# Pressure Vessel Design Problem Using Improved Gray Wolf Optimizer Based on Cauchy Distribution
## 4 Simulation Experiment and Results Analysis
### Validity Test of CGWO Inertia Weights: Comparison with Other Inertia Weight Strategies
For the vast majority of functions, the mean value and standard deviation of the CGWO results iarealso the best. Specifically, for \(f_{2}\) and \(f_{3}\), only CGWO converges to the theoretical optimal value, while other algorithms cannot achieve the theoretical optimal value. For \(f_{6}\), \(f_{7}\), \(f_{12}\), and \(f_{13}\), CGWO outperforms the other algorithms by up to seven orders of magnitude. Through the experimental results of the Cauchy weight strategy, under the unimodal benchmark function, multi-modal benchmark function, and fixed-dimension, multi-modal benchmark function, it can be concluded that the proposed Cauchy nonlinear weight strategy can improve the global search ability of the algorithm, enhance the local search performance of the algorithm, and reduce the possibility of the algorithm falling into the local optimal solution. This fully demonstrates the advantage of the inertia weight strategy of the proposed CGWO in terms of solution performance. In order to compare the optimization effects of different inertia weight strategies more intuitively on the algorithm, the convergence curves of the four inertia weight strategies on all test functions are shown in Figure 9.  
Figure 9: _Cont._  
It can be seen from Figure 9 that the performance of CGWO under most test functions has obvious advantages over other algorithms, especially functions \(f_{1}\) to \(f_{4}\), \(f_{9}\), \(f_{11}\), \(f_{14}\), and \(f_{15}\). For these functions, the convergence speed and solution accuracy of the CGWO algorithm are much higher than those of other algorithms. To be specific, for \(f_{6}\), \(f_{12}\), and \(f_{13}\), it can be clearly seen that, when the other three algorithms fall into the local optimum, CGWO breaks out of the local optimum in the iteration process, thus making the solution accuracy much higher than that of the other three algorithms. This fully reflects the advantages brought by introducing the long tail characteristic of the Cauchy distribution into the inertia weight, which improves the population diversity of the algorithm in the later stage. It also greatly improves the ability of the algorithm to jump out of the local optimum. For \(f_{7}\), \(f_{10}\), \(f_{14}\), and \(f_{15}\), although the four algorithms all tend to fall into the local optimum, the optimization ability of CGWO is still the best. For the multi-mode benchmark function \(f_{8}\) and the fixed dimension multi-mode benchmark functions \(f_{21}\) to \(f_{23}\), CGWO also has excellent performance, and the convergence speed is much faster than other algorithms, allowing for high solution accuracy and the ability to jump out of the local optimum. Figure 9: Comparison of different weight strategies. To sum up, CGWO outperforms the other three algorithms in terms of accuracy, stability, and convergence, and CGWO has a strong ability to jump out of the local optimum, which shows that, for GWO, the Cauchy nonlinear dynamic inertia weight strategy has better performance than the other three inertia weights, reflecting the effectiveness and improvement of the proposed weight strategy.