## 2 Differential Step-Scaling Antilion Algorithm  
### Antlion Algorithm  
Ant-lion algorithm is a kind of meta-heuristic algorithm, which is inspired by the unique predation behavior of antlion, and it belongs to bionics algorithms. Antlions typically spend their larval years hunting and saving up energy, only reproducing as adults. As larvae, antlions use their jaws to dig a funnel-shaped sand pit and lie in wait for prey to appear. The edge of the pit is sharp enough to make it easy for the prey to fall into the pit. And when there's prey, the antlion will try to catch it. But the prey will do anything to get out of the pit, and that's where the antlion's unique hunting behavior comes in: it moves towards the prey and throws sand at the prey, stopping it from escaping and speeding it down to the bottom of the pit. When they get close enough to capture their prey, the antlion drags it underground to digest it, then returns to the surface, where it dumps the residue while repairing its sandpit and waiting for the next victim to arrive. The probability of an antlion catching prey is related to the size of the pit, and studies have shown that the size of the pit is positively correlated with the degree of hunger of the antlion [22]. The hungrier the antilon is, the larger the pit it digs, then the greater the chance of catching prey it has, which is what antlions have evolved to ensure the survival of the colony. The ant-lion optimization algorithm is inspired by the interaction between predator antlion and prey ant in the sand pit. In order to describe this interaction, we need to first model the ant's walking route and the antlion's predation and reconstruction behavior respectively. Since ants typically conduct a random walk when searching for food, they can be modeled using the following vector:  
\[X(t)=[0,cumsum(2r(t_{1})-1),cumsum(2r(t_{2})-1)\cdots cumsum(2r(t_{max})-1)] \tag{1}\]  
where, \(cumsum\) represents the sum of the past time of the random function \(r(t)\), and \(t_{max}\) represents the maximum number of iterations. \(r(t)\) is a random function:  
\[r(t)= \begin{cases}1&rand>0.5\\ 0&rand\leq 0.5\end{cases}  \tag{2}\]\(rand\) generates a random number evenly distributed between 0 and 1. Although the path of a single ant is random, reference [13] gives an example that this vector can ensure that the path of a colony of ants cover the entire search space. It can be seen that for an individual ant, its walking route has already been determined when the algorithm is initialized. In addition, we need to define storage matrixes to store information about ants and antlions at each iteration:  
\[M_{A}=\left[ \begin{array}{cccc}A_{1,1}&A_{1,2}&\cdots&A_{1,d}\\ A_{2,1}&\vdots&\cdots&\vdots\\ \vdots&\vdots&\ddots&\vdots\\ A_{n,1}&\cdots&\cdots&A_{n,d}\end{array} \right] \tag{3}\]  
\[M_{AL}=\left[ \begin{array}{cccc}AL_{1,1}&AL_{1,2}&\cdots&AL_{1,d}\\ AL_{2,1}&\vdots&\cdots&\vdots\\ \vdots&\vdots&\ddots&\vdots\\ AL_{n,1}&\cdots&\cdots&AL_{n,d}\end{array} \right] \tag{4}\]  
where \(A_{ij}\) represents the \(j\)-th dimension information of the \(i\)-th ant and \(AL_{i,j}\) the corresponding information of the antlion. Each row in the two matrices represents a solution to the problem to be optimized. Moreover, we can define the position information of each ant and antlion as:  
\[P_{A,j}(t)=[A_{i,1},A_{i,2},\ldots,A_{i,d}] \tag{5}\]  
\[P_{AL,i}(t)=[AL_{i,1},AL_{i,2},\ldots,AL_{i,d}] \tag{6}\]  
To evaluate the value of each solution, we need to calculate their fitness functions one by one and store the values of fitness functions in the following two matrices:  
\[M_{OA}=\left[ \begin{array}{c}f(A_{1,1},A_{1,2},\ldots,A_{1,d})\\ f(A_{2,1},A_{2,2},\ldots,A_{2,d})\\ \vdots\\ f(A_{n,1},A_{n,2},\ldots,A_{n,d})\end{array} \right] \tag{7}\]  
\[M_{OAL}=\left[ \begin{array}{c}f(AL_{1,1},AL_{1,2},\ldots,AL_{1,d})\\ f(AL_{2,1},AL_{2,2},\ldots,AL_{2,d})\\ \vdots\\ f(AL_{n,1},AL_{n,2},\ldots,AL_{n,d})\end{array}