The basic SBOA initializes the secretary bird population randomly in the search space, which, while straightforward, can lead to uneven distribution of secretary bird positions, thereby affecting population diversity. Insufficient population diversity limits the algorithm’s global search capability and increases the risk of falling into local optima. To address this issue, this paper proposes the use of Iterative chaotic mapping to improve the population initialization process.

Iterative chaotic mapping, known for its excellent ergodicity and non-repetitiveness, is widely used in optimization algorithms to enhance initial population diversity. Chaos theory suggests that even minor changes in initial conditions can lead to significantly different long-term behaviors, making chaotic mapping an ideal choice for optimizing population initialization [15] .

The mathematical expression of iterative chaotic mapping is key to implementing this improvement. Through a carefully designed iterative formula, it ensures that the initial population is uniformly distributed in the search space, ensuring that the algorithm covers a broad search area in the early stages, laying a solid foundation for subsequent local searches and fine-tunings. The mathematical expression for Iterative mapping is:

$x_{k+1}=\sin \left(\frac{b\pi }{xk}\right)$ (7)

where $x_k$ is the state after the n-th iteration, and $x_{k+1}$ is the chaotic mapping function. In this paper, Iterative mapping is selected with control parameters $b\in (0,1)$. In experiments, iterative chaotic mapping ensures the uniform distribution of the initial population, thereby enhancing the algorithm’s global search capability.

Traditional SBOA relies primarily on random walks and simple neighborhood search strategies during the exploration phase, which limits the algorithm’s search capability when dealing with complex problem spaces. To overcome this limitation, this paper introduces an adaptive weighting strategy to enhance the algorithm’s global search capability and population diversity. This strategy integrates dynamic search intensity and random control with time decay, enhancing the algorithm’s exploration capability and exploitation efficiency in complex search spaces.

The introduction of the adaptive weighting strategy significantly improves the Secretary Bird Optimization Algorithm. Firstly, through dynamic search intensity $I$, the algorithm can conduct more effective searches within the solution space, reducing the likelihood of getting trapped in local optima. Secondly, by introducing a density factor $\alpha$, the algorithm can smoothly transition from exploration to exploitation during the search process, enhancing the algorithm’s convergence speed and solution quality. Furthermore, to further improve algorithm efficiency, the adaptive weighting strategy introduces a flag variable $F$ that dynamically adjusts the search direction based on the current solution quality. This not only helps the algorithm escape from local optima but also maintains a certain level of diversity during the search process, preventing premature convergence. The improved prey capture and prey attack phases are as follows:

Prey Capture Phase

$x_{new}=x_{best}+F\times \beta \times I\times x_{best}+F\times r_3\times \alpha \times d_i\times \left|\cos \left(2\pi r_4\right)\times \left[1-\cos \left(2\pi r_5\right)\right]\right|$ (8)

where $x_{best}$ is the historical best position, and $x_{new}$ is the improved prey capture phase population iteration formula, $r_3$, $r_4$, $r_5$ with random numbers between 0 and 1.

$\begin{array}{l}{I}_i=r_2\frac{S}{4\pi d_i^2},r_2\in \left(0,1\right)\\ S={\left(x_i-x_{i+1}\right)}^2\\ d_i=x_{\text{best}}-x_i\end{array}$ (9)

where $I_i$ is the search intensity of the i-th individual, $S$ is the dynamic search intensity, and $d_i$ is a small constant, ensuring that the search step length is inversely proportional to the distance between the individual and the best solution, enabling more effective searches within the solution space.

$\begin{array}{l}\alpha =C\times \exp \left(\frac{-t}{t_{\max }}\right),\\ C=2\\ F=\{\begin{array}{cc}1& if{r}_6\le 0.5\\ -1& else\end{array}\\ r_6\in \left(0,1\right)\end{array}$ (10)

where $C$ are constants (default values 2), and t is the current iteration number, $t_{\max }$is the maximum number of iterations.

The flag variable $F$ is used to change the search direction to help the algorithm escape from local optima.

Prey Attack Phase

$x_{new}=x_{best}+F\times r_7\times \alpha \times d_i,r_7\in \left(0,1\right)$ (11)

where $x_{best}$ is the historical best position, and $x_{new}$ is the control parameter for the improved prey attack phase population iteration formula $r_7\in (0,1)$.

Through this adaptive weighting strategy, the algorithm can effectively narrow the search range while maintaining population diversity, accelerating convergence to the optimal solution. This strategy endows the algorithm with the ability to dynamically adjust search behavior, making it more efficient and robust when solving complex optimization problems.

In the later stages of the SBOA algorithm’s evolution, all secretary bird individuals tend to converge towards the best individual, leading to a decrease in population diversity and an increased risk of the algorithm falling into local optima. To reduce this probability, this paper proposes a Cauchy mutation operation on the position of the best secretary bird individual in the current iteration, randomly on m-dimensions. The Cauchy distribution, compared to the normal distribution, is more likely to produce random numbers far from the origin, thus better preventing the algorithm from falling into local optima [16] .

Therefore, the improved mutation strategy proposed in this paper is as follows:

$x_{new}=x_{best}+Cauchy\times ones\left(1,\dim \right)$ (12)

where $x_{new}$ represents the new position after mutation, $x_{best}$ is the position of the current best individual, and $Cauchy$ is the parameter controlling the intensity of mutation. By introducing the Cauchy mutation operator, the algorithm can enhance the probability of jumping out of local optima while maintaining exploration capabilities.

Step 1: Initialization

Initialize the parameters of the secretary bird population. Utilize the improved iterative chaotic mapping to generate the initial positions of the secretary bird population, ensuring a uniform distribution of the population within the search space.

Step 2: Fitness Calculation

Calculate the fitness value of each secretary bird individual, defined according to the specific optimization problem.

Step 3: Update Density Factor

Update the density factor according to Equation (10) to reflect changes in the current search environment.

Step 4: Update Search Intensity

Update the search intensity I according to Equation (9), adjusting the balance between exploration and exploitation.

Step 5: Population Update

Perform differential evolution strategy to update population positions. Further adjust population positions according to Equation (8). Conduct the prey attack phase update according to Equation (11).

Step 6: Position Correction

Correct the new positions to ensure individual positions are within the preset search space boundaries. Calculate the fitness values of individuals after position updates.

Step 7: Escape Strategy

Update the positions of secretary bird individuals according to the escape strategy in the SBOA to prevent premature convergence of the population.

Step 8: Optimal Solution Update

Update the current best fitness value according to Equation (12), ensuring tracking of the global optimal solution.

Step 9: Termination Condition Check

Check if the stopping criteria of the algorithm are met (such as reaching the maximum number of iterations or fitness threshold). If satisfied, output the best fitness value and the corresponding secretary bird position information; otherwise, return to Step 2 and continue iterating.

The ten benchmark test functions are presented in Table 1 . The benchmark test functions used were all 20-dimensional, with a population of 30 individuals in the algorithm, and the algorithm was independently iterated 500 times. The HS-SBOA, SBOA, GWO, HHO, PSO, and WOA algorithms were used to solve the test functions listed in Table 1 . Thirty independent experiments were conducted using MATLAB to compare the average values and standard deviations of the two algorithms, mitigating biases due to the randomness of the algorithms and ensuring their rationality. The experimental results are shown in Table 2 , and the convergence curves are depicted in Figures 1-10 .

The experimental results presented in Table 2 provide an in-depth understanding of the performance of the HS-SBOA algorithm. By comparing with four other algorithms, the HS-SBOA demonstrated significant advantages in optimization performance on the test functions, especially in terms of convergence accuracy, where the HS-SBOA algorithm achieved the top rank in these test functions. This not only proves its efficiency in solving global optimization problems but also reflects the effectiveness of the multi-scale dynamic population strategy and honey-badger-inspired mutation operations introduced in the algorithm design.

The data in Table 2 indicates that the improved algorithm has strong search capabilities and faster convergence speed, quickly converging to the target value for most functions.

In the data-driven modern world, constraint optimization for engineering problems has become a key task. Although the previous section has conducted optimization analysis on standard test functions, real-world engineering challenges often come with specific constraints. This section will demonstrate the application of HS-SBOA, SBOA, GWO, and BWO in practical engineering cases such as the Gear Train Design Problem (GTDP) [17] , Spring Compression Design Problem (TCSD) [18] , and Pressure Vessel Design Problem (PVD) [19] , aiming to evaluate and prove the practicality and effectiveness of HS-SBOA in dealing with actual engineering problems. Through this practice, the performance and potential of these algorithms in real-world problems will be deeply explored.

The gear train design problem is a common optimization problem in the field of mechanical engineering, with its core objective being to reduce the manufacturing and operational costs of gear transmission systems.

In this problem, designers need to determine four key variables: the number of teeth for the input gear, intermediate gear, driver gear, and output gear. These variables directly affect the gear ratio, efficiency, strength, durability, and overall size and weight of the gear train. The optimization process requires a comprehensive consideration of gear material costs, manufacturing process complexity, system reliability, and maintenance needs. By applying mathematical modeling, computer-aided design, finite element analysis, and optimization algorithms, designers can explore different gear configurations to achieve maximum cost-effectiveness and optimal performance. The specific description is given by the following equation:

$\begin{array}{l}\min f\left(x\right)={\left(\frac{1}{6.931}-\frac{x_2{x}_3}{x_1{x}_4}\right)}^2\\ s.t.12\le x_i\le 60,i=1,2,3,4\end{array}$ (13)

Thirty independent experiments were conducted to validate the effectiveness of the algorithms, with the results shown in Table 3 .

Through Table 3 , the optimal solutions and optimal values for HS-SBOA, SBOA, GWO, and BWO can be obtained, indicating that the proposed HS-SBOA is feasible in solving the GTDP.

The spring compression design problem plays a fundamental and critical role in mechanical design, involving the precise calculation and design of springs to accommodate different compression loads and working environments.

It requires engineers to find the optimal spring design within a limited parameter space to meet specific functional requirements and performance standards. The main objective of the spring compression problem is to minimize weight by selecting three variables: wire diameter, mean coil diameter, and the number of active coils. The mathematical expression is given by:

$\begin{array}{l}\min f\left(x\right)=\left(x_3+2\right)x_2{x}_1^2\\ \text{s}\text{.t}\text{.}{g}_1\left(x\right)=1-\frac{x_2^3{x}_3}{71785x_1^4}\le 0\\ g_2\left(x\right)=\frac{4x_2^2-x_1{x}_2}{12566\left(x_2{x}_1^3-x_1^4\right)}+\frac{1}{5180x_1^2}\le 0\\ g_3\left(x\right)=1-\frac{140.45x_1}{x_2^3{x}_3}\le 0\\ g_4\left(x\right)=\frac{x_1+x_2}{1.5}-1\le 0\\ 0.05\le x_1\le 2,0.25\le x_2\le 1.3,2\le x_3\le 15\end{array}$ (14)

Thirty independent experiments were conducted to validate the effectiveness of the algorithms, with the results shown in Table 4 .

In Table 4 , the optimal solutions and optimal values for HS-SBOA, SBOA, GWO, and BWO can be obtained, indicating that the proposed HS-SBOA is feasible in solving the TCSD problem.

The pressure vessel design problem is a classic structural optimization problem that requires minimizing the weight or cost of the vessel while satisfying a series of engineering constraints and safety standards.

Such problems often involve complex geometric shapes, material properties, and loading conditions, requiring a comprehensive consideration of strength, stability, and durability. In the design process, engineers must ensure that the vessel can withstand internal and external pressures while also considering manufacturing processes and economic viability. To achieve this goal, advanced numerical methods and optimization algorithms are often employed to find the optimal design solution, which not only improves the performance of the vessel but also reduces material usage, thereby reducing costs and increasing efficiency. In both academic research and industrial applications, the pressure vessel design problem is an extremely challenging field that continues to drive the development of material science, computational mechanics, and optimization technology. The specific description is given by the following equation:

$\begin{array}{l}\min f\left(x\right)=0.6224x_1{x}_3{x}_4+1.7781x_2{x}_3^2+3.1661x_1^2{x}_4+19.84x_1^2{x}_3\\ s.t.g_1\left(x\right)=-x_1+0.0193x_3\le 0\\ g_2\left(x\right)=-x_3+0.00954x_3\le 0\\ g_3\left(x\right)=-\pi x_3^2{x}_4-\frac{4}{3}\pi x_3^3+1296000\le 0\\ g_4\left(x\right)=x_4-240\le 0\\ 0\le x_1,x_2\le 99,10\le x_3,x_4\le 200\end{array}$ (15)

Thirty independent experiments were conducted to validate the effectiveness of the algorithms, with the results shown in Table 5 .

Through Table 5 , the optimal solutions and optimal values for HS-SBOA, SBOA, BWO, and GWO can be obtained, indicating that the proposed HS-SBOA is feasible for solving the PVD problem.