Multi-Strategy Improved Northern Goshawk Optimization Algorithm and Application

The Northern Goshawk Optimization Algorithm (NGO) is a population-based meta-heuristic algorithm inspired by the hunting behavior of the northern goshawk. Compared with other algorithms, NGO have a certain competitiveness, but there is still an imbalance in development and exploration, and it is easy to fall into the local optimal. This paper proposes a multi-strategy Improved Northern Goshawk optimization algorithm (INGO) to address these shortcomings. INGO uses an improved tent chaos mapping strategy to generate the initial population and improve the quality of the initial solution set. The levy flight strategy is introduced in the hunting stage of the NGO to improve the search range of the solution and avoid the algorithm’s prematurity. In addition, we introduce a nonlinear convergence factor and heart-shaped search strategy to reduce the probability of the algorithm falling into the local optimal and improve the convergence speed of the algorithm. We evaluated INGO’s performance using 23 benchmark functions, CEC2017 test suite functions, and three constrained engineering optimization problems and validated its optimization using Wilcoxon rank sum tests. Experimental results show that the algorithm has higher convergence accuracy and better detection ability. Finally, INGO was applied to optimize the ensemble learning system.


I. INTRODUCTION
Optimization is about finding the best solution for a given system from all possible values.Current optimization methods can be broadly categorized into those based on mathematics and those involving stochastic optimization.The main techniques of mathematical optimization are gradient descent algorithm [1], newton method [2], conjugate gradient algorithm [3].With the continuous improvement of industrial production levels, the engineering optimization problem is becoming more and more complicated in reality.Traditional mathematical optimization techniques are inefficient and ineffective in solving these problems.To eliminate the limitations of mathematical optimization, researchers began to study random optimization techniques.In recent years, various random search strategies have been effectively integrated into meta-heuristic algorithms(MHA) [4], [5], [6], [7].Its core idea mainly comes from various biological The associate editor coordinating the review of this manuscript and approving it for publication was Jerry Chun-Wei Lin .
behavior laws in nature, through the simulation of these laws, to achieve the quest for the best solution within the realm of solutions.Relative to mathematical optimization methods, MHAs do not need to consider specific gradient information and only focus on input and output [8].Due to its unique flexibility, it is favored by researchers and has been effectively utilized in various intricate optimization challenges [9], [10], [11].At present, various meta-heuristic algorithms have been developed with roughly the same idea, the main difference being that they use different search methods in the optimization process.Much of it is about finding the right balance between diversity and searchability.Based on maintaining good convergence accuracy and convergence speed, the population diversity should be increased as much as possible.However, Numerous conventional meta-heuristic methods exhibit poor accuracy in convergence and sluggish speed in real-world scenarios, making them prone to achieving local optimal states.Therefore, many researchers have suggested various enhancement strategies to address these issues.
Zhang et al. [12] unveiled the genetic algorithm's adaptive weight and mutation approaches to enhance the Harris Hawk Algorithm, leading to better convergence precision and speed.Yu et al. [13] use the optimal point set method to improve the population richness, introduce the beetle antenna search mechanism to prevent the algorithm from descending into local extremes, and enhance the gray Wolf optimization technique.In [14], control parameters were implemented to balance the algorithm's exploration and development capabilities, the wavelet mutation approach was employed to boost the algorithm's capacity to leap beyond the local extreme value, and it was integrated with the quadratic interpolation method to hasten the algorithm's convergence rate.Chaotic mapping and levy flight are two commonly used improvement strategies.Reference [15] uses chaotic mapping initiating the population to ensure a more uniform distribution within the initial solution area.The implementation of levy-flying aimed to boost population diversity and augment the algorithm's capacity for local development in the later stage.
In 2022, Dehghani et al. suggested an innovative algorithm inspired by biology: the Northern Goshawk Optimization Algorithm(NGO) [16].MOHAMMAD DEHGHANI et al. tested the NGO algorithm using 68 different objective functions and 4 engineering design problems, and the results showed that the algorithm can maintain a balance between exploration and extraction capabilities, and has stronger optimization capabilities and fewer control parameters than similar algorithms.It has been effectively applied in feature selection [17] and medical diagnosis [18].However, a single algorithm cannot solve all optimization problems, just as in MOHAMMAD DEHGHANI et al. 's study NGO could not get the best results in all test functions.In some specific problems [19], [20], the NGO algorithm shows poor convergence accuracy.And like the traditional metaheuristic algorithm, there are still some problems such as decreasing convergence speed and falling into local extreme value in the late iteration of the algorithm.In addition, unlike many two-stage algorithms [21], [22], each individual in the NGO algorithm is explored and developed in each iteration, which means that the NGO algorithm consumes more computing resources under the same conditions.To improve its performance in practical problems, researchers began to adopt various strategies to enhance the NGO algorithm.Such as, Liang et al. [23] introduced polynomial interpolation strategies and multiple opposing learning strategies into NGO to improve diversity among populations and enable individuals to search more comprehensively within the realm of search.Chen et al. [19] introduced the information of the best individual into the original NGO algorithm to improve the convergence speed.Youssef et al. [24] introduced the MPA into NGO to resolve the issue of the algorithm trapped in local optimal.Wang et al. [25] integrated adaptive weights into the hunting process of the NGO algorithm to reduce its randomness.The introduction of levy flight in the process of chasing prey also enhances the algorithm's adaptability and variety during the search phase, thereby preventing early attainment of the local optimum.Li et al. [26] introduce a new exploration strategy and levy flight to reduce the risk of being trapped in a local optimum and use a non-linear convergence factor to balance exploration and exploitation.Relative to the NGO, the above studies have improved the performance to a certain extent, yet certain restrictions remain.For example, although the improvement measures in [23] and [24] significantly improve the performance of the NGO algorithm, they cost more computing resources.Overall, there is room for further improvement in NGO.The original NGO algorithm mainly has the following defects: • Random initialization in the population initialization phase does not guarantee a uniform distribution of individuals within the search area.
• The location of individuals updated in the first phase of the NGO algorithm influences the search outcomes directly in the second phase.Suppose the population falls into the local optimal position after the first stage.
In that case, it will be difficult to change this dilemma in the second stage, which will directly lead to the prematurity of the algorithm.
• In the chasing prey model of the NGO algorithm, a linear factor controls the size of the search step, which is too single to adapt to complex nonlinear problems.At the same time, the model of chasing prey is too simple, and it is easy to fall into a partially sighted search state relying only on the individual's information, eventually leading to slow convergence.
Addressing the abovementioned shortcomings, we suggest an enhanced Northern Goshawk optimization algorithm with multiple strategies.Initially, the improved tent mapping served to kickstart the population during the NGO algorithm's initialization stage, aiming to boost population diversity.Secondly, In the NGO algorithm's first phase, the levy flight is employed to seek the most effective solution proactively, make the algorithm random, and strengthen the algorithm solutions' diversity.Finally, a nonlinear weight factor was added in the exploration phase to harmonize the algorithm's capabilities for global exploration with its local development, and given the problem that the NGO's chasing prey model is too simple, the food source-oriented heart-shaped search strategy is used to improve the local development ability of the NGO algorithm.
The primary contributions presented in this article include: • A multi-strategy improved Northern Goshawk Optimization Algorithm (INGO) has been proposed, • The efficacy of the suggested algorithm is confirmed when contrasted with six fundamental algorithms across 23 benchmark functions, the CEC2017 test suite, and three engineering optimization problems.The findings reveal that the suggested algorithm achieves quicker convergence rates and greater accuracy in addressing functional optimization challenges.
• The application of the INGO in the optimization of Ensemble systems demonstrates its superiority in practical applications.The subsequent sections of this article are structured in this manner: Section II details the original NGO.Section III provides an overview of INGO using multi-strategy fusion.The outcomes of the INGO and additional comparison algorithms in addressing the 23 benchmark functions, the CEC2017 test suite, and three engineering optimization problems are presented in Section IV.Section V applies INGO algorithms to optimizing ensemble systems.Finally, the sixth part summarizes the work.

II. ORIGINAL NORTHERN GOSHAWK OPTIMIZATION ALGORITHM
While exploring, The northern goshawk selects its prey randomly and quickly initiates an assault.Since the game in the search area is randomly selected, this stage enhances the exploration capabilities of the NGO algorithm.Specific definitions are as follows: The prey chosen by the ith goshawk is denoted as P i , its function value is represented by F pi , X new i is the new position, is the value of its objective function, r is a arbitrary number within [0, 1], while I can be any integer between 1 and 2.
The development phase simulates the northern goshawk hunting prey in a chasing manner and simulates this trend to improve the algorithm's utilization of local search space.The definition is as follows: where the current and maximum iterations are t and T, respectively.r is an arbitrary number within [0, 1].

III. THE IMPROVED NORTHERN GOSHAWK OPTIMIZATION ALGORITHM
An enhanced NGO algorithm is suggested to meet the goal of creating an algorithm with superior performance, as elaborated further below.

A. TENT CHAOTIC MAPPING
The original NGO receives the initial solution randomly, making it challenging to guarantee uniform coverage of the entire solution spectrum by a single individual.Chaotic mapping frequently enhances the efficiency of scattered populations and minimizes aggregation processes.In numerous mappings, the chaotic map of the tent exhibits improved uniformity in traversal and quicker search velocities [27].However, tent mapping also has shortcomings; for example, Tent chaotic iterative sequences will appear in small periods 0.2,0.4,0.6,0.8 and fixed points 0,0.25,0.5,0.75.To solve this problem, [28] introduces adaptive parameter to improve tent mapping: In [28], k is 1/D, where D represents the dimension of the problem in question.r is a random number between [0,1].
Although the introduction of parameters can prevent tent mappings from falling into fixed points, for problems with lower dimensions, a larger k will break the laws and properties of the original mapping.Therefore, in this article, we set k to 0.001.Figure 1 shows the sequence distribution of different k values.It can be seen from the figure1 that a k value of 0.001 is a more uniform distribution.

B. THE LEVY FLIGHT STRATEGY
After the first stage of the NGO algorithm, the location of the population will have a significant impact on the search of the second stage.If the individuals in the population become aggregated after the first stage, it is likely to drive the position of the individuals in the second stage to update to a worse position.To weaken the problem and protect population diversity, we introduce Levy flight to perturb individual positions.The model is given in Eq 8.
where X m represents the average location of the population, the dimension space is represented by D.α and ρ are two constants, set to 0.01, 1.5,µ ∼ N 0, σ 2 , v ∼ N (0, 1).The following equation is used to calculate σ ::σ .

C. NONLINEAR REDUCTION AND HEART-SHAPED SEARCH STRATEGY 1) NONLINEAR CONVERGENCE FACTOR
During the development phase of the NGO algorithm, it used the linear factor 1 − t T to balance local and global search.However, linear attenuation can not reflect the real situation of the northern goshawk chasing prey, and it will lead to premature convergence of NGO in the context of intricate nonlinear issues.Consequently, a sine functionbased nonlinear parameter approach was employed to balance exploration with production efficiently.
The Eq (11) defines the convergence factor: a is a power exponent, and a is set to 2 in this paper.Figure 1 shows the variation curves of the nonlinear convergence factor and the original linear convergence factor with the number of iterations.
Comparing the two convergence curves, we find that the value of the nonlinear convergence factor is larger and changes slowly in the early stage.It indicates the individual goshawk carries out the global search with a giant search stride, which is conducive to quickly finding the global optimal region and improving the convergence speed.During the intermediate and advanced stages, the value of the nonlinear convergence factor decreases rapidly and finally maintains a low value.This suggests that a single goshawk began to reduce its search stride size rapidly and to search with a smaller search stride.In contrast, the change of the nonlinear convergence factor is single, and it is easy to fall into the local optimum.Import the nonlinear convergence factor into Eq(4):

2) HEART-SHAPED SEARCH
Heart-shaped search originated from the honey badger optimization algorithm(HBA) proposed by Egyptian scholar Hashim et al. [29].In the HBA, the honey badger forges in two main ways: First, the honey badger estimates the location of its prey through its sense of smell, and when it gets there, it moves around the prey, choosing the right place to dig.Second, honey badgers find their hives by following their guide birds.Hashimet calls the first pattern the digging pattern and the second the honey pattern.During the digging stage, the motion of the honey badger is similar to the heartshaped, so it is also called heart-shaped search, which is defined as follows: where X p is the global optimal position, θ represents the ability to hunt prey (default is 6), r 1 , r 2 , r 3 are random numbers on [0, 1], F is the sign of changing the search direction and a random number controls its value.I is the prey odor concentration, β is the density factor, and d i represents the distance between the prey and the ith honey badger.The specific expressions of F, I , β and d i are as follows: We introduce the heart-shaped search strategy into the chasing prey stage of the NGO and use the random variable r to control how the location is updated.Use Eq(12) when r is less than 0.5; otherwise, use Eq (13).This approach has two advantages: The heart-shaped search strategy is oriented towards food sources, which can improve the algorithm's convergence rate and enrich the mathematical model of NGO algorithms chasing prey.).Therefore, the improvement strategy proposed for NGO in this paper does not increase the operational cost of the algorithm, and the operational efficiency of the algorithm does not decrease.

IV. EXPERIMENTS AND ANALYSIS
The experiments were performed in Matlab 2021a with 64-bit, Windows 11, 2.50GHz i5-1155G7 processor, and 16GB memory.In the experimental section, 52 different test functions and 3 constrained complex optimization questions were used to test the effectiveness of various strategies and the performance of the INGO.The 52 test functions include 23 commonly used standard test functions and CEC2017 functions.The standard test functions were split into three

A. IMPROVE THE EFFECTIVENESS TEST OF THE STRATEGIES
In Section III, we use three strategies to improve the original NGO algorithm.To assess the influence of each strategy on the algorithm, four NGO-derived algorithms (INGO1, INGO2, INGO3, INGO) were developed according to Table 3.These NGO variants were tested using the 23 test functions in Table 2, with the dimension of f 1 − f 13 is set to 30.Table 4 5 reveals that INGO consistently ranks the best on various types of functions.The three NGO-derived algorithms hold a superior ranking compared to the NGO algorithm.Among them, NGO1 has a certain degree of improvement in most functions, but the improvement effect is insignificant, and other measures are still needed to strengthen further.NGO2 and NGO3 have a greater impact on NGO.This effect is primarily manifested in unimodal function and multimodal function.NGO2 obtained the optimal solution on the unimodal functions f 1 and f 3 and performed significantly better than NGO on the functions f 2 , f 4 , f 6 and f 7 .This shows that introducing levy flight can markedly enhance the algorithm's optimization capabilities, and the effect is significant.Although the improvement effect of NGO3 in unimodal function is not as good as that of NGO2, it is still a great improvement compared with NGO.From the results, the introduction of a heart-shaped search does improve the optimization ability of the algorithm.This is consistent with our original intention of introducing the heart-shaped search strategy.Among the multimodal functions, f 8 , f 10 , f 12 , f 13 are relatively difficult to optimize.These functions can fully reflect the capacity of the algorithm to leap out of the local extreme value.Although NGO2 and NGO3 do not perform as well on f 12 , the results on several other functions are better than the original algorithm.This indicates that strategy two and strategy three can adjust individual positions in time to avoid individuals being trapped in local extreme values and maintain good population diversity in the search process.In general, combining these three strategies has significantly improved the overall performance of NGO, further demonstrating the importance of balancing development and exploration capabilities to improve algorithm performance [31].

B. COMPARE WITH OTHER INTELLIGENT ALGORITHMS ON STANDARD BENCHMARK FUNCTIONS
To further verify the effect of INGO, it is compared with PSO [32], WOA [33], MPA [34], HBA [29], HHO [35], GWO [36], were compared experimentally.From the comparison methods selected, PSO, WOA, and GWO are influential heuristic algorithms with strong optimization abilities and have been widely used in the engineering field.HBA, MPA, and HHO are relatively new algorithms.The comparison with these algorithms can fully prove the overall effectiveness of the proposed method.Table 6 shows the result obtained by the selected algorithm over 30 independent runs.
Based on the average performance of the classical benchmark, INGO achieves absolute global optimality on 13 of the 23 test functions.INGO ranked first in 18 functions, second in 2 functions (f 13 , f 1 ), third in 1 function (f 5 ), fifth in 2 functions (f 8 , f 20 ), and none of the functions ranked last.INGO is considered to be the best performing of all comparison algorithms, as other algorithms rank first on a maximum of 10 functions.INGO ranks at least in the top three for 21 functions, far outpacing other algorithms in terms of the number of functions, indicating its strong performance in terms of global best achievements.In terms of the type of the benchmark functions,for f 1 − f 7 , the INGO performs the best and obtains the optimal solution on f 1 , f 2 , f 3 , and f 4 .Although INGO does not obtain the optimal value on f 6 and f 7 , its superiority over rival algorithms is notable.For f 5 , the HHO algorithm performs best and has a particular gap with other algorithms, indicating that INGO needs to be further optimized.Overall

1) CONVERGENCE CURVE ANALYSIS
The statistics discussed earlier show the robustness of INGO's algorithm.During the search process, the accuracy and convergence rate of the algorithm can be measured by the descent rate and final position of its curve.Figure 4 shows the best values found by the selected algorithm in every 500 iterations, which are averaged over 30 independent runs.According to the convergence curve, INGO has a satisfactory convergence rate.Most INGO curves converge to a minimum fitness value of around 5-60% of the entire iteration.Although INGO converges slightly more slowly on some benchmark functions (e.g: f 6 , f 12 , f 15 ) than some competing algorithms, it is clear that INGO can achieve better convergence accuracy.It is also evident from Figure 4 that INGO has superior convergence speed and convergence accuracy on unimodal functions.On multimodal functions f 8 , f 12 , and f 13 , INGO's convergence speed is relatively slow and the convergence accuracy is not ideal.However, 34254 VOLUME 12, 2024 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

2) WILCOXON RANK AND INSPECTION
In the above discussion, INGO is considered to be the most robust of all comparison algorithms in terms of classical benchmark functions.However, the swarm intelligence algorithm has strong randomness.To reflect the advantages of INGO, we use the Wilcoxon test to test the experimental results statistically.Where α = 0.05.'NAN' means a significant difference cannot be determined.The symbols '+', '−', and '=' indicate superior, inferior, and equal to the comparison algorithm, respectively.The results are shown in Table7.Most of the values in Table7 are less than 0.05, and there are more '+', which indicates that the performance of the INGO algorithm is significantly different from the other six algorithms, and INGO has better optimization performance.As seen from the last line of Table7, INGO's algorithm is significantly better than the competing algorithm on at least 60.9% (14/23) functions, and no worse than the competing algorithm on at least 86.9% (20/23) functions.Among the competitive algorithms, the HHO algorithm significantly outperforms INGO in three test functions and is the best among all the competitive algorithms.It can also be seen from Table7 that INGO's advantages are most prominent in unimodal functions.Although INGO's advantages in multi-modal functions and functions with fixed dimensions are less prominent than those in unimodal functions, they are still significantly superior to comparison algorithms.These results show that the proposed INGO algorithm has significant advantages in optimization capability.

C. COMPARE WITH OTHER INTELLIGENT ALGORITHMS ON CEC2017 FUNCTIONS
To further verify the feasibility and robustness of INGO, optimization tests were carried out in CEC2017 optimization functions.The CEC2017 functions are considered highly complex test platforms with complex characteristics that make it difficult for most algorithms to solve for optimal values, so they are used to evaluate the quality of INGO.In addition, we chose several of the latest swarm intelligence algorithms for comparison: DO [37], SCSO [38], POA [39], HHO, HBA, and NGO.The specific parameters are shown in table 1.

1) EVALUATION OF IEEE CEC2017
The results of different algorithms for solving CEC2017 test suite functions are shown in Table8.The results showed that the enhanced NGO algorithm provided better quasi-optimal solutions in 14 test functions, while the other comparison algorithms ranked first in up to 5 functions.The INGO algorithm obtained a better solution on 79.3% of the functions than the original NGO algorithm.Compared to other algorithms, INGO's algorithm has an average rank of 1.6896, which is lower than other algorithms.

D. APPLICATION TO PRACTICAL PROBLEMS
In addition to the test function, the real optimization problem with strong constraints can also reflect the performance of the algorithm.In this section, we will use INGO to solve several practical problems to demonstrate its usefulness in real problems.It is used primarily for three engineering problems: Tension/compression spring design problem(T/CSD), Pressure vessel design problem (PVD), Three-bar truss design problem (T-bTD).
There are many constraints in practical problems, and the methods of constraint processing are divided into three categories: penalty function method, feasible rule method, and multi-objective method.Because of its simple principle and easy implementation, the penalty function method is one of the most widely used constraint processing methods.By imposing a penalty on a non-feasible solution, the constrained optimization problem is transformed into an unconstrained optimization problem [40].The penalty formula is as follows: where m represents the number of constraints in the problem, lambda is the penalty constant, and the algorithm selects the candidate solution and the original solution in the iterative process can refer to [41].

1) T/CSD PROBLEM
T/CSD aims to reduce its mass f x within specific limitations, encompassing four inequality constraints: slightest deflection, shear stress, frequency of oscillation, and the maximum outer diameter.Spring wire diameter d (x 1 ), Spring coil average diameter D (x 2 ), and the adequate number of turns spring N (x 3 ), three design variables.Its specific mathematical model is as follows: Comparison of test results on CEC2017 functions.(The two rows of each function are the mean and standard deviation, respectively.)Constraints: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.The results show that INGO finds the optimal solution 0.012665, corresponding to the following variables:[0.0515700.353863 11.458263].

2) PVD PROBLEM
PVD aims to reduce the overall expense f x in alignment with production requirements, incorporating four key design elements: Shell thickness (x 3 ), head thickness (x 4 ), inner radius (x 1 ), and the container's length (x 2 , excluding the lead).The precise mathematical representation is outlined below: min f(x) = 0.6224x 1 x 3 x 4 + 1.7781x 2 x 2 3 + 3.1661x 2 1 x 4 + 19.84x 2  1 x 3 Constraints: The results show that INGO and HBA find the optimal solution 5885.33, corresponding to the following variables:[0.77810.3846 40.3196 200 ].

3) T-bTD PROBLEM
The issue aims to reduce the volume while adhering to the stress limitations on both sides of the truss component.The variable for this problem is x = (x 1 , x 2 ), and its mathematical model is: Constraints: Constraint on the boundary: According to the results, INGO has the lowest cost, and its value is 263.8958.The corresponding values of each variable are:[0.78860.4084].

V. OPTIMIZE ENSEMBLE SYSTEM
Incorporating ensemble learning is a straightforward and effective method to improve the classification result.On the other hand, ensemble learning unites multiple complementary classifiers to form a classifier system that can generate more reliable forecasts than individual classifiers.[42].Nowadays, ensemble learning has been widely used.However, despite the excellent performance of ensemble learning, ensemble systems suffer from three obvious drawbacks [43]: The use of multiple base classifiers leads to high computational costs; The prediction performance of the base classifier is poor, which hurts the whole system.The inclusion of base classifiers with high similarity renders the ensemble system redundant.With the development of big data, these three defects are becoming more and more serious.To address these three drawbacks, researchers have proposed the concept of ensemble pruning.Ensemble pruning aims to some classifiers from the initial classifier pool to obtain better classification and generalization abilities.Certain scholars convert the issue of ensemble pruning into one of optimization and identify the most suitable subset of base classifiers that meet predetermined requirements.[44].GASEN [45] is a typical example of an optimization-based method using the genetic algorithm for neural network ensemble classifier selection.It uses the verification error of the ensemble system as the objective function, randomly assigns a weight to the base classifier, and evolves the weight by genetic algorithm.Finally, the selection of base classifiers is determined by evolutionary weights and given thresholds.Optimizationbased ensemble pruning techniques rely on defining the objective function and the solution space search strategy.In the last three years, various bio-inspired algorithms, such as artificial fish swarm algorithm [46], bee algorithm [47], and salp swarm algorithm [48], have been applied to pruning problems and achieved good results.Recent research has shown that margin distribution is critical for better generalization.Inspired by the discovery, we use it as an optimization target and INGO as an optimization tool to optimize the ensemble system.Figure5 shows the process.

A. CONSTRUCT THE INITIAL CLASSIFIER POOL
The variety of base classifiers greatly influences the ensemble system's performance.The more dissimilar the base classifiers are, the more reliable the ensemble system will be and the more adeptly it can generalize since even slight changes in the data can majorly affect the formation of the decision tree.So, We use cart decision trees as the base classifier in our experiment and use self-service sampling in Bagging to generate an initial classifier pool of size T. The number of basic classifiers in the initial classifier pool is also something we need to consider.More classifiers are not always better for an ensemble system.To determine the initial number of classifiers, Figure 6 shows the trend of classification accuracy of different sizes of classifiers in the ensemble system on the data sets Wine and Sonar as the ensemble size grows.The classification accuracy increases with the number of classifiers.When the classifier reaches 225, the accuracy begins to level off.So, we set the number of initial classifiers to 225.

B. DEFINE THE OBJECTIVE FUNCTION
In this paper, the margin distribution is used as the objective function.Specific expression is as follows: Suppose the ensemble system T ens = {T 1 , T 2 , . . ., T M }, W = (w 1 , w 2 , . . ., w M ) T is a set of weight vectors.Training sample S = {(x i , y i ) , i = 1, 2, . . ., n} is used to generate the matrix H . h ij = ±1, h ij = 1 indicates that the classification result of the ith sample by the jth classifier is consistent with its true label.hij = −1 indicates inconsistent with its true label.
The margin of the ensemble system on the sample (x k , y k ) is defined as: where e k is a 1 × n unit vector, and the kth position is 1.
The average margin of the ensemble system is: where e is a 1 × n vector, and all elements are 1.The margin variance is: Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.To avoid overfitting, we use the voting method for classification.Therefore, the ensemble classifier design problem can be presented as:

C. CLASSIFIER SELECTION
We first encode the weight vector the INGO algorithm to solve (20).To satisfy the constraint that the weight vector in ( 20) is non-negative and the sum is one at the same time, this paper uses' parallel space mapping 'to transform the weight vector, namely: x i represents the position of the ith northern goshawk, and w i is the weight of the ith base classifier.We remove some weights with less weight from the transformed weight vector by the given threshold p.Thus, we reduce the ensemble scale.
where w ′ i = 1 denotes the ith classifier is included in the final ensemble, w ′ i = 0 denotes the ith classifier is not selected, and the threshold p = 1/T.

D. OPTIMIZATION RESULTS OF EACH ALGORITHM
To evaluate each method on each dataset, Each dataset is divided into three equal subsets: one subset for training cart trees to form the base classifier pool, one subset for optimizing the base classifier pool, and one subset for testing.Meanwhile, to further reduce the randomness of each algorithm, each method is executed 10 times on every partition of the data set, and the average performance is documented.Table 12 gives detailed descriptions of the UCI data sets.
As seen from table 13, Among all the compared methods, Bagging performs the worst in classification accuracy and ensemble size, which agrees with the fact that ensemble classifier design can enhance the efficiency of ensemble systems even more.The results in table13 also show that all four intelligent optimization algorithms can achieve better generalization performance while significantly reducing the ensemble size.Although the ensemble size obtained by INGO is not the smallest, it does achieve better classification performance compared to other intelligent algorithms.HBA and GWO do not improve the classification capability of the ensemble system as much as INGO, which is due to HBA and GWO eliminate too many basic classifiers, which reduces the diversity of the ensemble system and leads to poor generalization ability.In general, INGO's overall performance in optimizing ensemble systems is better.At the same time, more than 85% of cart trees can be eliminated, which means that we can use fewer classifiers to design ensemble systems with better performance and save many computing resources in practical applications.

VI. CONCLUSION
To improve INGO's performance, this article proposes three strategies.Improved tent mapping was used to enhance the diversity of the initial population.The initial phase of the NGO implemented a levy flight approach to expand the solution's search scope and prevent the algorithm from converging too soon.A nonlinear convergence factor and a heart-shaped search approach enhance the algorithm's capacity to leap out of local extreme values.To test the ability of the improved algorithm, we designed a four-part experiment.In the first part, we use 23 commonly used benchmark test functions to evaluate the effectiveness of the three improvement strategies.The experimental results of this part show that the single use of one improvement strategy has a weak improvement effect on the original algorithm, while the simultaneous use of three strategies can significantly improve the performance of the original algorithm.better search performance without increasing computational complexity, which is mainly reflected in faster convergence speed, higher convergence accuracy, and stronger global search capability.However, INGO also has some limitations.For example, INGO has absolute advantages on unimodal functions, but its performance on multi-modal functions is not outstanding.Compared with some classical two-stage algorithms, the NGO algorithm uses more computational resources, and the INGO algorithm does not propose an effective solution to this problem.In general, INGO significantly improves NGO algorithms while retaining the advantages of NGO algorithms.Finally, INGO was used to optimize the ensemble system and the optimized ensemble system was evaluated on several common UCI datasets.The experimental results show that the ensemble system optimized by INGO uses fewer base classifiers to achieve higher prediction accuracy.
In the future, we will make improvements based on the limitations of INGO to reduce the consumption of computing resources without diminishing search performance.The

FIGURE 2 .
FIGURE 2. The curves of convergence factor.
Concurrently, introducing the best individuals can prevent the algorithm from falling into a partially sighted search state.On the other hand, the heart-shaped search model and the northern goshawk chasing prey model can make up for their respective shortcomings and play their respective advantages.In the heart-shaped search, once the food falls into the local optimal position, other individuals will converge toward the food, eventually leading to the algorithm's prematurity.The model of the northern goshawk chasing its prey is unaffected by the food to avoid other individuals approaching it.In the model of the northern goshawk chasing prey, once an individual is trapped in the local optimal position, it is difficult to solve the dilemma only by slight movement near the individual position, especially in the late iteration of the algorithm.It can be found from Eq(13) that the heart-shaped search position update is not only limited to the individual position but also affected by the food source, which can help an individual in trouble get out of the dilemma.Using random variables to adjust the two search methods dynamically can give full play to their respective advantages and improve the algorithm's performance.D. THE PROPOSAL OF INGO This is the procedure for the INGO algorithm: step1: Initialize parameters: population size N, max_ iteration T, ub and lb; step2: Initialize the population using tent mapping.step3: Calculate the fitness value and randomly select a prey for each northern goshawk; step3: If rand<0.6,use Eq (2) to update the individual location.Otherwise, use Eq (8) to calculate the individual location; step4: Update the location of the individual according to Eq(3); step5: Calculating the convergence factor; step6: Calculate the adaptation value of the current population and select the best one as the current food source; step7: If rand<0.5, use Eq (12) to calculate the individual location.Otherwise, use Eq (13) to calculate the individual location; step8: Update the location of the individual according to (6); step9: Update the best candidate solution and determine whether the maxi_iterations has been reached.If it has been reached, output the best candidate solution; otherwise, return to step3.

TABLE 6 .
Comparison of test results on standard benchmark functions.(The two rows of each function are the mean and standard deviation, respectively.)INGO's convergence curve on f 12 f 13 shows a continuous downward trend, indicating that INGO has greater potential to jump out of local extreme values.In general, INGO has a significant advantage over competing algorithms regarding convergence speed.At the same time, INGO's convergence curve on multimodal functions also shows its great potential to jump out of local extreme values.

FIGURE 6 .
FIGURE 6.The classification accuracy of different sizes.
In the second part, we use INGO to solve 23 common benchmark functions and compare it with 6 classical meta-heuristic algorithms.The results show that INGO has an absolute advantage on unimodal functions.Overall, INGO achieves global optimality on 56.5% (13/23) functions and outperforms competing algorithms on 78.3% (18/23) functions.
Wilcoxon test is used to detect the differences between various algorithms, and the results show that INGO is significantly different from competitive algorithms, that is, INGO has excellent optimization performance.The average convergence curve can also reflect the effect of different algorithms.Compared to other classical algorithms, INGO's curve drops rapidly.This is because the introduction of chaotic mapping makes the whole population more evenly distributed in the solution space, and the introduction of heart-shaped search enables the population to fully learn the information of dominant individuals so that individuals can determine the appropriate search direction and quickly move to the optimal solution.It can be seen from the convergence curve that theINGO  algorithm can effectively avoid search stagnation, which reflects that levy flight helps the algorithm expand the search scope and avoid local optimality.To further demonstrate INGO's superiority, we use INGO to solve the CEC2017 test suite in the third part of the experiment and compare it with six recently proposed metaheuristic algorithms.The results of the test show that INGO outperforms the NGO algorithm on 79.3% of the functions, and INGO has the lowest average ranking among the 7 comparison algorithms, which means INGO has superior optimization ability.Finally, INGO is evaluated using three high-intensity constraint problems, and the test results show that INGO's algorithm remains competitive for complex real-world optimization problems.The four-part experiments fully demonstrate that INGO performs better than competing algorithms and can effectively solve complex problems in the search space.Compared with NGO, INGO achieves

TABLE 1 .
Parameter setting.groups:f 1 − f 7 (unimodal) are used to evaluate the local development ability of the algorithm.f8−f13(multimodal) are used to test the algorithm's ability to eliminate the local optimal and find a global optimal position.f14−f23 (Fixed-dimensional multimodal) are used to evaluate the equilibrium of algorithm exploration and development.For a fair comparison, each intelligent optimization algorithm operates independently 30 times, maintaining a population of 30 and capping the iteration count at 500.Table1and table2give detailed descriptions of the algorithm's parameters and the test functions.
listed test results.As shown in table 4 and Table 5, INGOs with a complete improvement strategy performed best overall.According to the test results in Table 4, INGO performs better on 73.9% (17/23) functions than NGO.It serves as well as an NGO on 21.7% (5/23) of the functions.INGO also performs slightly weaker only on f 5 and f 20 compared to the other three derived algorithms.Further digging into the information in Table , INGO shows better search capability for single-peak test functions than other algorithms.For f 8 − f 13 , the INGO algorithm gets the optimal solution on f 9 and f 11 .The INGO algorithm performs well on f 10 and f 12 .Unfortunately, INGO performs poorly on f 8 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

TABLE 3 .
NGO variants with different improvement strategies.andhas a significant gap with HHO on f 13 .In general, INGO performs poorly on multimodal functions but is second only to HHO among the six compared algorithms.The results reveal that INGO has room for further improvement.For the fixed-dimensional functions f 14 − f 23 , INGO obtains optimal solutions on seven of them.For f 15 , INGO performs best.For f 19 , the PSO algorithm performed best, and the INGO algorithm followed the optimal result.Among the ten fixed-dimension functions, INGO only performs worst on f 20 .From a standard deviation perspective, INGO has a smaller standard deviation on 82.6% (19/23) of the functions, showing the most stable results on the largest number of functions compared to competing algorithms.Compared with

TABLE 4 .
Comparison of test results.(The two rows of each function are the mean and standard deviation, respectively.)

TABLE 5 .
Results of the Friedman test for the NGO variant.

TABLE 9 .
Comparison results of the INGO for T/CSD.

TABLE 10 .
Comparison results of the INGO for PVD.

TABLE 11 .
Comparison results of the INGO for T-bTD.

TABLE 13 .
Different intelligent algorithms generate classification accuracy (mean±std) and ensemble size (mean) (* indicates that the statistical test result is less than 0.05).