strategies, self-adaptive population size strategies have also been proposed [42; 43; 44; 45; 46; 47; 48; 49; 50]. It is, however, worth mentioning that a significant number of studies, including those focused on adaptive DE strategies, have adopted the practice of determining the initial population either as a function of the problem dimension [27; 28; 34; 37; 38; 40; 41; 43; 44; 47; 48; 50; 51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61], or a fixed value [19; 20; 21; 22; 23; 24; 25; 26; 29; 30; 32; 35; 36; 39; 42; 46; 49; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71]. Moreover, while dynamic population size adjustment offers a certain level of flexibility, setting a fixed initial population size may impose limitations of a predetermined population and could potentially restrict the ability of the algorithm to effectively respond to the dynamic nature of the search process. This limitation also arises in adaptive strategies that impose lower and upper limits for the population size, thereby becoming constrained by these bounds [16]. [PERSON] [16] emphasizes that the performance of self-adaptive strategies with lower and upper limits for the population size depends largely on these bounds. In a study by [PERSON] et al. [31], the population size increased beyond the upper bounds when the population stagnated. Notably, the emergence of adaptive population size strategies did not resolve the problem of initial population size selection since the user is not relieved from the hassle of the determination of the initial populations. Furthermore, the adaptive strategies for the population size in the DE algorithm often introduce additional control parameters, and thus are difficult to implement and tune [72]. Despite the acknowledged significance of \(N_{P}\), the guidelines for its determination exhibit a notable lack of clarity and precision, making it an ongoing research question that necessitates further investigation and refinement. The available guidelines for the determination of the population size in the DE algorithm have led to a one-size-fits-all setting that may not always align with the optimal parameters selection for specific problems. Such guidelines, although being widely applied, often fall short of accommodating the dynamic nature of different optimization scenarios, including the problem-specific characteristics and interrelationship between DE control parameters. A thorough examination of the presented papers reveals that the majority of studies focus on the adaptation of \(F\) and \(CR\) together while adapting the population size individually [72]. The interaction between the population size and other DE control parameters is under-researched. Specifically, the side-effects introduced by individually changing the values of one control parameter on other DE control parameters have yet to be explored. In light of the aforementioned challenges, the main objectives of this study are as follows:  
1.