A Nonlinear Convex Decreasing Weights Golden Eagle Optimizer Technique Based on a Global Optimization Strategy

: A novel approach called the nonlinear convex decreasing weights golden eagle optimization technique based on a global optimization strategy is proposed to overcome the limitations of the original golden eagle algorithm, which include slow convergence and low search accuracy. To enhance the diversity of the golden eagle, the algorithm is initialized with the Arnold chaotic map. Furthermore, nonlinear convex weight reduction is incorporated into the position update formula of the golden eagle, improving the algorithm’s ability to perform both local and global searches. Additionally, a final global optimization strategy is introduced, allowing the golden eagle to position itself in the best possible location. The effectiveness of the enhanced algorithm is evaluated through simulations using 12 benchmark test functions, demonstrating improved optimization performance. The algorithm is also tested using the CEC2021 test set to assess its performance against other algorithms. Several statistical tests are conducted to compare the efficacy of each method, with the enhanced algorithm consistently outperforming the others. To further validate the algorithm, it is applied to the cognitive radio spectrum allocation problem after discretization, and the results are compared to those obtained using traditional methods. The results indicate the successful operation of the updated algorithm. The effectiveness of the algorithm is further evaluated through five engineering design tasks, which provide additional evidence of its efficacy.


Introduction
The earliest approaches to optimization problems were mathematical or numerical [1], where the goal was to arrive at a zero-derivative point to arrive at the final answer. The search space expands exponentially as the number of dimensions rises, making it nearly impossible to use these techniques to solve nonlinear and non-convex problems with many variables and constraints. Additionally, numerical approaches may enter local optimality when the derivative is also zero. Such numerical approaches cannot guarantee the discovery of a globally optimal solution, since real-world problems usually exhibit stochastic behavior and have unexplored search areas [2].
To overcome the limitations of numerical approaches, more sophisticated meta-heuristic algorithms are routinely utilized to address difficult optimization issues. These techniques have the benefits of being easily applied to continuous and discrete functions, not requiring any extra complicated mathematical operations such as derivatives, and having a low incidence of falling into local optimality points. Single-solution-based methods and population-based methods are the two categories of meta-heuristic techniques. A solution (typically random) is generated and iteratively improved until the stopping criterion is reached in single-solution-based methods. Based on the interaction of information between the solutions, population-based methods create a random set of solutions in a cognitive radio spectrum. The enhanced whale method outperforms competing algorithms thanks to a nonlinear convergence factor and a hybrid reverse learning strategy. Similar to this, the same authors enhanced the Grey Wolf Algorithm (IBGWO) [20] for usage in this field and employed three strategies: the nonlinear convergence factor, the Cauchy perturbation approach, and adaptive weight, achieving improved outcomes. Yin Dexin et al. [21] adopted a novel strategy in which the industrial internet of things was recognized as an environment, and applied the improved sparrow search algorithm (IBSSA) to the problem model. Similar to this, Wang Yi et al. [22] applied the enhanced mayfly algorithm (GSWBMA) to the cognitive heterogeneous cellular network spectrum allocation problem, offering helpful guidance for future researchers. Numerous researchers have also enhanced numerous swarm intelligence algorithms in the context of traditional engineering applications [23][24][25].
A novel algorithm called the golden eagle optimizer (GEO) [26] was put forward in 2021, which was motivated by two hunting behaviors of golden eagles. The cruising and attacking phases are changed in order to catch prey more quickly. The shift from the golden eagle's cruise behavior to attack behavior has a direct impact on the end result. Cruise behavior mostly reflects the algorithm's exploration function, while attack behavior primarily reflects the algorithm's exploitation function. Because the original algorithm cannot balance the exploration and exploitation phases, it randomly switches from cruise behavior to attack behavior, which causes GEO to have a slow convergence speed, poor search accuracy, and weak robustness. There are some strategies to improve this. Few people, however, have previously used an improved approach to simultaneously take into account the cognitive radio allocation and traditional engineering design constraints, as described in this paper. Based on the advantages of the golden eagle approach, which include fewer optimization parameters, robustness to local optima, and superior handling of high-dimensional problems, this algorithm model is suggested. This paper suggests a nonlinear convex decreasing weights golden eagle optimization technique based on a global optimization strategy to handle the problem. Arnold chaotic map initialization, nonlinear convex weight reduction, and a global optimization strategy make up the three components of the improved algorithm. An Arnold chaotic map with decent ergodicity is used to initialize the population of golden eagles in chaos maps. On the one hand, after examining several inertia weight types, nonlinear convex decreasing weights are utilized to help the golden eagle achieve improved convergence during the optimization process. On the other hand, in order to better achieve optimization, the golden eagle location update formula now includes the global optimization method, which connects each golden eagle to the generation's most optimized individual. Under 12 benchmark functions and CEC2021, IGEO is contrasted with other sophisticated optimization algorithms to demonstrate its better performance. The results of the numerical experiment demonstrate that IGEO has better optimization performance than other techniques. Using a single-strategy method, we assess the numerical optimization abilities of the original GEO algorithm, IGEO, and GEO for 12 benchmark functions. The outcomes of the numerical experiment show how well these two methods function as compared to the original GEO. Eventually, IGEO is transformed into a binary algorithm and used for cognitive radio spectrum allocation. Compared with other improved algorithms, IGEO achieves higher spectral efficiency. Five classic engineering cases were also used to further test the performance of the algorithm.
The main contributions of this paper are as follows: (i) The Arnold chaotic map strategy is employed to help the golden eagle population obtain a better initial position. (ii) The nonlinear convex reduction strategy is used to coordinate the cruising and attacking behavior of the golden eagle, thereby improving its local and global search capabilities.
(iii) The global optimization strategy is utilized to assist the golden eagle population in complex environmental optimization, enhancing the possibility of escaping from local optima. (iv) Twelve benchmark test functions and the CEC2021 test set are employed to assess the capabilities of the improved algorithm, including comparisons with eight other algorithms. (v) Various statistical tests, including the Wilcoxon rank-sum test and Friedman ranking, as well as Holm's subsequent verification, are conducted to determine the overall ranking of the comparison algorithms and the improved algorithm on the twelve benchmark test functions and CEC2021, thereby validating the final performance of the algorithm. (vi) The individual effects of the nonlinear convex reduction strategy and the last global optimization strategy are tested separately. (vii) Algorithm complexity analysis is performed. (viii) The improved algorithm is applied to the classical problem of cognitive radio spectrum allocation and the results compared with those of commonly used algorithms. (ix) Five common engineering applications are utilized to test the search capabilities of the algorithm.
The following is a breakdown of this article. The second section goes over the fundamental ideas and the current state of GEO research. The third section offers a thorough explanation of the improved technique proposed in this article, and an analysis of its complexity. The fourth section, which mostly focuses on certain application components, contains simulation experiments and engineering best practices. In simulation experiments, there are compared experiments with a single improvement approach as well as comparative experiments between improved algorithms and alternative algorithms. The CEC2021 test set and 12 benchmark test functions served as the basis for these tests, and the experimental findings underwent statistical analysis. In engineering experiments, the enhanced algorithm is used to solve the well-known cognitive radio spectrum allocation problem, and the results compared with those of many other algorithms. Five classic engineering application cases are also used to verify the feasibility of the algorithm in this paper. Finally, the conclusions of this article are presented.

Prey Selection
The exploration and exploitation phases make up the two sections of the GEO method. During the exploration phase, the golden eagle may be seen to fly around its target, before attacking it during the exploitation phase. As a result, each golden eagle must select a victim before cruising in and attacking it. Currently, the best solution is that which locates the prey. Every golden eagle is supposed to remember the best answer thus far. Each golden eagle hunts and flies in search of better prey. The golden eagle's attack and cruise vectors are depicted in a two-dimensional trajectory diagram in Figure 1. Golden Eagle's attack direction is shown by the left arrow, while its cruise direction is indicated by the downward arrow.

Attack and Cruise
The procedure of catching the prey after the golden eagle gets close to it constitutes its attack behavior. A vector with a direction that goes to the prey and that extends from the golden eagle's current location to the location of the prey in its memory can be used to replicate the attack behavior. Equation (1) can be used to describe how the golden eagle attacks.
where i A is the attack vector of the th i golden eagle, * f X is the best position found by golden eagle f so far, and i X is the current position vector of golden eagle i . The cruise behavior of the golden eagle is a process that starts with the eagle's current location, turns around the prey, and continues to move toward the prey. In this procedure, the attack vector, the hyperplane, and the cruise vector are all connected. The cruise vector is in the tangent hyperplane of the golden eagle cruise trajectory, perpendicular to the assault vector. To compute the cruise vector, the hyperplane formula equation must first be obtained. A crucial reference plane in cruise behavior, the hyperplane is a linear subset with n − 1 dimensions in the n−dimensional linear space. An arbitrary point on the hyperplane and its normal vector can be used to identify it. Equation (2) displays the scalar form of the hyperplane equation in n−dimensional space.
where H is the hyperplane's normal vector, X is a variable vector, and d is a constant. Therefore, for the arbitrary point P of the hyperplane, Equation (3) can be deduced.
If the attack vector A is regarded as a normal vector and the location of the eagle X is regarded as an arbitrary point, Equation (4) can clearly be obtained.
The starting point of the cruise vector C is the position of the golden eagle. The cruise vector can be chosen freely in n − 1 dimensions, but the hyperplane equation specifies the last dimension, as illustrated in Equation (4). There must be a free variable and a fixed variable, which are easy to determine using the following methods: randomly select a variable from among the variables as a fixed variable, denoted as k ; and assign the Attack Prey Golden Eagle Cruise random value to all variables except the th k variable. Therefore, the cruise vector of golden eagle i in an n−dimensional space can be expressed using Equations (5) and (6).
where j a is the th j element of attack vector A . The relationship between the golden eagle's cruise behavior and attack behavior and the hyperplane is shown in Figure 2. A represents the Golden Eagle's attack vector, while C represents a potential cruise vector for the Golden Eagle on this hyperplane.

Move to New Location
The golden eagle i cruises around the prey to obtain the cruise vector i C , and when it reaches an appropriate position, it pounces on the prey to obtain the attack vector i A . After the golden eagle has performed its cruise and attack, the displacement of golden eagle i can be expressed as shown in Equation (7), while the position of golden eagle i at the 1th t + generation can be expressed as shown in Equation (8).

Related Works on GEO
GEO is widely utilized by researchers, as it is an outstanding meta-heuristic method. Aijaz et al. [27] presented a two-stage photovoltaic residential system with electric vehicle charging capability, where the performance of the system was improved by an optimized proportional and integral gain selection bidirectional DC/DC converter (BDC) proportional-integral controller via the GEO algorithm. The golden eagle algorithm also outperformed the particle swarm and genetic algorithms in terms of performance. Magesh et al. [28] presented improved grid-connected wind turbine performance using PI control strategies based on GEO to improve the dynamic and transient stability of grid-connected PMSG-VSWT. In addition, GEO has also been applied to fuzzy control. Kumar et al. [29] set out to improve the performance of a nonlinear power system using a hierarchical golden eagle architecture with a self-evolving intelligent fuzzy controller (HGE-SIFC). GEO seems to be popular in wind and power systems. Sun et al. [30] accurately predicted wind power generation by means of the GEO algorithm, which they first improved, before proposing an extreme learning machine model that combined the improved GEO with an extreme learning machine for the prediction of wind power from numerical weather forecast data. In solar photovoltaic power systems, the GEO algorithm has another predictive feature. Boriratrit et al. [31] addressed the instability of machine learning and combined the GEO with a machine learning model, proposing a new machine learning model that yielded a smaller minimum root mean square error than the comparison model. Huge electricity demands have proved to be a tough challenge for power companies and system operators due to the increasing number of consumers in the electricity system and the unpredictability of electricity loads. Therefore, Mallappa et al. [32] proposed an effective energy management system (EMS) named golden eagle optimization with incremental electricity conductivity (GEO-INC) to meet demands with respect to load. Zhang et al. [33] proposed an improved GEO with improved strategies including an individual sample learning strategy, a decentralized foraging strategy and a random perturbation strategy, and these improved strategies were applied to a hybrid energy storage system with a complementary wind and solar power storage system for energy optimization. GEO has also been used in forecasting using learning machines; for example, a meta-learning extreme learning machine (MGEL-ELM) based on GEO and logistic mapping, and a samedate time-interval-averaging interpolation algorithm (SAME) [34] have been proposed in the literature to improve the forecasting performance of incomplete solar irradiance time series datasets. Profit Load Distribution (PLD) is a typical multi-constrained nonlinear optimization problem, and is an important part of energy saving and consumption reduction in power systems. Group intelligent optimization algorithms are an effective method for solving nonlinear optimization problems such as PLD. For the actual operational constraints of the power system in the PLD model, a novel GEO-based solution was proposed in [35]. A series of segmented quadratic polynomial sums were modelled for the fitness function as the cost function used to calculate the optimization by the first GEO. The results showed that the GEO was able to effectively solve the power system PLD problem. In addition to the prediction and power system aspects, GEO has also been applied in the field of image research. Al-Gburi et al. [36] applied GEO to the disc segmentation of human retinal images using pre-processing based on the golden eagle algorithm-guided geometric active Shannon contours and post-processing based on regular interval cross-sectional segmentation. Dwivedi et al. [37] proposed a novel medical image processing technique for analyzing different peripheral blood cells such as monocytes, lymphocytes, neutrophils, eosinophils, basophils and macrophages, using fuzzy c-mean clustering (MLWIFCM) based on GEO to perform cell nucleus segmentation. Justus et al. [38] proposed a hybrid multilayer perceptron (MLP)-convolutional neural network (CNN) (MLP-CNN) technique to provide services to SUs even under active TCS constraints in order to address the spectrum scarcity problem in cognitive radio. Eluri et al. [39] improved the GEO algorithm by using the transfer function to transform GEO into discrete space, and used time-varying flight duration to balance the cruise and attack parts of GEO. Moreover, their improved GEO was applied to feature selection, achieving good results. GEO has been used for multi-objective optimization in the context of the problem of reducing robot power consumption, obtaining a better Pareto frontier solution [40]. Xiang et al. [41] proposed PELGEO with personal example learning in combination with the grey wolf algorithm. Pan et al. [42] proposed GEO_DLS with personal example learning to enhance the search ability and mirror reflection learning to improve the optimization accuracy. Both PELGEO and GEO_DLS were applied for the 3D path planning of a UAV during power inspection. Ilango R et al. [43] proposed S2NA-GEO combined with a neural network learning algorithm. Later, a model for the uncertainty associated with renewable energy based on GEO was developed to relate the negative effects of variations in RES output for electric vehicles and intelligent charging [44].

Cognitive Radio Model
The demand for spectrum resources has significantly increased as a result of the quick expansion of wireless communication services brought on by the introduction of 5G communications. However, low spectrum utilization and the wastage of important resources have been caused by the fixed spectrum allocation mechanism and the limited supply of spectrum resources. In order to address these issues, Ghasemi et al. proposed Cognitive Radio (CR) [45] as a means to improve spectrum utilization.
Cognitive radio allows wireless communication systems to be aware of their surroundings, to secondarily utilize the unused spectrum holes of authorized users, to adapt to environmental changes and to automatically adjust system parameters, and to utilize the spectrum in a more flexible and efficient way. Existing spectrum allocation models include graph-theoretic coloring models [46,47], bargaining mechanism models [48], game theory models [49], etc.
Models for spectrum distribution can optimize system advantages, but it is challenging to precisely manage user fairness; therefore, they cannot guarantee the absence of flaws like those related to user fairness and unethical user competition. Scholars have used different swarm intelligence algorithms to optimize spectrum allocation, such as genetic algorithms (GA) [50], particle swarm algorithms (PSO) [4], butterfly optimization algorithms (BOA) [13], cat swarm algorithms (CSO) [51] and other intelligent optimization algorithms, for spectrum allocation problems in cognitive radio. Most of the above algorithms suffer from premature convergence, stagnation and iterative solution drawbacks when solving the problem. To overcome these problems and maintain population diversity, an improved golden eagle algorithm based on a graph-theoretic coloring model is applied to the optimization of the spectrum allocation model in this paper.
A conversion function is used to convert the search space from a continuous search space to a discrete search space. A suitable conversion function is required to increase the performance of the binary algorithm. Firstly, the conversion function is algorithm independent and does not affect the search capability of the algorithm; secondly, the computational complexity of the algorithm does not change. For continuous-to-discrete conversions, this work employs the Sigmoid function, which can be written as follows: generation. The function of spectrum allocation is to build a reasonable allocation of available spectrum (channels) for cognitive users, which can satisfy the resource demand of cognitive users while effectively avoiding interference with primary users and maximizing the use of resources. Assuming that there are N secondary users (SUs) and M available channels in a certain region, the relevant definition of the graph-theoretic coloring model is as follows:

Population Initialization Using the Arnold Chaotic Map
An individual golden eagle's position is generated at random in the golden eagle optimization method. The existing population of golden eagles is relatively random and weak in terms of diversity, making it difficult to research and develop algorithms for algorithm optimization. In this paper, the initial location of the golden eagle is generated using an Arnold chaotic map to address the aforementioned issues. A Russian mathematician named Vladimir Igorevich Arnold suggested the Arnold chaotic map. This chaotic mapping technique can be used for repetitive folding and stretching transformation in a constrained space. The precise procedure is to first map the variables to the chaotic variable space using the cat mapping relationship, and then to use linear transformation to map the resulting chaotic variables to the space that needs to be optimized. Equation (11) describes the golden eagle's starting position in the search space.
where max X and min X are the upper and lower bounds for variables, 0 X is the initial position of the golden eagle and Arnold is an Arnold chaotic map in the range of 0 to 1. The Arnold chaotic map has a simple structure and great ergodic uniformity. Its expression is shown in Equation (12).

Nonlinear Convex Decreasing Weight
The nonlinear convex decreasing weight represents the degree of inheritance of the current location. The algorithm has strong global optimization capability (i.e., better exploration) when this value is large, because the population individuals inherit a large portion of the particle positions from the previous generation, and the algorithm has strong local optimization capability when this value is small, because the population individuals inherit little (i.e., better exploitation). To balance exploration and exploitation, a nonlinear convex decreasing weight is added in GEO. The greater inertia weight enables the golden eagle to cruise and explore more effectively in earlier phases of GEO. Golden eagles can attack prey more effectively because of their reduced inertial weight in later stages of GEO. Equation (8) shows that the location and moving step of the previous generation determine the golden eagle's updated position. In order for the previous person to affect the current individual, a nonlinear convex decreasing weight is added. Introducing the nonlinear convex declining weight is Equation (13).  (14) represents the position update formula of the algorithm after introducing nonlinear convex decreasing weight.
The value of m influences the weight's decreasing mode and further impacts the algorithm's performance during optimization, most notably its rate of convergence. Figure  3 depicts the nonlinear convex decreasing weight when m rises from 1 to 7. The following tests are conducted to investigate the value of m that is most advantageous to the algorithm. Six benchmark functions are chosen as experimental objects, 30 unique experiments are carried out, and seven different m values explain seven algorithms. The evaluation was conducted after generating the average convergence curve for these six benchmark test functions. The precise method involves finding the smallest number of iterations for each of the seven methods under the conditions of best convergence accuracy. The algorithm with the fewest iterations receives seven points, and so on, until the algorithm with the most iterations, which receives one point. The results of the seven algorithms for the six benchmark test functions are displayed in Table 1. Figure 3 and Table 1 show that the lowest score for m = 1, which reflects the linearly decreasing inertia weight, shows the algorithm's slowest convergence pace. In this paper, the value of m is chosen to be 4, because it has the highest score. It also exhibits the fastest convergence speed.   m  1  2  3  4  5  6  7  F1  2  4  7  6  5  3  1  F2  2  3  4  7  6  5  1  F3  2  4  3  5  6  7  1  F4  2  6  7  5  4  3  1  F6  2  4  5  7  6  3  1  F7  2  3  5  7  6  4  1  Total  12  24  31  37  33  25  26 3.

Global Optimization Strategy
Without significant social contact, the golden eagle's position update formula in the golden eagle optimization method simply refers to the historical memory of the golden eagle's position and move step from the previous iteration. Such a blind random search in space not only fails to speed up the population's search, but also frequently hinders its ability to swiftly identify the best option. The golden eagle position update formula is enhanced by the introduction of a global optimization strategy. For each iteration, the best-fitting individual in the population is picked so that it can interact with the current individual and quickly find the ideal solution. The improved formula is shown in Equation (15).

Detailed Steps for the Improved Golden Eagle Optimizer Algorithm
The three strategies mentioned above can significantly increase the algorithm's convergence speed and search precision, balance global exploration with local exploitation, and improve the performance of the original GEO. The IGEO implementation method is depicted in Figure 4, and Algorithm 1 presents the pseudo-code for the suggested IGEO. The Golden Eagle's ideal hunting position is depicted in Figure 4 by * x . Calculate w (t) based on Eq.13

fitness(x)>fitness(x*)?
Calculate step vector based on Eq.7 Evaluate the fitness and find xbest The pseudo-code of the proposed IGEO is shown in Algorithm 1. Update Pc and Pa based on Equation (9)  7 For i = 1:N 8

Algorithm 1 Pseudo-Code of IGEO
Calculate A based on Equation (1)  9 If the length of A is not equal to zero 10 Calculate C based on Equations (5) and (6)  11 Calculate ω(t) based on Equation (13)  12 Calculate ∆ based on Equation (7)  13 Calculate the population fitness and selected fitness optimal individual xbest 14 Update new position x t+1 based on Equation (15) and calculate its fitness 15 If fitness(x t+1 ) is better than fitness of position in golden eagle's memory 16 Replace the new position with position in golden eagle's memory 17 End If 18 End If 19 End For 20 End While

Algorithm Complexity Analysis
The running process of the IGEO is mainly divided into the following two parts: population initialization and the algorithm main cycle. For the initialization part, the Arnold chaotic map is used to initialize the golden eagle population. N golden eagles are initialized in the D dimension, with a complexity of O (N × D). In the main part of the algorithm, for the boundary setting of N golden eagle individuals, the complexity is O (N). In the golden eagle location update formula, the complexity is O (N × D). For the whole main loop part, there are T iterations, so the algorithm complexity of the whole main loop part is O (N) + O (N × D) + O (N × D × T). Therefore, the total complexity of the entire algorithm is the sum of the complexity of the initialization part and the complexity of the main loop part, which is O (N × D × T). The complexity of the improved algorithm is not more complex than the original, but the optimization effect is much better.

Experimental Settings and Test Functions
The experimental operating environment was made up of the Windows 10 64-bit operating system, an Intel(R) Core (TM) i5-7200U processor, 2.5 GHz main frequency, and 4.00 GB of RAM. The algorithm was created using MATLAB 2021a as a basis.
Experiments were conducted on 12 benchmark functions in order to confirm the usefulness of the IGEO in solving diverse optimization challenges. Two continuous unimodal benchmark functions (F1 and F2) were used to assess the algorithm's speed and accuracy of convergence. To assess the algorithm's ability to perform a global search and the likelihood of leaving the local optimum, six complex multimodal benchmark functions (F3-F8) were utilized. The comprehensiveness of the approach was assessed using four fixed lowdimensional benchmark functions (F9-F12). Table 2 displays the specifications of the benchmarking functions.
f* represents the theoretical optimal value of each test function. D represents the dimension of the test function.

Comparative Analysis of Performance with Other Algorithms
In order to verify the overall performance of the IGEO, eight algorithms were selected for comparison, including the butterfly optimization algorithm (BOA) [13], the grey wolf optimizer (GWO) [10], particle swarm optimization (PSO) [4], the sine cosine algorithm (SCA) [52], the slap swarm algorithm (SSA) [12], the whale optimization algorithm (WOA) [9], the golden eagle optimizer algorithm (GEO) [26], and the golden eagle optimizer with double learning strategies (GEO_DLS) [42]. To ensure fairness, the population number for all algorithms was 50, and the maximum number of iterations was 1000. In addition, the parameters of each original algorithm were the best, as shown in Table 3. Table 3. Parameter settings of each algorithm.

Algorithms
Parameter Settings  Table 4 displays the analytical results in relation to the other eight algorithms, along with the mean, standard deviation (std), and average running time. These results show the accuracy of the convergence and optimization of the algorithms. From the results for mean value and standard deviation, it can be seen that IGEO has the best search accuracy: the IGEO was able to readily obtain an excellent advantage that could be matched by no other algorithm on F1, F2, F4, F5, F7, F8, and F11, and the only other algorithms that achieved the same grades as IGEO were GEO_DLS on F3, WOA on F6 and F9, GWO on F10, and PSO on F12. An MAE ranking [53] on the best outcomes of 50 experiments was performed in this work in order to more accurately assess the optimization performance of each method on the test function.
In a quantitative analysis of all algorithms, the average absolute error of 12 benchmark functions is used to rank all algorithms. The algorithm's performance increases as MAE decreases. The formula for determining the MAE ranking of these benchmark functions is given in Equation (16), and Table 5 displays the ranking of these algorithms.  Table 5 shows that IGEO has the lowest MAE value and the best performance. In Figure 5, the average convergence curve is also presented in logarithmic form for a better and more precise comparison of the experimental data. The absolute values of all findings are plotted, because −1 is the ideal value of the function F8. In Figure 5, the test results for the convergence speed and search accuracy of the algorithms are presented. The results demonstrate that IGEO outperforms the comparison algorithms on F1 to F8. Specifically, for F1, F2, F3, F6, and F7, IGEO shows significant advantages in terms of average value, indicating an overwhelmingly high convergence speed and search accuracy. For F4, F5, and F8, IGEO exhibits better performance than the comparison algorithms, and the ability to accurately capture the theoretical optimal values of 0.9 and −1 for F5 and F8, respectively. While IGEO shows inferior performance on F9, F10, F11, and F12, its standard deviation is 0, demonstrating superior robustness compared to other algorithms. Regarding F9, IGEO's performance is second only to WOA. The convergence speed of GWO and the search accuracy of WOA on F9 are comparable to those of IGEO. For F10, GWO outperforms both IGEO and WOA in terms of performance. For F11 and F12, although the average convergence curves of PSO and IGEO are similar, the results presented in Table 4 clearly indicate that IGEO exhibited a higher search accuracy than PSO. Table 4 and Figure 5 present the optimization results for the benchmark functions F1 to F8 in general dimensions (dim = 30), where it is evident that IGEO achieved the best results in terms of both convergence speed and search accuracy. To verify the comprehensive performance and robustness of IGEO, experiments were conducted on the eight benchmark functions under 100-dimensional conditions using the comparison algorithms. The conditional parameters of each algorithm were consistent with those shown in Table 3, and each algorithm was independently run 50 times for each function. Table 6 presents the results of the comparison with the other seven original algorithms, including mean and standard deviation (std). Table 6 clearly demonstrates that IGEO outperformed the other algorithms in terms of both average value and standard deviation, indicating its superior exploration and exploitation ability. Here, the optimization outcomes for the CEC2021 test set are also displayed to further demonstrate IGEO's efficacy. Recent winners are contrasted with IGEO. Both traditional improvements to particle swarm optimization, such EIW_PSO [54], and advances in new swarm intelligence algorithms, like FMMPA [15], are included among the comparison algorithms. Additionally, IGEO is contrasted with upgraded GEO algorithms, including GEO_DLS [42], CMA-ES-RIS [55], and L-SHADE [56]. Table 7 displays the specific outcomes. The data in the table were derived by deducting these values, because none of the 10 functions in the CEC2021 test set have an ideal value of 0. The table includes the minimum value, maximum value, standard deviation, and average value of the algorithm optimization. The table shows that IGEO's performance is still fairly strong, overall, despite the fact that it is not the best for every function. In this work, the Wilcoxon statistical test and the Friedman test are performed on the best results of the 50 experiments, and further follow-up verification was conducted using the Holm method to more accurately evaluate the optimization performance of each method on the test function.
(i) Wilcoxon test for statistics. If IGEO obtains the best results, it is then evaluated against the eight comparison algorithms using the rank-sum test, which is carried out at the 5% significance level. The best results from 50 separate tests make up the data vectors for comparison.
When p is less than 5%, the original hypothesis cannot be ruled out at a significance level of 100 × 5%, indicating that the enhanced algorithm produces better results than the comparison algorithm does, and vice versa. Table 8 displays the rank-sum test p values for each algorithm. Among these, "NAN" denotes that the numerical comparison is irrelevant and that the ideal value of the two procedures is 0. "+", "−", and "=" signify that IGEO's performance is better than, worse than, or equal to that of the comparison algorithm, respectively. Table 8 shows that when compared to the original algorithms GEO and SSA, IGEO achieves a 100% optimization rate; when compared to BOA, PSO, and SCA, IGEO achieves a 91.2% optimization rate; and when compared to GWO, WOA, and GEO DLS, IGEO achieves an 83.3% optimization rate. These results illustrate IGEO's thorough performance across the 12 benchmark functions.  (17), the Friedman rankings for the benchmark functions and CEC2021 are determined, and the results obtained are reported in Tables 6 and 7. To solve the algorithm ranking in a specific test function fairly, an average of 30 separate runs is used. Table 9 shows that IGEO's comprehensive rating comes in top place. In addition, Table 10 shows that IGEO ranks lower than FMMPA. (iii) Holm verification. Tables 11 and 12 present the data acquired after further Holm verification of the results by comparing the results obtained with the original and improved algorithms. Assuming that each algorithm's distribution is equal to that of the classical original algorithm, the system uses Friedman to perform a bidirectional rank variance analysis on the pertinent samples at a significance level of 0.05. After further verification usign Holm, the average chi square value after testing is 34.296, the critical chi square value of the system is 15.507, the degree of freedom gained through data testing is 9, and the p-value is significantly below 0.05. The original presumption is disproved, i.e., the presumption that all optimization results follow a uniform distribution. Table 11 demonstrates that the WOA, GEO_DLS, PSO, and GWO models reject the null hypothesis and significantly deviate from the enhanced method. SCA, BOA, GEO, and SSA all support the initial theory and differ little from the initial method. The final order of preference for the algorithms is IGEO > WOA > GEO_DLS > PSO > GWO > SCA > BOA > GEO > SSA.
Similarly, in the experiment using CEC2021 as the test object, the critical chi square value of the system is 19.943, which therefore also rejects the original hypothesis. Table 12 demonstrates that the L-SHADE, GEO_DLS, CMA-ES-RIS and FMMPA models reject the null hypothesis and significantly deviate from the enhanced method. Only EIW_PSO rejects the hypothesis that there is a significant difference from IGEO. The final order of preference for the algorithms is FMMPA > IGEO > L-SHADE > GEO_DLS > CMA-ES-RIS > EIW_PSO.

Comparative Analysis of Different Strategy Algorithms
To explore the effectiveness of the proposed algorithm and study the impact of different strategies on the golden eagle optimizer algorithm, ablation experiments were conducted. GEO with a single strategy includes GEO with nonlinear weights (wGEO) and GEO with global optimization strategy (pGEO). The experimental parameter settings were consistent with those presented in Table 3. The experimental algorithms included GEO, wGEO, pGEO, and IGEO. Each algorithm was independently run 50 times on each of the 12 test functions selected in this article. The experimental results are presented in Table 13.
The results presented in Table 13 indicate that the optimization accuracy when using a nonlinear convex decreasing weight strategy or a global optimization strategy individually is inferior to that when using the mixed strategy, with a significant difference in optimization accuracy despite having the same running time. For F1, F2, F7, F9, and F11, IGEO demonstrated absolute superiority, achieving the highest optimization accuracy. For F3 to F6, wGEO, pGEO, and IGEO exhibited similar search accuracy, which was significantly better than that of GEO. For F8, F10, and F12, pGEO and IGEO exhibited similar performance.
To verify the performance of IGEO in terms of convergence speed and search accuracy, Figure 6 presents the average convergence curves of the benchmark functions. Similarly, since the optimal value of function F8 is −1, the absolute value of all results is taken when drawing the visualization. Compared with wGEO and pGEO, IGEO exhibits faster convergence on F3 to F6, although the convergence precision is the same. IGEO and pGEO achieve identical performance on F8. For F10 and F12, IGEO demonstrates a slight advantage in terms of convergence speed, while the average optimal value and standard deviation are close to the theoretical optimal value. Furthermore, IGEO performs optimally on the other functions.
Combining the exploration of the value of m described in Section 3.2 and the two numerical experiments reported in Sections 4.1.2 and 4.1.3, it can be concluded that the nonlinear convex decreasing weight mainly affects the convergence speed of the algorithm, indicating the algorithm's ability to perform exploration. Conversely, the global optimization strategy mainly affects the search accuracy of the algorithm, indicating the algorithm's ability to perform exploitation. In this article, these two capacities are effectively balanced by combining these two strategies to improve GEO, leading to improved algorithm performance. The theoretical numerical experiment for the revised algorithm is presented in this section. In Sections 4.2 and 4.3, specific application experiments are provided.

IGEO Problem Solving Model
In the proposed model for solving the spectrum allocation problem, each golden eagle's binary-coded position represents a feasible spectrum allocation strategy. To determine the optimal spectrum allocation scheme, it is necessary to solve the interference-free allocation matrix A to maximize the total system benefit. Based on the status of the sec-  [45] for the idle matrix L involves exclusively numbering those elements for which the array is 1. Furthermore, the number of elements in matrix L that equal 1, representing the coding length D of the golden eagle individuals, is recorded. The formula for calculating this is as follows:  Calculate A(attack vector) based on Equation (1)  16 If the length of A is not equal to zero 17 Calculate C based on Equations (5) and (6)  18 Calculate ω(t) based on Equation (12)  19 Calculate ∆ based on Equation (7)  20 Calculate the population fitness and selected fitness optimal individual xbest 21 Update new position x t+1 based on Equation (14) and calculate its fitness 22 Discretize the current position of the golden eagle individual based on Equation (7)

The Solution Results of the Problem Model
In this experiment, the number of populations is set to 50, the number of iterations is set to 1000, and the number of primary users (number of spectrums) is set to M and N, respectively. After 30 experiments, the average of all outcomes is calculated. This investigation used the genetic algorithm (GA) [50], the particle swarm algorithm (PSO) [4], the butterfly optimization algorithm (BOA) [13], and the cat swarm optimization (CSO) [51] as comparison methods. Figure 8 shows the system solution's average maximum system benefit for N = 10 and M = 10. The figure indicates that GA exhibits the best system performance, followed by the IGEO algorithm, while the GEO algorithm exhibits the worst performance. This observation highlights the effectiveness of the improved strategy when utilized in the spectrum allocation model. Figure 9 presents the fairness of each algorithm in 30 distinct channel environments. The results demonstrate that the IGEO algorithm yields the highest user fairness, whereas the GA algorithm results in the worst user fairness. Overall, the IGEO algorithm achieves the best maximization of both system benefits and user fairness. These results suggest that IGEO can more effectively be used to resolve the spectrum allocation problem, compared to the other algorithms. To investigate the impact of different numbers of users on the average system benefit, in this study, the number of available channels is maintained at a constant M = 10 in the environment, while increasing the number of SUs N from 10 to 30, with an increment of 2. The relationship between the number of users and the average benefit is analyzed, as shown in Figure 10. The results indicate that the average benefit of the cognitive radio system gradually decreases with increasing numbers of SUs. However, the IGEO algorithm outperforms the GA, PSO, BOA, CSA, and GEO algorithms in terms of the average system benefit value obtained.
On the other hand, this study looks into how different channel counts affect the overall value of the system. The average advantage of the system for different numbers of channels is obtained while increasing M from 10 to 30, with an increment of 2, as shown in Figure 11. Our results show that when the number of possible spectra in the region rises, the average benefit of the channel gradually increases. Notably, with the exception of BOA, the average benefit of the IGEO algorithm is higher than that of other methods. This underlines once more how well the improved algorithm allocates spectrum.

Engineering Applications
To further validate the performance and practical effectiveness of the improved algorithm, in this article, five classic engineering problems were selected, and comparative experiments were conducted with other algorithms. All five engineering problems are static single-objective constrained optimization problems, which can be generally expressed as follows: To more effectively handle the constraint conditions, in this article, penalty functions are employed, which can be expressed as: where ( ) represents the final objective function, and i l and j o represent the penalty coefficients. All algorithms are independently tested 30 times.

Piston Rod Design Problem
A rarely encountered static, single-objective constrained problem is the piston rod optimization problem. By maximizing the locations of the piston components H, B, D, and X as the piston rod advances from 0 degrees to 45 degrees, as indicated in Appendix A, its primary goal is to reduce fuel consumption. The basic model is expressed in Appendix B.
In the piston rod design problem, the IGEO algorithm is compared with GWO, PSO, SCA, SSA, WOA, GEO, and GEO_DLS. The minimum cost and corresponding optimal variable values obtained by the above algorithms can be found in Table 14. Among the algorithms, IGEO achieves the best performance in the piston rod design problem, with the lowest cost of 0.0003717. The goal in the I beam structural design problem is to minimize vertical deflection by optimizing the length, height, and two thicknesses. Appendix A contains a structural schematic diagram, which depicts the left and major views of the I beam. For ease of computation, let The performance of IGEO when designing I beams is compared to that of other algorithms, including GWO, PSO, SCA, SSA, WOA, GEO, and GEO_DLS. The comparison results are presented in Table 15, where it can be observed that both GEO and GEO_DLS yield an optimal value of 0.001644. However, the key difference lies in the very small standard deviation of IGEO, indicating its high stability in solving the problem.

Car Side Impact Design Problem
The car side impact design problem is a static constrained optimization problem with a single objective and multiple variables. Its main objective is to minimize the total weight of the vehicle. The mathematical model is presented in Appendix B, and the meanings of the specific parameters can be found in [57].
The performances of different algorithms when designing car side collisions are compared in Table 16. Among the compared algorithms, which included GWO, PSO, SCA, SSA, WOA, GEO, and GEO_DLS, IGEO achieved the best result, with the minimum value of 21.8829. The cantilever beam optimization problem involves five variables and a vertical displacement constraint, where each variable has a constant thickness, and the objective is to minimize the weight of the beam. A schematic diagram for this problem is presented in Appendix A. The mathematical model for this problem is given in Appendix B.
After applying various optimization algorithms such as GWO, PSO, SCA, SSA, WOA, GEO, and GEO_DLS, IGEO was used to optimize the cantilever beam design problem. The results obtained using these algorithms are presented in Table 17, where it can be observed that IGEO achieved the best result, with a value of 1.179635. In the classic three-bar truss engineering design problem, the aim is to minimize the weight of a symmetrical light bar structure that is subject to constraints on stress, deflection, and buckling. The mathematical model for this problem is presented in Appendix B, and the structural diagram can be found in Appendix A. Table 18 presents the results of various optimization algorithms, including GWO, PSO, SCA, SSA, WOA, GEO, and GEO_DLS, as well as the comparison algorithm GEO_DLS, and IGEO for the three-bar truss design problem. The table indicates that IGEO achieved the best results, with a minimum value of 259.8111.

Conclusions
The principle and position update formula of the original golden eagle optimization method were explored, and a hybrid golden eagle optimization algorithm (IGEO) based on Arnold mapping and nonlinear convex decreasing weight was proposed. Using 12 benchmark functions, CEC2021, and the Wilcoxon, Friedman, and Holm tests, it was confirmed that the proposed IGEO has better search performance and stronger resilience. Additionally, the impact of the single-strategy method on the algorithm was also studied. After conducting simulation tests, the enhanced IGEO was applied to the model of the traditional cognitive radio spectrum allocation problem. It was discovered that IGEO had the best overall performance, and was able to allocate the spectrum well when compared to GA, PSO, and other methods. To test the proposed IGEO's problem-solving abilities, five real-world technical design problems were also addressed, and the solutions were contrasted with other methods. The IGEO methodology is supported by all of the findings, feedback, and analyses as being a superior approach that can be used to tackle challenging engineering optimization problems. In future research, IGEO is expected to be applied to more practical spectrum allocation problems in order to investigate the algorithm's additional capabilities.

Data Availability Statement:
The study did not report any data.

Acknowledgments:
The author is extremely thankful to anonymous referees and the editor for their valuable comments towards improving the manuscript.

Conflicts of Interest:
The authors declare no conflicts of interest.