## 2 Differential Step-Scaling Antilion Algorithm
### Differential Step-Scaling Ant-Lion Algorithm
The difference is that a genetic algorithm uses the fitness value of individual population to control the parent population for hybridization, and then carries out a mutation operation to obtain the probability value of offspring being selected. In a differential evolution algorithm, each individual in the population is regarded as a vector. Through vector calculation, the parent vector is calculated by difference to generate a mutation vector, and then the mutation vector is hybridized with the parent vector to generate a new vector, which is regarded as the child and selected with the parent directly, as shown in Figure 2. Differential evolution algorithms have strong robustness, fast convergence speed, and the most important thing is that they are easy to implement. The calculation steps of a differential evolution algorithm mainly include three stages: mutation, crossover, and selection, and \"DE/x/y/z\" is usually used to distinguish and represent different evolution methods and operators. In \"DE/x/y/z\", x specifies how to choose a basis vector; y specifies the number of difference vectors in evolution; z is a way of crossing operations. In addition to selecting strategies for specific problems, the performance of a differential evolution algorithm is also related to three key parameters: population size NP, scaling factor F, and crossover probability CR. A reasonable evolutionary strategy and a set of appropriate key parameters can greatly improve the convergence speed and accuracy of the algorithm. In this paper, we choose the \"DE/rand/1/binDE\" mode, initialize all the individuals of the population into \(n\)-dimensional vectors, and then randomly generate individual positions in the search area. In the mutation stage, two vectors are randomly selected for difference operation to obtain the mutation vector, and then the mutation vector is scaled and added with a third parent vector to obtain a new child vector. The binomial crossover operator was used to ensure that at least one dimension of the final test individual comes from the mutant individual, so as to avoid being the same as the current individual and ensure the diversity of the population. A random number conforming to uniform 0-1 distribution is generated and compared value with the crossover probability CR. If the value of the random number is less than CR, the value of the mutant individual at this dimension is given to the test individual, or the value of the test individual at this dimension will come from the current individual. In the selection stage, the fitness values of the new individuals and the original individuals after mutation and crossover are compared, and the excellent ones are retained while the bad ones are discarded to ensure that the individuals in the population are currently optimal before entering the next iteration. Figure 2: Basic idea of differential evolution algorithm. Specifically in this paper, the combination points of the differential evolution algorithm and the ant-lion algorithm are as follows:  
In each iteration, after obtaining all antlions at the current iteration, the antlions were variated and differentiated. Mutation: select several pairs of antlions randomly, regard them as the parent vector, calculate the difference between them and scale them according to the scaling factor, then add the scaled difference vector and the vector of the third antlion individual to obtain a new test body, the formula is defined as follows:  
\[P^{\prime}(t)=P^{d}_{AL,r1}(t)+F\cdot(P^{d}_{AL,r2}(t)-P^{d}_{AL,r3}(t)) \tag{17}\]  
where \(F\) is the scaling factor, \(r_{1},r_{2},r_{3}\) is the three different random numbers in the interval \([1,\text{N}]\), \(P^{\prime}(t)\) is the test object obtained by mutation operation in the \(t\)-th iteration, and \(P^{d}_{AL,r1}(t)\), \(P^{d}_{AL,r2}(t)\), \(P^{d}_{AL,r3}(t)\) is \(r_{1}\)-th, \(r_{2}\)-th, \(r_{3}\)-th antlion vector of this iteration.