Improved Grasshopper Algorithm Based on Gravity Search Operator and Pigeon Colony Landmark Operator

The grasshopper optimization algorithm (GOA) is a new meta-heuristic algorithm inspired by the behavior of grasshopper groups. Aiming at the shortcomings of poor development ability and low convergence accuracy of GOA, this paper introduces the gravity search operator into the optimization process of GOA to improve the grasshopper’s global exploration and avoid falling into local optimum in advance. At the same time, a pigeon search operator-landmark operator is introduced to improve and balance the algorithm’s exploration and development capabilities. In order to verify the validity of the improved algorithm, this paper will adopts the gravity search operator and a deterrent landmark operator hybrid grasshoppers algorithm (HGOA) with basic grasshopper algorithm (GOA), particle swarm optimization (PSO) algorithm, sine and cosine algorithm (SCA), moth-flame optimization (MFO) algorithm, salp swarm algorithm (SSA), and bat algorithm (BA) to optimize 28 test functions. And the analysis and comparison of the obtained statistical data results finally show that the proposed improved grasshopper algorithm has better optimization ability.


I. INTRODUCTION
Optimization is the process of finding the best solution for a particular problem. As the complexity of the problem increases, the need for new optimization techniques has become more apparent in the past few decades than before. Before heuristic optimization was proposed, mathematical optimization was the only tool used to optimize problems. Mathematical optimization methods are mainly deterministic and have a major problem: local optimal notch waves, which makes them extremely inefficient in solving practical problems. Natural heuristic algorithms can effectively help solve many optimization problems encountered in actual processes, which makes natural heuristic algorithms a research hotspot for optimization problems in recent years [1]. Natural heuristic algorithm includes heuristic algorithm and meta-heuristic algorithm. Heuristic algorithms are the process of finding The associate editor coordinating the review of this manuscript and approving it for publication was Shaojun Wang.
solutions through trial and error [2]. Many heuristic algorithms, such as simulated annealing (SA) and artificial neural network (ANN), have been widely used in the field of optics and combination optimization. The meta-heuristic algorithm is a stochastic optimization algorithm with prior knowledge of random search. It is an optimization process that starts with a random solution and then randomly explores and utilizes the search space with a specific probability. These algorithms can use the useful information of the population to find the optimal solution [3]. These efficient and robust algorithms are used to solve a variety of problems, such as path planning [4], economic scheduling problem [5], inverter parameter identification [6], backpack problem [7] and location problem [8]. So far, various researchers have conducted in-depth research on these algorithms, and introduced many naturally-inspired meta-heuristic algorithms, such as particle swarm optimization (PSO) algorithm [9], bacterial foraging algorithm (BFA) [10], artificial fish swarm algorithm (AFSA) [11], artificial bee colony (ABC) algorithm [12], cuckoo search (CS) VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ algorithm [13], bat algorithm (BA) [14], ant lion optimizer (ALO) [15], moth-flame optimization (MFO) algorithm [16], and salp swarm algorithm (SSA) [17], etc. Grasshopper algorithm is a new meta-heuristic algorithm proposed by Saremi et al. in 2017 [18], which is inspired by the swarm behavior of locusts. Since the algorithm was proposed, it has been widely used in many fields, such as optimal power flow [19], feature selection [20], financial stress prediction [21], and image segmentation [22]. In fact, this swarm based meta-heuristic algorithm avoids the stagnation of local optimality to some extent, but it is found in the test function that GOA has the disadvantages of poor global search ability and slow convergence speed. Therefore, in some cases, GOA cannot find a global optimal solution. The search strategy used in the base GOA is primarily based on minimums and random walks, so it cannot always successfully solve the optimization problems. Thus many researches focus on improving the performance of meta-heuristic algorithms by promoting exploration and exploitation.
Wu et al. [23] introduced natural selection strategy and democratic decision-making mechanism into GOA to avoid falling into local optimization. In addition, based on the dynamic feedback mechanism, the 1/5 principle is also introduced to adjust the parameter c to better balance global and local search. Liu et al. [24] proposed an integration strategy combining linear weighting and grasshopper optimization algorithm to solve the problem of energy management, improve the comprehensive energy efficiency and realize the regional coordination optimization. Ewees et al. [25] proposed an improved GOA. The proposed OBLGOA algorithm includes two stages: the first stage uses the OBL strategy to generate the initial population, and the second stage uses OBL as an additional stage to update the GOA population in each iteration. Luo et al. [21] combined three strategies to achieve a more appropriate balance between exploration and exploitation. The Gaussian mutation was adopted to increase population diversity and make GOA have stronger exploitation capabilities. Then, Levy flight was used to enhance the randomness of the search agent's movement and make GOA have a stronger global exploration ability. In addition, the opposition-based learning was introduced into GOA to achieve a more efficient search solution space. Arora and Anand [3] introduced chaos theory into the optimization process of GOA, which improved the global convergence speed of GOA. In the optimization process, the use of chaotic maps effectively balanced the exploration and exploitation among locusts, and reduced the repulsion and attraction between locusts. Sulaiman et al. [26] updated GOA with a better initialization strategy to balance GOA's search capabilities. Yan and Ye [27] proposed a hybrid locust optimization algorithm based on the idea of quantum computing in order to overcome the shortcomings of the algorithm with low convergence accuracy and used quantum revolving gate operation to improve it. Liang et al. [28] used Levy flight to balance the exploration and development of GOA so as to improve the convergence speed and accuracy of the algorithm. Taher et al. [19] applied random mutations of GOA to enhance the new search area to avoid local optimal value stagnation. Jia et al. [22] proposed an effective satellite image segmentation method based on hybrid GOA and minimum cross entropy (MCE), and they combined GOA with adaptive differential evolution (DE) to improve search efficiency, especially to preserve population diversity in subsequent iterations. An improved grasshopper algorithm based on gravity search operator and pigeon colony landmark operator was proposed. Firstly, this paper introduces the gravity search operator (GGOA) on the basis of the original GOA, which improves the randomness of the algorithm, increases the global search ability of the algorithm, and avoids falling into local optimization. On this basis, pigeon landmark operator is added to improve and balance the exploration and exploitation ability of the algorithm. The selection strategy is adopted in the location update strategy, which further increases the selection diversity of the algorithm and improves the convergence accuracy of the algorithm.

II. BASIC PRINCIPLES OF GRASSHOPPER ALGORITHM
The grasshopper optimization algorithm is inspired by the foraging behavior of grasshoppers [18]. Grasshoppers are a type of insect that are usually seen alone, but they are one of the largest groups of organisms. The life cycle of grasshopper is divided into two main stages: larval stage and adult stage. The larval stage is characterized by the slow movement and small movements of locusts. The adult stage is characterized by a large range of sudden movements. Similarly, naturally inspired algorithms logically divide the search process into two trends: exploration and exploitation. In the exploration phase, grasshoppers tend to move quickly, while in the exploitation phase, they are encouraged to move locally. Both of these functions, as well as finding food sources, are naturally realized by grasshoppers. Theoretically speaking, the swarm behavior of grasshoppers can be summarized as: where, X i represents the location of the i − th grasshopper, S i represents social interaction, G i represents the gravity of the i − th grasshopper, and A i represents wind advection. The S component in Eq.
(1) can be calculated by: where, d ij represents the distance between the i−th grasshopper and the j − th grasshopper, and the expression is d ij = x j − x i . s is the strength of the social force, and it is calculated by using Eq. (3). The unit vector between the i − th grasshopper and the j − th grasshopper is calculated in terms ofd ij = (x l − x i ) d ij . The social power is defined by the s function, which can be calculated by:  [1,4].
where, f is the intensity of attraction and l is the scale of attraction length. According to literature [18], when f = 0.5, l = 1.5, the algorithm has best optimization performance. Fig. 1 shows the influence of s function on grasshopper social interaction (attraction and exclusion). It can be seen form Fig. 1 that a distance from 0 to 15 is considered. Rejection occurs in the interval [0, 2.079]. When one grasshopper is 2.079 units away from another, it is neither attractive nor repulsive. This is called a comfort zone or comfort distance. Fig. 1 also shows that the attraction increases from 2.079 to close to 4 and then decreases gradually. The component G in Eq. (1) can be calculated by: where, g stands for the gravitational constant, andê g is the unit vector pointing to the center of the earth. The component A in Eq. (1) can be calculated by Eq. (5).
where, u represents the constant drift, andê w is the unit vector of the wind direction. Substitute S, G and A into Eq. (1) to obtain: (6) where, N represents the number of grasshoppers.
In the optimization algorithm, Eq. (6) is not used because it prevents the optimization algorithm from exploration and exploitation the searching space near the solution [18]. In fact, this grasshopper model is designed for colonies of grasshoppers that live in free space. In addition, the mathematical model is not directly used to solve the optimization problem because the grasshopper quickly reaches the comfort zone, while the grasshopper does not converge to the specified point. Therefore, the modified Eq. (6) is used to solve the optimization problem.
where, ub d and lb d represent the upper and lower limits respectively.T d represents the optimal position. c 1 and c 2 represent coefficients of contraction comfort zone, exclusion zone and attractiveness. It can be seen from Eq. (7) that the location of the grasshopper next time is determined according to its current position, target position and the location of all other grasshoppers. Eq. (7) consists of two parts: the first part considers the location of the current grasshopper relative to other grasshoppers in the region; The second part reduced the movement of grasshoppers around the target, allowing for the exploration of the entire population around the target. Specifically, the parameter c 1 is responsible for reducing the movement of locusts around the target, which means that c 1 basically balances the exploration and exploitation of the entire population around the target; The parameter c 2 reduces the attraction zone, comfort zone and exclusion zone between grasshoppers, that is to say that c 2 linearly reduces the space to guide grasshoppers to find the optimal solution in the search space. It is worth noting that the adaptive parameter c 1 helps reduce the repulsion or attraction between grasshoppers, which is proportional to the number of iterations, while c 2 reduces the search range around the target as the number of iterations increases. In order to balance exploration and exploitation, c 1 decreases proportionally as the number of iterations increases [3]. Similarly, as the number of iterations increases, the value of c 2 decreases in order to minimize the comfort zone. In [18], the parameters (c 1 and c 2 ) are considered as a single parameter and modified using the following methods: where, c max and c min represent the maximum and minimum values, t represents the current number of iterations, and T is the maximum number of iterations. The algorithm flowchart of the grasshopper algorithm is shown in Fig. 2. The pseudocode of the grasshopper algorithm is described as follows [18]. Initialize the swarm X i (i = 1, 2, · · · , n) Initialize c max , c min and maximum number of iterations Calculate the fitness of each search agent Target = the best search agent while (l<Max number of iterations) Update c using Eq.(8) for each search agent Normalize the distances between grasshoppers in [1,4] Update the position of the current search agent by the Eq. (7) Bring Inspired by the law of universal gravitation in physics, Rashedi et al. [29] proposed the Gravitational Search Algorithm (GSA) in 2009. GSA believes that each particle in the search space has a mass, and can actually perform undamped objects, and the objects are attracted to each other by gravitation. The law of motion follows Newton's second law, so that it moves toward the optimal position under the action of this law movement [30]. The standard gravitational search algorithm is described as follows. Suppose the dimension of the search space is D, the population size is N , and the initial population is N is the position of the i − th particle, and x k i represents the position of the i − th particle on the k − th dimension.
(2) The gravity of particle j on the k − th dimension of particle i can be defined as: where, Mass i (t) and Mass j (t) are the inertial masses of particles i and j respectively. ε is a very small constant, R ij (t) is the Euclidean distance between particles i and j, and G(t) is the universal gravitational constant at time t, which decreases with the age of the universe and is calculated by: where, G(0) is the universal gravitational constant at time t = 0, α is the attenuation coefficient, and T is the maximum number of iterations.
(3) The resultant force of particle i in the k − th dimension can be calculated by: where, rand j is the random number within the interval [0,1], K best is the document storing the elite particle of the population and its scale decreases linearly with the number of iterations. According to Newton's second law, at time t, the acceleration of particle i in the k − th dimension can be defined as: Therefore, the velocity and position of the particle at time t + 1 are respectively described as: x k

B. PIGEON-INSPIRED ALGORITHM
Pigeon-inspired optimization (PIO) algorithm is a bionic intelligent optimization algorithm proposed by Duan et al in 2014 [31]. Each pigeon in PIO algorithm corresponds to a feasible solution, which has two attributes of position and speed, which respectively represents the current position and moving speed of the pigeon in the solution space, and evaluates the quality of the pigeon through the fitness function.
The PIO algorithm uses a compass operator and a landmark operator to imitate navigation tools in different flight phases, and completes evolution and screening operations in two independent loops to achieve optimization. The landmark operator can be described as: where, R is the map and guide factor . t represents the current number of iterations. X i (t) and V i (t) represent the position and velocity of pigeon i in generation t, X gbest represents the global best position obtained by comparing the positions of all pigeons after the iteration t − 1, and rand is a random number on the scope [0,1]. The landmark operator is used to imitate the role of landmarks in individual pigeon flock evaluation and correction when the pigeons are flying close to the destination [31]. If the pigeons are familiar with landmarks, they can fly straight to their destination. Conversely, if the pigeons are unfamiliar with the landmark and stay away from the destination, they will follow the familiar pigeon to reach the destination. In the landmark operator, the number of pigeons in each generation is halved by Np. However, the pigeons are still far from their destination and they are not familiar with landmarks. Let X c (t) be the center of a pigeon's position at the t −th iteration, assuming that each pigeon can fly directly to its destination. The position update rule of pigeon i at the t − th iteration can be described as follows. (20) where, fitness() represents the fitness value of pigeon.

C. IMPROVED GRASSHOPPER ALGORITHM
According to Eq. (7), after a given initial population, the search individual will converge to the target location, and continuously adjust the location according to the influence of interaction forces during the convergence process, and finally make the population more evenly distributed around the target location. This unique way of interacting with other search individuals is a significant feature that distinguishes it from other algorithms and allows the algorithm to have better exploitation capabilities. However, this remarkable feature also leads to the shortcomings of the basic locust optimization algorithm, such as rapid population diversity decline and insufficient global search ability, which leads to the algorithm falling into local optimization and insufficient convergence precision. In order to improve the performance of the algorithm, the acceleration in gravity search is introduced to enhance the global search ability, which is described as follows.
where, a d i is the acceleration in the gravitational search. The purpose of a d i is to increase the randomness of grasshopper movement, improve grasshopper's global exploration and avoid falling into local optimization. Suppose indicates the new position of the grasshopper formed by the effect of the force between grasshoppers. Eq. (7) can be simplified as: At the same time, in order to improve the convergence speed of the algorithm, this paper introduces the search operator in the pigeon algorithm after introducing the gravity search operator, which is described as follows.
where, R represents the map and guide factor. The research results of literature show that the smaller R is, the stronger the search ability is and the greater the development potential is [45]. In order to solve the balanced development problem, the linear variation strategy is adopted in this paper, and the calculation formula is defined as: where, R max represents the maximum map compass factor value, R min represents the minimum map compass factor value, and p r represents the probability of variation. VOLUME 8, 2020 Therefore, the operation flow chart of the improved grasshopper algorithm is shown in Fig.3, and the pseudocode is described as follows: Initialize the swarm X i (i = 1, 2, · · · , n) Initialize c max , c min , R max , R min , G(0), α and maximum number of iterations T Calculate the fitness of each search agent Target = the best search agent while (l < T ) Update c using Eq. (8) for each search agent Normalize the distances between grasshoppers in [1,4] if rand>0. The computational complexity of the algorithm is an important index to evaluate its running time, which can be defined according to the structure and implementation of the algorithm. This section uses the big oh notation, which is the worst-case time complexity. The time complexity of HWOA depends on the number of grasshoppers, the dimension of the variable, the maximum number of iterations, and the location-updating mechanism of grasshoppers in each iteration. The calculation of the specific time complexity is as follows: x To initialize and assign the value of n grasshoppers in d-dimensional search space, dn times should be run.
y Computing the fitness values of each grasshoppers, and inertial mass to run n times, and selecting the optimal fitness value needs to run n(n−1) 2 times. z To calculate the distance between grasshoppers and normalize it to [1,4] requires running n(n−1) 2 times. { Updates of parameters c, R and G need to be run once, respectively.
| According to Eq.13, the acceleration of n grasshoppers in the d dimension needs to be run (n(n − 1)d) times.
} According to Eq.23 or Eq.22, updating the position of the grasshopper requires running n 2 times.
The optimal value and the judgment algorithm need to run once respectively.
In summary, the time complexity of HWOA is simplified as:

IV. SIMULATION EXPERIMENTS AND RESULT ANALYSIS ON FUNCTION OPTIMIZATION PROBLEMS A. TEST FUNCTIONS
In order to verify the performance of the proposed HGOA in this section, 28 most widely used benchmark functions are adopted for evaluation. The expressions of the 28 benchmark functions are shown in Table 1. In order to prove the superiority of the improved algorithm from multiple angles, the 28 functions are divided into three groups. F1 ∼ F12 are unimodal functions [33]- [35], that is to say that there is only one global optimal solution in the solution space of the function, so these unimodal functions are used to test the convergence rate of the algorithm. F13 ∼ F22 are multimodal functions [35], [36]. There is more than one extremum in multimodal function, so there are many local optima in multimodal function, so the algorithm is easy to fall into local optima. The test results of multimodal function can well show the function optimization ability of the algorithm. F23 ∼ F28 are combination functions [33], which are formed by rotation, displacement and offset of various benchmark functions. Such functions have low dimensions and a small number of local optimal values [36]. Both unimodal and multimodal functions are tested in 30 dimensions, and composite functions are tested in their own dimensions.

B. SIMULATION EXPERIMENTS AND PERFORMANCE ANALYSIS WITH OTHER SWARM INTELLIGENT ALGORITHMS 1) TEST FUNCTION RESULTS AND ANALYSIS
In order to verify the performance of the improved algorithm, this paper compares the improved hybrid grasshopper algorithm (HGOA) with the original grasshopper algorithm (GOA), particle swarm optimization (PSO) algorithm, sine cosine algorithm (SCA), moth flame optimization (MFO) algorithm, salp swarm algorithm (SSA) and bat algorithm (BA). The parameter settings of each algorithm are shown in Table 2 . The simulation results of 28 test functions are shown in Fig.4, and the performance comparison simulation results under 10 runs are shown in Table 4. The optimal value, mean value and variance of the simulation results for 10 times are given in Table 3 , in which bold data is the best result. From the above experimental results, we can see that the function optimization ability of HGOA is better than other optimization algorithms. For the test results of the unimodal functions F1 ∼ F12, HGOA algorithm gets the best results. In most single-mode test functions, the performance of the improved algorithm has been significantly improved, showing the high performance of the algorithm. The single-modal test function has only one global optimum, and the results show that HGOA has a high exploitation ability. It can be seen from the data table of test results, HGOA greatly improves the accuracy of optimization results. In function F 10 , HGOA improves the accuracy by 58% compared with GOA, and in function F 8 , it improves the accuracy by 37%. However, according to the optimal convergence curve, it can be seen that although HGOA effectively improves the convergence accuracy of the algorithm, for some functions, its convergence speed needs to be further improved. For the multimodal functions F13 ∼ F22, the optimization results of HGOA are all near the optimal value. For the function F13, the optimal value is basically found. In other functions, the optimization accuracy is also improved compared with other algorithms in different degrees. Because there are a large number of local solutions in the multi-modal test function, these results quantitatively show the effectiveness of the algorithm to avoid local solutions in the optimization process. For the test function with fixed dimension, HGOA shows a good optimization effect. Except for the test function F25, the optimal value of the function can be found, and the convergence speed of the algorithm can be improved to a large extent. At the same time, it can be seen from the difference column of the test data that HGOA is very stable in the 10 times of optimization process. The fixed dimension function is more challenging than the multimodal test function, which requires a proper balance between expansion and development. Therefore, it can be said that HGOA can well balance exploration and development and solve such challenging problems. According to all the test results, HGOA has been improved in most cases in terms of convergence speed, and has shown good optimization effect in terms of convergence accuracy. Especially, in the fixed dimension test functions, the algorithm's robustness is improved while ensuring the convergence speed and optimization accuracy.

2) NONPARAMETRIC TEST ANALYSIS
In order to verify the stability of the algorithm, the average value of each algorithm has been carried out the nonparametric tests. In this work, Wilcoxon rank sum test is used as nonparametric statistical test to determine the importance of the results. Table 4 depicts the 5% p value obtained from this test. It can be seen from Table 4 that the p-value of unimodal functions and multimodal functions are far less than 0.05, highlighting the significant advantages of HGOA compared with other algorithms based on p-value (less than 0.05).

C. SIMULATION EXPERIMENTS AND PERFORMANCE ANALYSIS OF IMPROVED ALGORITHM
In order to more accurately express the influence of gravity search operator and pigeon search operator on the algorithm, this section carries out simulation tests on the basic grasshopper algorithm (GOA), the grasshopper algorithm (GGOA) with gravity search operator and the grasshopper algorithm VOLUME 8, 2020 (PGOA) with pigeon search operator. In the simulation experiment, it was found that both GGOA and PGOA had different degrees of improvement in convergence speed and convergence accuracy, so the curves and data with relatively obvious effect were given. Partial convergence curves were shown in Fig.5, and the convergence results were shown in Table 5. It can be seen from the convergence curve that for unimodal functions, the influence of GGOA and PGOA on the convergence speed of the algorithm is not obvious. For multimodal functions, the convergence speeds of GGOA and PGOA have been improved to varying degrees, especially the effect of PGOA is the most obvious. It can be seen from the table 5 that PGOA has greatly improved the convergence accuracy of the algorithm, especially by 40 orders of magnitude in both functions F8 and F9. At the same time, combined with Fig.4 and Table 3, it can be found that HGOA has further improved convergence speed and accuracy compared with GGOA and PGOA.

V. SIMULATION AND RESULT ANALYSIS OF PRESSURE VESSEL DESIGN PROBLEM
In recent years, using stochastic optimization technology to solve structural design problems is a research hotspot in the field of structural design [37], [38]. In order to further verify   the effectiveness of the improved algorithm, the design of pressure vessel is optimized in this section. The objective of pressure vessel design problem is to minimize the total cost of material, forming and welding of a cylindrical vessel. Fig. 6 shows the pressure vessel and parameters involved in the design [39], [40]. In Fig. 6, T s represents the thickness of the shell, T h represents the thickness of the head, R represents the inner radius, and L represents the length of the cylindrical section without considering the head. The problem has four constraints, which are described as follows.
Consider  The algorithm used to test this problem are the algorithms for test function in Section 4.2. The convergence curve of each algorithm is shown in Fig. 7. The optimal value and corresponding variable value are shown in Table 6. In order to test the stability of the algorithm, run the problem 10 times, and obtain the optimal value, average value and variance as shown in Table 7. Obviously, the inspection algorithm showed that HGOA showed very competitive results compared with GOA, PSO, SCA, MFO, SSA and BA in this problem, and had better optimization ability. Moreover, according to the variance data in the table, the robustness of the algorithm is obviously better than other algorithms. Because the search space of this problem is unknown, these results provide strong evidence for the applicability of HGOA in solving practical problems. In addition, due to the limitations of case studies, it can be said that HGOA algorithm can also optimize the search space with infeasible areas. The results show that HGOA algorithm can effectively deal with the problem of limited search space.

VI. CONCLUSION
Grasshopper algorithm, as a new kind of biologicallyinspired algorithm, can avoid the local optimal stagnation to some extent. However, under certain circumstances, grasshopper algorithm always fails to show the global search ability and solve the problem successfully. This paper firstly introduces the gravity search operator, which greatly improves the global search capability of GOA, and then introduces the pigeon landmark operator, which well improves and balances the exploration and development capability of the algorithm. In order to verify the effectiveness of the improved algorithm, 28 test functions were used for testing. Among them, the unimodal function indicates that HGOA has a higher mining ability; the multimodal function indicates the effectiveness of HGOA in avoiding local solutions in the optimization process; the fixed-dimensional function indicates that HGOA can balance exploration and development well. At the same time, in order to test the influence of the two operators on the improved algorithm, the grasshopper algorithm (GGOA) with only gravity search operator, the grasshopper algorithm (PGOA) with only pigeon landmark search operator and the original grasshopper algorithm (GOA) are compared. The experimental results show that PGOA plays a great role in the convergence speed and accuracy. However, combined with the data of HGOA, it can be found that HGOA has better optimization results than GGOA and PGOA. Therefore, while increasing the exploration ability of the algorithm, the balance between the exploration and development ability of the algorithm must be ensured to achieve the optimal effect of the algorithm. Of course, in the convergence curve of some functions, it can be seen that the convergence speed of HGOA is slow, so the convergence speed of the algorithm needs to be further studied in the next study. This paper also applies the improved HGOA to the design of pressure vessels. Experiment shows that HGOA has a good performance in solving the problem of unknown search space. In the future study, we should pay attention to improving the global search ability of the algorithm, while maintaining the balance between the exploration and development ability of the algorithm, and apply the idea to other algorithms to improve the optimization effect of the algorithm.