# A Multi-Strategy Improved Honey Badger Algorithm for Engineering Design Problems
## 2 Algorithm Design
### Proposed Algorithm
#### 2.2.2 Water Wave Dynamic Density Factor  
In HBA, the value of the density factor \(\partial\) controls the balance of the algorithm during the exploration and development phases. The primitive decreasing mechanism often leads the algorithm to fall into a local optimum, making it difficult to effectively cover  
Figure 1: Population initialization. (**a**) Random distribution map, (**b**) Halton distribution map, (**c**) Random initialized particle swarm distribution histogram, (**d**) Halton initialized particle swarm distribution histogram.  
and explore the entire search space, thus limiting the accuracy of the algorithm in the development stage, especially when dealing with complex and high-dimensional problems. To solve this problem, researchers have proposed a variety of improved convergence factors: reference [32] proposes a convergence factor based on sinusoidal functions, which can help the algorithm to search for the optimal value in the global range, as shown in Equation (10). Reference [33] presents the power-of-five nonlinear convergence factor, which can enhance the search ability of the algorithm, as shown in Equation (11). Reference [34] proposes a convergence factor that utilizes the uncertainty property in water wave dynamics, which improves the algorithm's adaptability and solving ability for complex functions, as shown in Equation (12). Figure 2 illustrates the above four convergence factors, where (a) is the original convergence factor shown in Equation (3), (b) is the sinusoidal convergence factor shown in Equation (10), (c) is the power-of-five convergence factor demonstrated in Equation (11), and (d) is the water wave dynamic density factor shown in Equation (12).  
\[\partial=1+\sin(\pi/2+\pi\times t/t_{\rm max}), \tag{10}\]  
\[\partial=2-2\times(t/t_{\rm max})^{5}, \tag{11}\]  
\[\partial=2\times rand\times S\times\exp(-t/t_{\rm max})^{3}, \tag{12}\]  
where \(t_{\rm max}\) is the maximum number of iterations, \(t\) is the current number of iterations, \(S\) is a random integer, and \(S\in[0,1]\).  
Figure 2: Four different convergence factors. (**a**) The original convergence factor shown in Equation (3), (**b**) the sinusoidal convergence factor shown in Equation (10), (**c**) the power-of-five convergence factor demonstrated in Equation (11), and (**d**) the water wave dynamic density factor shown in Equation (12).  
HBA often lacks accuracy when dealing with complex multimodal functions. To solve this problem, the dynamic density factor of water waves is introduced. The uncertain fluctuation characteristics in the water wave dynamics can dynamically adjust the density factor, realizing the adaptive switching of the algorithm between the exploration and development phases. This self-adjustment mechanism helps to improve the overall search efficiency of the algorithm and avoid falling into local optimization. The multi-wave interference characteristic in water wave dynamics is very similar to the characteristics of complex multi-peak optimization problems. Using this characteristic to design the density factor helps the algorithm to better cope with the complex multi-peak problem and reduce the risk of falling into the local optimum. Compared with the traditional methods, such as sinusoidal function and power-of-five convergence factor, the water wave dynamics density factor can better capture the uncertainty characteristics in the complex environment, realize the adaptive equilibrium of exploration and development, enhance the ability to cope with the complex multi-peak problems, and improve the adaptability of the algorithm in the complex environment.