# Hybrid Improved PSO Algorithm for Soil Property Parameter Estimation
## 3 Improvement of the PSO Algorithm
### Comparative Test Experiment  
To validate the performance of the PSO-EDO, we selected six widely-used benchmark test functions (F1-F6) from the 2017 IEEE Congress on Evolutionary Computation (IEEE CEC) [22], as shown in Table 2. These functions, encompassing both unimodal and multimodal types, are designed to evaluate the algorithm's performance across diverse optimization scenarios. While standard test functions are commonly employed in algorithm comparisons, their selection here is intentional and grounded in the specific challengesof soil parameter estimation, particularly for multivariate, high-dimensional, nonlinear, and multimodal optimization problems. For example, F1 and F2, as classical unimodal functions, simulate idealized conditions in soil parameter optimization where a single global optimum exists, such as estimating parameters like cohesion modulus (\(k_{c}\)) under homogeneous soil conditions. In contrast, F3-F6, as multimodal functions, emulate complex real-world scenarios where soil parameters (e.g., soil friction angle (\(\varphi\)) and sinkage coefficient (\(n\)) may exhibit multiple local optima depending on soil heterogeneity, necessitating the algorithm to avoid premature convergence. By optimizing these standard functions, we comprehensively assess PSO-EDO's global search capability, ability to escape local optima, and computational efficiency in addressing intricate soil parameter estimation challenges. Figure 3 shows the plots of F1-F6 for the two-dimensional case. Six related algorithms were selected for comparative experiments: basic PSO [23], SPSO, PSO-GA, PSO-WOA, EDO, and the improved PSO algorithm. To ensure fairness, the basic parameters of the different PSO algorithms and the improved PSO algorithm were kept consistent, including the population size \(N=30\), the dimensionality \(D=30\), and the maximum number of iterations \(k_{max}=1000\). Each algorithm was run independently 30 times, and the mean, best value, standard deviation (std), and computation time of the optimal results were calculated as performance metrics for algorithm optimization. From Table 3, it can be observed that the mean and standard deviation of the PSO-EDO optimization results for F1 and F2 reach the theoretical optimal values. This result indicates that the improved algorithm offers good solution accuracy and stability for these unimodal optimization problems. Although the EDO algorithm also achieved the theoretical optimal values for the mean and standard deviation in F1 and F2, its average computation time was longer than that of PSO-EDO. The PSO-WOA algorithm also yielded good optimal values in the 30 iterations; however, its mean optimal value and standard deviation were relatively high, indicating low stability. For the multimodal problems F4, F5, and F6, the optimization results of EDO and PSO-EDO were similar, both achieving the theoretical optimal values. However, the improved algorithm was 0.1 to 0.2 s faster on average than EDO in terms of computation time. The PSO-WOA algorithm also produced desirable optimization results with a relatively short average computation time, but its robustness was inferior to the improved algorithm.