# An Analysis of Differential Evolution Population Size
## 2 Background and Related Work
### Population Size as a Problem Dimension-Related Factor  
In their original research published in 1997, [PERSON] and [PERSON] articulated that effective tuning of the DE control parameters to achieve desirable outcomes is not a tedious task and recommended the adoption of a population size ranging from \(5\times D\) to \(10\times D\) (where D is the problem dimension) [9]. Subsequently, in 1999, [PERSON] and [PERSON] further proposed enlarging the population size to avoid premature convergence, advocating for a population size of \(20\times D\)[73]. Thus, a general perception of setting the population size as a dimensionality-dependent parameter has been widely acknowledged and applied [74]. As a result, a population size that is equal to \(10\times D\) has been frequently used [75; 76; 77; 78; 79; 80; 81].  
The general guidelines provided by [PERSON] and [PERSON] regarding the determination of population size in the DE algorithm have been argued in various studies. Particularly, several studies have highlighted that the selection of appropriate values for the population size in the DE algorithm is more challenging than expected [82]. [PERSON] et al. [74] mentioned that the guidelines provided by [PERSON] and [PERSON] lack generality and are, therefore, unsuitable for practical applications. Instead, [PERSON] et al. suggested employing population sizes ranging from \(3\times D\) and \(8\times D\). Nevertheless, the conclusions drawn by [PERSON] et al. were based on limited experimental studies that focused on two unimodal and two multimodal problems. Additionally, several studies have indicated that a population size lower than or equal to the dimensionality of the problem could be advantageous in specific instances [17; 83; 84]. Figure 1 provides a visual summary of population size setting strategies within the DE algorithm, illustrating fixed-value, problem dimension-related, and adaptive/self-adaptive population size approaches.  
[PERSON] et al. [17] critiqued the practice of linearly increasing the population size as the problem dimension increases, referring to the principle of the \"curse of dimensionality\". The challenge posed by the curse of dimensionality was first demonstrated by [PERSON] in 1957 [85], revealing that the search space volume grows exponentially with increased problem dimensionality in continuous domains. [PERSON] et al. [17] argued that despite the well-known effects of the curse of dimensionality, the population sizes employed to explore higher dimensional problems usually grow only linearly and further concluded that employing a population size smaller than the problem dimension had a detrimental impact on DE convergence and resulted in the algorithm becoming trapped within a subspace of  
Figure 1: A Venn diagram illustrating the different approaches to setting the population size in the DE algorithm: problem dimension-related factor, fixed initial population size, and adaptive/self-adaptive population sizes. Adaptive/self-adaptive approaches may increase, decrease, or dynamically adjust the population size during the optimization process.  
the overall search space. However, it is worth noting that the experimentation conducted by [PERSON] et al. was based solely on the unimodal spherical problem, indicating a possible limitation in generalizing their conclusions. Notably, the conventional guidelines that suggest setting the population size to be linearly proportional to the problem dimension have never been justified, and empirical evidence to establish a solid linear correlation between population size and problem dimensionality has never been reported. [PERSON] et al. [18] identified a nonlinear logarithmic relationship between population size and problem dimension within the context of the PSO algorithm based on experimental studies.