# Pressure Vessel Design Problem Using Improved Gray Wolf Optimizer Based on Cauchy Distribution
## 3 The Proposed Algorithm
### Dynamic Nonlinear Inertia Weights Based on Cauchy Distribution  
[PERSON] and [PERSON] [37] proposed an inertial weight method in particle swarm optimization that made outstanding contributions to the balance between algorithm exploration ability and development ability. Since GWO can easily fall into local optima, the idea of inertia weight has been introduced into GWO to improve its performance [38; 39]. In order to improve the optimization ability of GWO, and inspired by the improved inertia weight of the PSO algorithm [40], this study designs a dynamic nonlinear inertia weight that combines the Cauchy random number and logarithmic function. In order to balance the exploration ability and development ability, previous research shows that the inertia weight should be decreasing [41] and that the early \(\omega\) value is large, which improves the global search  
Figure 4: Comparison of random initialization and Cauchy random initialization: (**a**) random initialization; (**b**) Cauchy random initialization. ability of the algorithm. The value of \(\omega\) decreases in the later period, which is aimed at the local search ability of the algorithm. However, when \(\omega\) is very small in the later period, there is a loss in population diversity and the algorithm is susceptible to easily falling into the local optimum. In order to solve this problem, this paper introduces the Cauchy random value and proposes a dynamic and more flexible weight that not only realizes the nonlinear reduction in inertia weight but also improves population diversity and prevents premature convergence. The inertia weight proposed is shown in Equation (11):  
\[ \begin{array}{rl}\omega(t)=&\omega_{min}+\left(\frac{\omega_{max}-\omega_{ min}}{2}\right)\frac{1}{\ln\left[e+(k\cdot\tau)^{2}\right]}\\ &+\left(\frac{\omega_{max}-\omega_{min}}{2}\right)\cdot Cauchy(x_{0},\gamma) \end{array}  \tag{11}\]  
where \(\omega_{max}\) and \(\omega_{min}\) represent the maximum and minimum inertia weights, respectively; \(\tau=t/T_{max}\), that is, the number of current iterations \(t\) divided by the maximum number of iterations. \(k\) is a constant used to regulate the trend of weight reduction. This formula combines the logarithmic function and Cauchy random number to optimize the algorithm, which not only realizes the overall trend of nonlinear decline but also increases the population diversity of the algorithm in the later stage because of the introduction of the Cauchy random number, which meets the necessary conditions for convergence and will not converge prematurely. The first half of the equation ensures that the inertia weight decreases nonlinearly through the logarithmic function \(1/\ln\left[e+(k\cdot\tau)^{2}\right]\), and the weight decreases gradually as \(\tau\) increases. Cauchy random numbers are introduced into the second half of the formula. Due to the strong disturbance of the Cauchy distribution, the diversity of the population in the later stage is improved, the local optimal solution is avoided, the accuracy of the algorithm is improved, and the convergence speed of the algorithm in the later stage is accelerated. The position update formula based on inertia weight is shown in Equation (12):  
\[ \begin{array}{l}\mathop{\stackrel{{\rightarrow}}{{X}}}(t+1)= \frac{\omega\cdot\left(\mathop{\stackrel{{\rightarrow}}{{X}}}_{1} +\mathop{\stackrel{{\rightarrow}}{{X}}}_{2}+\mathop{\stackrel{{ \rightarrow}}{{X}}}_{3}\right)}{3}\end{array}  \tag{12}\]  
Equation (12) is modified by the application of a non-linear inertia weight \(\omega\) governed by the Cauchy distribution, building upon the foundational position update Equation (6). This alteration transcends the realm of simplistic average calculations. During the position update procedure, the dynamic non-linear decay of the inertia weight enhances the equilibrium between exploration and exploitation, thereby expediting the algorithm's iteration speed.