# Pressure Vessel Design Problem Using Improved Gray Wolf Optimizer Based on Cauchy Distribution
## 2 Related Work
### Cauchy Distribution  
The Cauchy distribution, also known as the Cauchy--Lorentz distribution, is a continuous probability distribution [28]. It has some special properties, such as heavy tails. This makes Cauchy distribution better able to capture the occurrence of extreme values and is suitable for describing some abnormal situations or rare events [29]. The probability density function of the Cauchy distribution is shown in Equation (7):  
\[f(x,x_{0},\gamma)=\frac{1}{\pi}\cdot\left[\frac{\gamma}{\left(x-x_{0}\right)^{ 2}+\gamma^{2}}\right] \tag{7}\]  
where \(x_{0}\) is the location parameter, which can control the location of the distribution. \(\gamma\) is the scale parameter [30], which can control the shape of the distribution. In order to understand the Cauchy distribution more intuitively, the distribution curve of the Cauchy distribution with different parameters and the comparison between the Cauchy distribution and Gaussian distribution curves are drawn, as shown in Figure 3.  
It can be seen from Figure 3 that the Cauchy function curve is a symmetric bell-shaped curve, which slowly decreases from the top to the two ends. The peak distribution of the Cauchy distribution at the origin of the coordinates is shorter, and the rest of the peak distribution is longer. In comparison to the Gaussian distribution, the Cauchy distribution exhibits a broader spread and a more pronounced propensity for dispersion [31], endowing it with the ability to produce outlier values at considerable distances from the mean. [PERSON] [32] has observed that, while the Gray Wolf Optimizer (GWO) demonstrates exemplary efficacy in resolving optimization problems with global optima situated at the coordinate origin, its performance is marred by a marked search bias in scenarios where the optima deviate from this central point. Such an association demonstrably underscores the efficacy  
Figure 3: Cauchy distribution graph: (**a**) Graphs of different parameters; (**b**) Comparison of Cauchy distribution and Gaussian distribution among them. The red curve in Figure 3 is the standard Cauchy distribution.  
and logical foundation underlying the employment of the Cauchy distribution as a strategic enhancement of the GWO algorithm.  
Building on the foregoing analysis, the incorporation of the Cauchy distribution within the Gray Wolf Optimizer (GWO) algorithm is hypothesized to bolster its explorative prowess while simultaneously diminishing the tendency for premature convergence to local optima. Consequently, the heavy-tailed nature of the Cauchy distribution is of considerable significance in the sophisticated augmentation of the established GWO paradigm.  
The cumulative distribution function of the Cauchy distribution is shown in Equation (8):  
\[f(x,x_{0},\gamma)=\frac{1}{2}+\frac{1}{\pi}\cdot\tanh^{-1}\!\left(\frac{x-x_{0 }}{\gamma}\right) \tag{8}\]  
The formula [33] for generating random numbers using Cauchy distribution is shown in Equation (9):  
\[Cauchy(x_{0},\gamma)=x_{0}+\gamma\cdot\!\tan[\gamma\!\cdot\!(u-0.5)] \tag{9}\]  
where \(u\) is a random number in the range from 0 to 1.  
Since the slope of the tangent function tan increases at a very fast rate, Cauchy random numbers have a probability of some extreme values, and the randomness is greater. If the optimal solution of the problem occurs, in some extreme cases, the use of Cauchy random numbers is advantageous. When the algorithm falls into the local optimum, the mutation of random numbers has a greater probability of jumping out of the local optimal solution. Based on the above characteristics of Cauchy distribution and Cauchy random numbers, the basic GWO algorithm is improved.