## 2 Differential Step-Scaling Antilion Algorithm
### Algorithm Idea and Specific Steps  
The improvement of the algorithm mainly focuses on the enrichment of population diversity and improvement of local search ability. The pseudo-code corresponding to the Algorithm 1 is shown as follows:```
Initialize the first population of ants and antlions randomly  Calculate the fitness of ants and antlions  Find the best ant or antlions, then set it as the initial elite antlion  While the maximum iteration is not reached  For each ant  Select an antlion using Roulette wheel  Update boundaries using Equations (12) and (13)  Make a random walk using Equation (1)  Normalize and update the position of ant using Equations (9) and (16)  End for  Calculate the fitness of all ants  Replace an antlion if its corresponding ant becomes fitter  Apply Mutation, Crossover, and Selection operator to antlions  Update the elite antlion End while Return the elite antlion
```

**Algorithm 1** Pseudo-Code of DSALO

## 3 Performance Evaluation of Differential Step-Scaling Ant-Lion Algorithm

### Algorithm Evaluation Criteria

When we use a test function, the optimal value of each test function is already known. Because of the random walk strategy of the algorithm, the results of the algorithm in each run may be different. Considering the stability and accuracy of the algorithm, the mean value of fitness, standard deviation, maximum and minimum are selected as evaluation criteria of the algorithm. For an algorithm, if it can obtain a fitter mean value closer to the optimal value, a smaller fitness standard deviation, a smaller fitness maximum value as well as a smaller fitness minimum value, it indicates that the algorithm is excellent, or better than other algorithms in the problems applied in this comparison. ### Test Function

The test functions, corresponding solution intervals and optimal values used in this paper are shown in Table 1. Of the four test functions, F1 is a unimodal function, F2 and F3 are multimodal functions, and F4 is a composite function [13]. ### Analysis of Test Results

In this paper, DSALO is compared with ALO, PSO, and OEALO (Opposition based Exploratory differential Lion-based Optimization [24]). The dimension of test functions is set to 50. The test results are shown in Table 2. It can be seen from the table that \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Functional Expression** & **Solution** & **The Optimal Value** \\ \hline F1 & \(f_{1}(x)=\sum\limits_{i=1}^{n-1}\left[100(x_{i+1}-x_{i}^{2})^{2}+(x_{i}-1)^{2}\right]\) & \([-30,30]\) & 0 \\ F2 & \(f_{2}(x)=-20\exp\left(-0.2\sqrt{\frac{1}{n}\sum_{i=1}^{n}x_{i}^{2}}\right)-\exp \left(\frac{1}{n}\sum_{i=1}^{n}\cos(2\pi x_{i})\right)+20+e\) & \([-32,32]\) & 0 \\ F3 & \(f_{3}(x)=\frac{1}{4000}\sum\limits_{i=1}^{n}x_{i}^{2}-\prod\limits_{i=1}^{n}\cos (\frac{\lambda_{i}}{\sqrt{4}})+1\) & \([-600,600]\) & 0 \\  & \(f_{1},f_{2},\ldots,f_{10}=F_{3}\) & & \\ F4 & \([\sigma_{1},\sigma_{2},\ldots\sigma_{10}]=[1,1,\ldots,1]\) & \([-5.12,5.12]\) & 0 \\  & \([\lambda_{1},\lambda_{2},\ldots,\lambda_{10}]=[0.05,0.05,\ldots,0.05]\) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Test functions. DSALO has significantly improved the search accuracy, which indicates that DSALO has some certain advantages. Since the original ant-lion algorithm randomly selects the initial value when the algorithm starts to run, the dependence of the ant-lion algorithm on the initial value is low, while the DSALO algorithm inherits this characteristic of the original ant-lion algorithm. The DSALO algorithm not only improves the accuracy, but also improves the convergence speed and stability. To intuitively demonstrate this point, the convergence curves of the DSALO algorithm, original ALO algorithm, PSO algorithm, and OEALO algorithm with 5 times of running on test functions F1 to F4 are shown in Figures 3-6 respectively. The test functions are all set as 100 dimensions and the maximum iteration is set to 800.