# Research on Hard Rock Pillar Stability Prediction Based on SABO-LSSVM Model
## 2 Materials and Methods
### Introduction of the Subtraction Average Optimization Algorithm  
The subtraction average-based optimizer (SABO) is an intelligent optimization algorithm based on mathematical behavior, proposed by [PERSON] and [PERSON] (2023) in 2023. It updates the positions of individuals in the search space using the subtraction average of individuals, exhibiting strong optimization capabilities and fast convergence speed. Currently, there are a limited number of cases in which this algorithm has been effectively developed and applied. [PERSON] et al. (2024) addressed the issues of insufficient harmonic data support and inadequate harmonic monitoring capabilities in power grid harmonic pollution. They proposed an improved SABO-BP algorithm for harmonic prediction, which achieved good results and had engineering significance.  
The SABO algorithm refers to the solution space in different optimization problems as the search space, which is a subset of the dimensional space with dimensions consistent with the number of features in the given training set. The individuals in this algorithm are called search agents, contain information about the decision variables, and are mathematically modeled using vectors. Their collection forms the overall algorithm. Similar to other optimization algorithms, the SABO algorithm requires a random initialization of the population in the optimization space:  
\[x_{i,j}=lb_{j}+r\big{(}ub_{j}-lb_{j}\big{)} \tag{11}\]  
where \(x_{\mathrm{ij}}\) represents an individual in the population, \(lb_{\mathrm{j}}\) represents the lower bound of the optimization interval, \(ub_{\mathrm{j}}\) represents the upper bound of the optimization interval, and \(\mathrm{r}\in[0,1]\).  
[PERSON] and [PERSON] incorporated mathematical concepts, such as the average value, differences in search agent positions, and differences in the signs of the objective function values, when designing the SABO algorithm. In traditional approaches, search agents are updated by calculating the arithmetic mean positions of all the individuals.  
This unique \"\(-v\)\" operation is applied throughout the process of updating the search agents in the algorithm. In the search space, the displacement of each search agent, \(X_{i}\), is calculated using the arithmetic mean of \"\(v\)-subtraction\" from search agent \(X_{j}\). The formula is as follows:  
\[X_{i}^{new}=X_{i}+\overset{\rightarrow}{r}_{i}\frac{1}{N}\sum_{j=1}^{N}(X_{i} -vX_{j}),i=1,2,\ldots,N \tag{13}\]  
\[X_{i}=\left\{\begin{array}{l}X_{i}^{new},F_{i}^{new}<F_{i}\\ X_{i},\ \ else\end{array}\right. \tag{14}\]  
where \(N\) represents the population size, \(r\in[0,1]\), and \(X_{new}\) represents the updated position of the search agent. The update condition is shown in Equation (14), where the new position is only replaced with the update value if it is superior (i.e., has a lower fitness value). Otherwise, the position remains unchanged.  
It is worth mentioning that the SABO optimization algorithm requires fewer control parameters than other algorithms. It only requires setting the population size \(N\) and maximum number of iterations, making it simple and efficient. In performance tests on unimodal, high-dimensional multimodal, fixed-dimension multimodal types, and the CEC 2017 test suite, the SABO optimization algorithm showed good solving effectiveness for most benchmark functions.