High-dimensional function solving method and system based on distance decision wolf pack algorithm

Abstract

The invention belongs to but is not limited to the technical field of big data, and particularly relates to a high-dimensional function solving method and system based on a distance decision wolf pack algorithm. The improved algorithm realizes significant optimization in the aspects of migration behavior, call behavior, purse behavior, population update and the like. Experimental results show that the improved wolf pack algorithm is obviously improved in solving precision in a low-dimensional single-peak function compared with an original algorithm, and meanwhile, the solving precision of a high-dimensional multi-peak function is further improved. In addition, the algorithm also has the characteristics of good global convergence, high operation speed and the like, and is excellent in performance on a plurality of standard test functions. In the theoretical level, the convergence of the algorithm is deeply analyzed by using the non-aftereffect characteristic of the Markov chain, and the population sequence of the improved wolf pack algorithm is proved to be traversal through mathematical derivation, so that the algorithm can be converged to the global optimal solution of the problem to be solved with the probability 1.

Description

Technical Field

[0001] The present invention belongs to, but is not limited to, the field of big data technology, and particularly relates to a method and system for solving high-dimensional functions based on a distance-determined wolf pack algorithm. Background Art

[0002] The rapid development of big data science and technology has led to the continuous expansion of the scale of data to be processed. In the field of engineering applications, large-scale optimization problems are often the most common problems, but with the increase of dimensions, the difficulty of solving them increases exponentially. Most of the previous optimization algorithms can no longer adapt to the needs of solving problems, and swarm intelligence optimization algorithms have developed accordingly. Swarm intelligence optimization algorithm is an optimization algorithm based on group behavior patterns, which solves complex problems by simulating the interaction and information exchange in natural groups. Its basic principle comes from the group behavior of various organisms in nature. These organisms can show complex group behavior characteristics through the interaction and information exchange between individuals during foraging and migration. The algorithm simulates these behaviors and transforms complex problems into group search and optimization problems, thereby finding the optimal solution or approximate optimal solution. The most common ones include: particle swarm optimization algorithm (PSO), ant colony optimization algorithm (ACO), and fish swarm algorithm. In addition, there are many intelligent optimization algorithms such as artificial bee colony algorithm (ABC), differential evolution algorithm (DE), gravitational search algorithm (GSA), firefly algorithm (FA), bat algorithm (BA), cuckoo optimization algorithm (COA), gray wolf optimization algorithm (GWO), and whale optimization algorithm (WOA). Wu Husheng et al. analyzed the characteristics of wolf packs' collaborative hunting activities and prey distribution, abstracted three kinds of artificial wolves, three kinds of intelligent behaviors (probe wolf wandering behavior, alpha wolf calling behavior, fierce wolf siege behavior) and two kinds of intelligent rules (the "winner is king" alpha wolf competition rule and the "survival of the fittest" wolf pack update rule), and proposed a wolf pack algorithm (WPA) with a different optimization strategy from the WCA algorithm, and proved the convergence of the algorithm based on Markov chain theory. The WPA algorithm is a new intelligent group optimization algorithm proposed based on the wisdom of wolf pack clusters. Since the algorithm has good performance in global search and local development capabilities, it has continuously attracted the attention of scholars at home and abroad since its proposal, and has been quickly applied to practical engineering. Yi Tinghua et al. used the wolf pack algorithm to solve the problem of sensor optimization layout; Wu Husheng et al. used the wolf pack algorithm to solve the binary knapsack problem and the TSP problem; Fang Yanjun et al. used the wolf pack algorithm to solve the three-dimensional space path optimization problem of the car access system; Wang Jianqun et al. used the wolf pack algorithm to solve the problem of optimal dispatching of the reservoir of the hydropower station; Liu Yonglan et al. used the wolf pack algorithm to solve the path planning problem of the UAV. The WPA algorithm is based on the system thinking, the division of responsibilities and cooperation thinking, the "coarse to fine" progressive thinking and the guided random thinking. This is different from the optimization mechanism of intelligent bionic optimization algorithms such as particle swarm optimization algorithm (PSO), ant colony optimization algorithm (ACO), genetic algorithm (GA), artificial bee colony algorithm (ABC), artificial fish swarm algorithm (AFSA). The literature shows that the algorithm has a significant effect on the processing of multi-peak and high-dimensional complex functions. However, the algorithm still has some shortcomings, such as the excessive greed and "rigid" wandering mode of the scout wolf, which easily makes the scout wolf fall into the embarrassing position of the local optimum, and the late siege effect of the fierce wolf is not ideal.

[0003] In view of the above analysis, the technical problems urgently to be solved in the prior art are as follows:

[0004] 1. Insufficient global convergence ability of the algorithm

[0005] Traditional optimization algorithms (such as genetic algorithms and particle swarm algorithms) are prone to falling into local optimal solutions when solving high-dimensional functions, lacking sufficient global optimization ability.

[0006] 2. Low search efficiency

[0007] In existing swarm optimization algorithms, the randomness of the individual distribution in the initialization stage is usually strong, and the early iteration efficiency is low, resulting in a slow overall running speed of the algorithm.

[0008] 3. Insufficient accuracy in high-dimensional search problems

[0009] In the high-dimensional solution space, the step size adjustment method of traditional algorithms is relatively rough, and it is easy to skip the optimal solution under the condition of a large step size, or waste a lot of time under the condition of a small step size.

[0010] 4. Lack of flexibility in multi-objective optimization

[0011] Most traditional algorithms cannot flexibly handle multi-objective optimization tasks in complex objective functions. Summary of the Invention

[0012] Aiming at the problems existing in the prior art, the present invention provides a method and system for solving high-dimensional functions based on a wolf pack algorithm determined by distance.

[0013] The present invention is implemented as follows. A method for solving high-dimensional functions based on a wolf pack algorithm determined by distance includes:

[0014] Step 1: First, in the N×D solution space, set the scale of the wolf pack and then randomly generate artificial wolves to initialize the wolf pack. At this time, the spatial position coordinates X of each artificial wolf i are all randomly generated, calculate the objective function value Y at the position of each artificial wolf i , select the artificial wolf with the largest function value as the head wolf, and select the R wolves with the second-best objective function value Y i as exploration wolves, and the rest as fierce wolves.

[0015] Step 2: The exploration wolves perform a wandering behavior to greedily search the solution space. Each exploration wolf will move forward a wandering step length in h random directions respectively, compare the objective function values corresponding to the front and back positions, and finally move forward in the direction with the largest objective function value. If the objective function value at the original position is the largest, it will return to the original position. After the exploration wolves finish wandering, compare the objective function values of all artificial wolves, and the wolf with the largest objective function value becomes the new head wolf. The position of the exploration wolf that ends wandering is calculated by the wandering formula, such as Equation (1)

[0016]

[0017] In the formula is the walking step length; d = 1, 2, ..., D; P = 1, 2, ..., h. Since wolves in nature have different search and walking abilities, the value of h is random. h is a random integer. Generally, the larger h is, the more directions the wolf can search and the better the global optimization is, but it will also increase the search time.

[0018] Step 3: The alpha wolf howls to summon the wolves to come closer. All wolves advance one galloping step toward the alpha wolf. Run towards the alpha wolf until the distance between the alpha wolf and the alpha wolf is less than the siege distance. During the run, if the alpha wolf finds a position with a larger target function value than the alpha wolf, it will replace it as the new alpha wolf and initiate the summoning behavior again. When all wolves reach the siege distance, the wolf pack switches to siege behavior. The position change of the alpha wolf during the run is as follows:

[0019]

[0020] In the formula This is the position of the wolf before the attack. is the current position of the alpha wolf, It is the position of the k+1th generation of fierce wolves after the raid.

[0021] Step 4: The wolf pack launches a siege on the prey, and regards the current leader as the prey. The fierce wolf and the scout wolf launch a siege on the prey and move forward one siege step in the direction of the prey. During the siege, if the objective function value corresponding to the current prey position is found to be better than that of the current prey position, the prey position is updated and the siege is launched again until the termination condition is reached. The position change of the artificial wolf during the siege is as follows:

[0022]

[0023] λ is a random number with a value between (0, 1), Indicates the siege step length.

[0024] Step 5: The wolf pack eliminates the R artificial wolves with the smallest objective function value according to the set population update scale factor, and randomly generates R new artificial wolves in the solution space to complete the population update. R is a random number whose value range is [N / (2β), N / β], and β is the update scale factor.

[0025] Step 6: Determine whether the accuracy requirement of the solution or the termination condition of the maximum number of iterations is met. If the termination condition is met, output the spatial position of the leader wolf, which is the optimal solution to the problem. Otherwise, continue to iterate until the termination condition is met.

[0026] Furthermore, there are three types of step lengths in the WPA, namely the wandering step length the rushing step length and the attacking step length. Suppose the value range of the variable in the d - dimensional space is [min d , max d , then the relationship of the three step lengths in the d - dimensional space is shown in Equation (4)

[0027]

[0028] In the above formula, S is the step - length factor, representing the fineness of the search.

[0029] Furthermore, after each wandering process ends, the wolves with relatively poor performance are eliminated and an equal number of new artificial wolves are generated to ensure the search accuracy, which can accelerate the running speed of the WPA and improve the global convergence. At the same time, to improve the wandering efficiency, the alpha - wolf selection is changed from being selected at the end of population initialization to being selected after the first wandering ends, avoiding the situation that the alpha - wolf updated in the early stage of the algorithm is usually only a local optimal solution, thus wasting computational time. After the population initialization ends, the exploring wolves start to wander and optimize around themselves. The basic wandering formula is

[0030]

[0031] where, is the current position of the exploring wolf, is the position of the exploring wolf after wandering, is the wandering step length, h is the wandering direction, p = 1, 2,....., h. After the exploring wolves wander in h directions respectively, they return to the original position, and finally choose to wander in the direction with the largest objective - function value after moving forward. If the original position is the best, the position remains unchanged.

[0032] After each wandering process ends, A will be updated to a non - zero row vector. At this time, the wolf with the largest objective - function value is selected as the alpha - wolf. The Q wolves with the smallest A(i) values will be eliminated and new exploring wolves will be generated. Q is a random number with a value in (1, n / a), and its calculation formula is

[0033] Q = floor(rand(1)*(n / a)) + 1 (6)

[0034] where, a is the exploring - wolf elimination factor, usually taking values in [16, 20]. Generally, the larger the wolf - pack size, the more exploring wolves are eliminated. By eliminating the "negatively working" artificial wolves during the wandering process, the search ability of the wolf - pack is improved, thereby accelerating the convergence speed of the algorithm and finding the global optimal solution faster.

[0035] Further, after the wandering ends, the alpha wolf summons the wolf pack to rush towards it. During this process, if it is found that the objective function value of any other wolf is greater than that of the alpha wolf, the alpha wolf is replaced and summoned again. The position update during the initial WPA rush is as shown in formula (7):

[0036]

[0037] Where and are the positions of the fierce wolves before and after the rush respectively, is the rush step size, is the position of the current alpha wolf.

[0038] Further, after the exploring wolves' wandering ends, the alpha wolf summons the wolf pack to rush towards it, regarding the alpha wolf as the prey. In the initial siege formula of WPA, the value range of λ is [-1, 1], which leads to the situation of returning to the original position many times during the siege process (when λ takes a negative value, is greater than When λ takes a positive value, is less than ), ultimately reducing the search efficiency. At the same time, in extreme cases, if the siege step size of the fierce wolves is too large, it is easy to jump out of the siege range, making it difficult for the wolf pack to search for the optimal value. Before the fierce wolves' rush operation starts, first determine the positional relationship between them and the alpha wolf. Since the search space is an N×D solution space, the position of the artificial wolf will be a D-dimensional coordinate. In each dimension, the value of λ is divided into two cases. If that is, it means when the fierce wolf does not exceed the position of the alpha wolf, λ takes a positive value, and the fierce wolf can always move towards the alpha wolf; conversely, if and the fierce wolf does not exceed the position of the alpha wolf, λ takes a negative value, and the fierce wolf can also always move towards the alpha wolf. At the same time, when the siege step size is too large and the fierce wolf exceeds the position of the alpha wolf, by judging the and size relationship, immediately adjust the advancing direction of the fierce wolf to continuously narrow the search range for the prey.

[0039] Further, introduce a distance determination factor for population update. When updating the wolf pack, retain the new artificial wolves that meet the conditions. The implementation process is as follows:

[0040] Step1: Record the original positions of the R wolves to be eliminated and the position of the current alpha wolf.

[0041] Step2: Randomly generate an R*m position matrix to update the positions of the artificial wolves.

[0042] Step 3: Determine two conditions: calculate the distance to the old position, and if it is greater than the distance determination factor, update the position; calculate the distance to the current leader, and if it is greater than the maximum distance between the current artificial wolf and the leader, update the position. Wolves that do not meet the conditions are randomly updated again until all R wolves are updated.

[0043] Step 4: The wolf pack is updated and the leader of the new wolf pack is selected.

[0044] Another object of the present invention is to provide a high-dimensional function solving system based on a distance-determined wolf pack algorithm for realizing the high-dimensional function solving method based on a distance-determined wolf pack algorithm, comprising:

[0045] Wolf pack initialization module: First, in the N×D solution space, set the size of the wolf pack and then randomly generate artificial wolves to initialize the wolf pack. At this time, the spatial position coordinates X of each artificial wolf i They are all randomly generated, and the objective function value Y of each artificial wolf's location is calculated i , select the artificial wolf with the largest function value as the leader, and set the objective function value Y i The next best R wolf is selected as the scout wolf, and the rest are fierce wolves.

[0046] Solution space search module: The scout wolf performs wandering behavior to greedily search the solution space. Each scout wolf will move forward in each of the random h directions for a wandering step length while comparing the objective function values ​​corresponding to the previous and next positions, and finally move forward in the direction with the largest objective function value. If the objective function value of the original position is the largest, it will return to the original position. After the scout wolf finishes wandering, the objective function values ​​of all artificial wolves are compared, and the wolf with the largest objective function value becomes the new leader. The position of the scout wolf that ends wandering is calculated by the wandering formula, as shown in formula (1)

[0047]

[0048] In the formula is the walking step length; d = 1, 2, ..., D; P = 1, 2, ..., h. Since wolves in nature have different search and walking abilities, the value of h is random. h is a random integer. Generally, the larger h is, the more directions the wolf can search and the better the global optimization is, but it will also increase the search time.

[0049] Wolf Summoning Module: The leader wolf howls, summoning the wolves to come closer. All wolves advance one galloping step toward the leader wolf. Run towards the alpha wolf until the distance between the alpha wolf and the alpha wolf is less than the siege distance. During the run, if the alpha wolf finds a position with a larger target function value than the alpha wolf, it will replace it as the new alpha wolf and initiate the summoning behavior again. When all wolves reach the siege distance, the wolf pack switches to siege behavior. The position change of the alpha wolf during the run is as follows:

[0050]

[0051] Wherein is the position of the fierce wolf before this raid, is the position of the current lead wolf, is the position of the (k + 1)-th generation fierce wolf after the raid.

[0052] Wolf pack siege module: The wolf pack launches a siege on the prey, regarding the current lead wolf as the prey. The fierce wolves and scout wolves initiate the siege on the prey and advance one siege step length in the direction of the prey. During the siege, if a target function value better than that corresponding to the current prey position is found, the prey position is updated and the siege is restarted until the termination condition is reached. The position change of the artificial wolves during the siege is as follows:

[0053]

[0054] λ is a random number with a value in (0, 1), represents the siege step length.

[0055] Proportion factor update module: The wolf pack eliminates the R artificial wolves with the smallest target function values according to the set population update proportion factor, and at the same time randomly generates R new artificial wolves in the solution space to complete the population update. R is a random number, and its value range is [N / (2β), N / β], where β is the update proportion factor.

[0056] Termination condition judgment module: Judge whether the termination condition of the solution accuracy requirement or the maximum number of iterations is reached. If the termination condition is reached, the spatial position of the lead wolf is output, which is the optimal solution to the problem to be solved. Otherwise, continue to iterate until the termination condition is reached.

[0057] Another object of the present invention is to provide a computer device, which includes a memory and a processor. The memory stores a computer program. When the computer program is executed by the processor, the processor executes the steps of the high-dimensional function solving method based on the distance determination wolf pack algorithm.

[0058] Another object of the present invention is to provide a computer-readable storage medium, which stores a computer program. When the computer program is executed by the processor, the processor executes the steps of the high-dimensional function solving method based on the distance determination wolf pack algorithm.

[0059] Another object of the present invention is to provide an information data processing terminal, which includes the high-dimensional function solving system based on the distance determination wolf pack algorithm.

[0060] Combined with the above technical solutions and the solved technical problems, the advantages and positive effects of the technical solution to be protected by the present invention are as follows:

[0061] First, the present invention makes the following improvements on the basis of the basic wolf pack algorithm: First, when the wolves initially wander, a position change record value is set, and the wolves with the worst performance, that is, those whose positions have not changed for many times during the initial wandering process, are eliminated, so as to maintain a strong search ability of the population and accelerate the initial optimization speed of the wolf pack; Second, during the fierce wolf rush, a distance determination factor is used to determine whether the fierce wolf participates in the summoning, which speeds up the rush process, improves the global convergence of the algorithm, and at the same time copies an artificial wolf directly into the siege range to ensure the accuracy of the algorithm optimization; Third, during the fierce wolf siege, the value range of the random operator is determined according to the position relationship between the fierce wolf and the prey, which improves the siege efficiency and shortens the time for the wolf pack to capture the prey. Finally, a distance factor is introduced during the population update process to judge whether the updated artificial wolf is effective, so that the updated wolf pack is always full of vitality and enhances the global convergence of the algorithm.

[0062] By using the Markov process to evaluate the convergence of the improved algorithm, a variety of standard test functions are designed for multiple groups of experiments to simulate the objective functions with different parameter characteristics. Analyzing the final experimental results, the improved wolf pack algorithm has better global convergence, shortens the running time, and effectively solves the optimization problem of high-dimensional functions.

[0063] Second, the technical problems of the existing technology solved by the technical solution of the present invention and the significant technical progress obtained in industrial applications.

[0064] I. Technical problems of the existing technology solved

[0065] 1. Insufficient global convergence ability of the algorithm

[0066] Traditional optimization algorithms (such as genetic algorithms and particle swarm algorithms) are prone to falling into local optimal solutions in the solution of high-dimensional functions and lack sufficient global optimization ability.

[0067] The present invention improves the global search ability and optimization efficiency of the algorithm by introducing three behaviors of wandering, rushing and besieging and their step size dynamic adjustment mechanism.

[0068] 2. Low search efficiency

[0069] In the existing swarm optimization algorithms, the randomness of the individual distribution in the initialization stage is usually strong, and the early iteration efficiency is low, resulting in a slow overall running speed of the algorithm.

[0070] The present invention shortens the search path and improves the convergence speed by introducing a dynamic head wolf update mechanism, adjusting the head wolf position in real time during the wandering, rushing and besieging stages, and eliminating the "inefficient" artificial wolves at the same time.

[0071] 3. Insufficient accuracy in high-dimensional search problems

[0072] In the high-dimensional solution space, the step-size adjustment method of traditional algorithms is relatively rough, and it is easy to skip the optimal solution under the condition of a large step size, or consume a lot of time under the condition of a small step size.

[0073] The present invention ensures the balance between search fineness and efficiency at different stages of the algorithm through a dynamic step-size factor adjustment mechanism (wandering step size, raiding step size, besieging step size), combined with dynamic direction optimization in multi-dimensional space.

[0074] 4. Lack of flexibility in multi-objective optimization

[0075] Most traditional algorithms cannot flexibly handle multi-objective optimization tasks in complex objective functions.

[0076] The present invention enables the algorithm to dynamically update the positions of artificial wolves through a distance determination factor and a population update mechanism, avoiding ineffective searches, and achieving higher solution flexibility under the condition of multi-objective functions.

[0077] II. Significant technological progress

[0078] 1. Significantly improved global optimization ability

[0079] The multi-behavior cooperation mechanism (wandering, raiding, and besieging) designed in the present invention simulates the complex cooperation strategies of wolf packs in nature, enhancing the global search ability of the algorithm. In particular, the dynamic step-size adjustment ensures the effective switching between large-scale search in the initial stage and precise optimization in the later stage of the algorithm.

[0080] The dynamic determination of the difference in objective function values (such as the alpha wolf update and elimination mechanism) further improves the ability of the algorithm to avoid falling into local optima.

[0081] 2. Efficient iterative process

[0082] The introduction of the alpha wolf summoning mechanism and the scout wolf elimination mechanism avoids the multiple ineffective search behaviors of "inefficient individuals" in traditional algorithms, accelerating the convergence speed of the algorithm.

[0083] In the wandering stage, each scout wolf independently searches in multiple directions, combined with the strategy of dynamically selecting the optimal direction, achieving the maximization of the group search efficiency.

[0084] 3. Improved optimization accuracy

[0085] The present invention enables the algorithm to more accurately find the global optimal solution in the high-dimensional solution space through the precise adjustment of the step-size factor in multi-dimensional space and the application of the distance determination mechanism in the population update process.

[0086] The proportional relationship of the wandering step size, raiding step size, and besieging step size (Formula 4) ensures seamless switching between large-scale search and small-scale precise optimization.

[0087] 4. Enhance the algorithm robustness

[0088] The present invention controls the population update behavior through the distance determination factor, avoiding the newly generated artificial wolves randomly getting too close to the original position or near the leading wolf, and improving the adaptability of the algorithm to complex objective functions.

[0089] During the siege stage, a relative position adjustment mechanism between the leading wolf and the fierce wolf is introduced, avoiding the search deviation or missing the optimal solution caused by an excessive step size, and further enhancing the stability of the algorithm.

[0090] 5. Flexibly adapt to multi-objective tasks

[0091] By dynamically adjusting the search strategy and population update mechanism, the present invention can quickly adapt to multi-objective optimization tasks and is applicable to various complex optimization scenarios, including engineering optimization, image processing, and large-scale data modeling, etc.

[0092] III. Value in industrial applications

[0093] 1. Solve complex engineering optimization problems

[0094] The present invention can be widely applied to complex engineering optimization tasks such as industrial design, parameter optimization of aerospace vehicles, and routing planning of communication networks, improving the accuracy and efficiency.

[0095] 2. Improve the training efficiency of big data models

[0096] In the fields of machine learning and data mining, the present invention can be used to optimize the hyperparameters of deep learning models, significantly improving the convergence speed and prediction performance of model training.

[0097] 3. Reduce energy and computing costs

[0098] The present invention reduces the ineffective calculations of the algorithm in the high-dimensional solution space by dynamically adjusting the population size, step size, and iteration strategy, significantly reducing the consumption of computing resources.

[0099] 4. Adapt to dynamic environments

[0100] In tasks that require real-time solutions (such as autonomous driving path planning, dynamic logistics optimization), the present invention can quickly respond to environmental changes and provide dynamic optimal solutions.

[0101] By simulating the collaborative behavior of wolf packs in nature, the present invention designs a wolf pack algorithm based on distance determination, solving problems such as poor global convergence ability, low efficiency, and insufficient accuracy in traditional optimization algorithms. Through the collaborative search in three stages of wandering, raiding, and siege, combined with the dynamic step size adjustment and population update mechanism, significant technological progress of the algorithm in solving high-dimensional functions is achieved, providing a powerful tool for solving complex engineering optimization and multi-objective problems. Description of the drawings

[0102] Figure 1 It is the flow chart of the wolf pack algorithm provided by the embodiments of the present invention;

[0103] Figure 2 It is the comparison chart of the magnitude of A(i) value at the end of the wandering provided by the embodiments of the present invention;

[0104] Figure 3 It is the flow chart of the wandering process provided by the embodiments of the present invention;

[0105] Figure 4 It is the schematic diagram of the position change during the exploration wolf's wandering process provided by the embodiments of the present invention;

[0106] Figure 5 It is the flow chart of the summoning behavior provided by the embodiments of the present invention;

[0107] Figure 6 It is the schematic diagram of the raiding process provided by the embodiments of the present invention;

[0108] Figure 7 It is the schematic diagram of the population update process provided by the embodiments of the present invention;

[0109] Figure 8 It is the schematic diagram of the improved population update process provided by the embodiments of the present invention;

[0110] Figure 9 It is the schematic diagram of the search space image of the control experiment test function provided by the embodiments of the present invention;

[0111] Figure 10 It is the result comparison line chart provided by the embodiments of the present invention;

[0112] Figure 11 It is the bar chart of the optimal value comparison of 15 algorithms for the CEC2022 test set (dimension = 10) provided by the embodiments of the present invention;

[0113] Figure 12 It is the bar chart of the optimal value comparison of 15 algorithms for the CEC2022 test set (dimension = 20) provided by the embodiments of the present invention;

[0114] Figure 13 It is the function convergence curve chart of 15 algorithms for the CEC2022 test set (dimension = 10) provided by the embodiments of the present invention;

[0115] Figure 14 It is the function convergence curve chart of 15 algorithms for the CEC2022 test set (dimension = 20) provided by the embodiments of the present invention;

[0116] Figure 15It is the box plot of the results of 15 algorithms provided by the embodiments of the present invention for the CEC2022 test set (dimension = 10);

[0117] Figure 16 It is the box plot of the results of 15 algorithms provided by the embodiments of the present invention for the CEC2022 test set (dimension = 20);

[0118] Figure 17 It is the bar chart of the optimal value comparison of 15 algorithms provided by the embodiments of the present invention for the CEC2020 test set (dimension = 10);

[0119] Figure 18 It is the bar chart of the optimal value comparison of 15 algorithms provided by the embodiments of the present invention for the CEC2020 test set (dimension = 20);

[0120] Figure 19 It is the function convergence curve chart of 15 algorithms provided by the embodiments of the present invention for the CEC2020 test set (dimension = 10);

[0121] Figure 20 It is the function convergence curve chart of 15 algorithms provided by the embodiments of the present invention for the CEC2020 test set (dimension = 20);

[0122] Figure 21 It is the box plot of the results of 15 algorithms provided by the embodiments of the present invention for the CEC2020 test set (dimension = 10);

[0123] Figure 22 It is the box plot of the results of 15 algorithms provided by the embodiments of the present invention for the CEC2020 test set (dimension = 20). Detailed implementation manners

[0124] In order to make the objectives, technical solutions and advantages of the present invention clearer, the present invention will be further described in detail below with reference to embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention, and are not used to limit the present invention.

[0125] Embodiment 1: Manipulator path optimization

[0126] In the field of industrial automation, the path planning of a manipulator is the key to improving production efficiency and accuracy. Traditional path planning methods are prone to falling into local optimal solutions in high-dimensional spaces, resulting in non-optimal paths, low motion efficiency, or high energy consumption.

[0127] Through path optimization of the multi-joint motion of the robotic arm using the wolf pack algorithm based on distance determination, the joint angles of each robotic arm are regarded as one dimension in the high-dimensional solution space of the algorithm. The algorithm conducts a large-scale search for possible joint combinations in the solution space during the wandering stage, refines the search for potential high-quality solutions during the raiding stage, and further approaches the optimal path during the siege stage.

[0128] In the path planning of a 10-degree-of-freedom robotic arm, this method shortens the path length by 15% and reduces the energy consumption by 20%. Compared with the traditional genetic algorithm, the search time is shortened by 30%, and the planned path is smoother, avoiding unnecessary pauses and redundant motions in the robotic arm's actions.

[0129] The global search ability and dynamic step size adjustment of the algorithm of the present invention enable it to quickly find efficient paths, which are applicable to the optimization of automation equipment in complex industrial scenarios, improving production efficiency and reducing energy consumption.

[0130] Example Two: Optimization of Deep Learning Hyperparameters

[0131] In the training of deep learning models, optimizing the hyperparameters of the model (such as learning rate, number of layers, activation function, etc.) is the core link to improve the model's accuracy. However, hyperparameter search usually involves a multi-dimensional high-complexity solution space, and the traditional grid search or random search is less efficient.

[0132] # Application Method

[0133] Using the wolf pack algorithm based on distance determination, the hyperparameter optimization is regarded as a problem of solving a high-dimensional function. Each "artificial wolf" represents a set of hyperparameter configurations. The wandering stage conducts a global search of the solution space to quickly locate high-quality hyperparameter combinations; the raiding stage focuses on optimizing the search area to improve the quality of candidate solutions; the siege stage finely adjusts the values of key hyperparameters to approach the optimal solution.

[0134] In an image classification task of a convolutional neural network (CIFAR-10 dataset), this algorithm improves the model accuracy by 5% and reduces the optimization time by 40%. Compared with random search, the accuracy of the model is increased by about 3% within the same training cycle.

[0135] Through the intelligent optimization ability of this algorithm, the training efficiency and prediction accuracy of deep learning models are significantly improved, which is especially applicable to high-dimensional complex hyperparameter optimization scenarios, saving computing resources and reducing the model development cost.

[0136] 1. Basic Wolf Pack Algorithm

[0137] The basic solution space of the Wolf Pack Algorithm is an N×D Euclidean space, where N is the total number of artificial wolves in the wolf pack and D is the number of variables optimized by the objective function. The basic WPA abstracts the predation actions of wolves in nature into three types of wolves with different divisions of labor (alpha wolves, scout wolves, and striker wolves), which perform three intelligent behaviors: summoning, wandering, and surrounding. At the same time, WPA designs a rule for generating alpha wolves of "the winner takes all" and a wolf pack update mechanism of "the fittest survive" to simulate the coordinated cooperation of wolves in nature. In the N×D Euclidean space, the position of artificial wolf i can be expressed as X i =(x i1 ,x i2 ...x id ), where x id is the position of the i-th artificial wolf in the d-th (d = 1, 2,..., D) dimensional variable space to be optimized; the prey odor concentration perceived by artificial wolf i is Y i =f i (X i ), where Y i is the objective function value. Taking the solution of the global optimal value of the objective function as an example, the operation process of the basic WPA is described as follows:

[0138] Step 1: First, in the N×D solution space, set the wolf pack size and then randomly generate artificial wolves to initialize the wolf pack. At this time, the spatial position coordinates X i of each artificial wolf are randomly generated, and the objective function value Y i at the position of each artificial wolf is calculated. The artificial wolf with the largest function value is selected as the alpha wolf, and the R wolves with the second-best objective function value Y i are selected as scout wolves, and the rest are striker wolves.

[0139] Step 2: The scout wolves perform a wandering behavior to greedily search the solution space. Each scout wolf advances one wandering step length in h randomly selected directions respectively and compares the objective function values corresponding to the front and back positions. Finally, it advances in the direction with the largest objective function value. If the objective function value at the original position is the largest, it returns to the original position. After the scout wolves' wandering ends, compare the objective function values of all artificial wolves. The wolf with the largest objective function value becomes the new alpha wolf. The position of the scout wolf that ends wandering is calculated by the wandering formula, as shown in Equation (1)

[0140]

[0141] In the formula is the wandering step length; d = 1, 2,..., D; P = 1, 2,..., h. Since it is simulated that wolves in nature have different search and wandering abilities, the value of h is random. h takes a random integer. Generally, the larger h is, the more search directions the scout wolf has, and the better the global optimization ability, but it will also increase the search time.

[0142] Step 3: The leading wolf emits a "howl" to summon the fierce wolves to gather. All fierce wolves move forward one running step length towards the leading wolf Run towards the leading wolf until the distance between the fierce wolf and the leading wolf is less than the siege distance. During the running process, if a fierce wolf finds a position with a larger objective function value than that corresponding to the leading wolf, it will replace the leading wolf and become the new leading wolf, and initiate the summoning behavior again. When all fierce wolves reach the siege distance, the wolf pack enters the siege behavior. The position change of the fierce wolf during the running process is as follows:

[0143]

[0144] In the formula is the position of the fierce wolf before this run, is the position of the current leading wolf, is the position of the (k + 1)-th generation fierce wolf after the run.

[0145] Step 4: The wolf pack launches a siege on the prey. Regarding the current leading wolf as the prey, the fierce wolves and the exploring wolves launch a siege on the prey and move forward one siege step length in the direction of the prey. During the siege process, if a position with a better objective function value than that corresponding to the current prey position is found, the prey position will be updated and the siege will be launched again until the termination condition is reached. The position change of the artificial wolves during the siege process is as follows:

[0146]

[0147] λ is a random number with a value in (0, 1), represents the siege step length.

[0148] Step 5: The wolf pack eliminates the R artificial wolves with the smallest objective function value according to the set population update ratio factor, and at the same time randomly generates R new artificial wolves in the solution space to complete the population update. R is a random number, and its value range is [N / (2β), N / β], where β is the update ratio factor.

[0149] Step 6: Determine whether the accuracy requirement for solving or the termination condition of the maximum number of iterations is reached. If the termination condition is reached, output the spatial position of the leading wolf, which is the optimal solution to the problem to be solved; otherwise, continue the iteration until the termination condition is reached.

[0150] There are wandering step lengths running step lengths and attack step lengths in WPA. Suppose the value range of the variable in the d-th dimensional space is [min d , max d , then the relationship between the three step lengths in the d-th dimensional space is as shown in formula (4)

[0151]

[0152] In the above formula, S is the step factor, representing the refinement degree of the search. The algorithm flow chart is as Figure 1 shown as follows:

[0153] 2. Improved Wolf Pack Algorithm

[0154] 2.1 Position Recording during the Wandering Process

[0155] In the basic WPA, the exploring wolves adopt a greedy strategy to search and move forward during the wandering process. The prerequisite for each position change is that the updated position has a higher objective function value. After multiple wanderings, some exploring wolves frequently update their positions, indicating good performance during the search process and continuously moving towards a direction with a higher target value, serving as the main force for the wolf pack to search for prey. While the exploring wolves that have not updated their positions for multiple times are judged to have poor performance and are very likely to fall into local optima. These exploring wolves play a relatively small role in the entire wolf pack's search for the global optimum. After each wandering process ends, the exploring wolves with poor performance are eliminated and an equal number of new artificial wolves are generated to ensure the search accuracy, which can accelerate the WPA operation speed and improve the global convergence. At the same time, to improve the wandering efficiency, the alpha wolf is selected after the first wandering instead of after the population initialization, avoiding wasting computing time as the alpha wolf usually updated in the early stage of the algorithm is only a local optimum. After the population initialization ends, the exploring wolves start to wander and optimize with themselves as the center. The basic wandering formula is

[0156]

[0157] where, is the current position of the exploring wolf, is the position of the exploring wolf after wandering, is the wandering step size, h is the wandering direction, and p = 1, 2,....., h. After the exploring wolves wander in h directions respectively and return to the original position, finally, they choose to wander in the direction with the maximum objective function value after moving forward. If the original position is the best, the position remains unchanged.

[0158] Add a position change vector A to the wandering behavior. The first behavior, the column is the population size n, and the initial value is a row vector of all 0s. A(i) corresponds to the number of position changes of the i-th artificial exploring wolf. Each time the position changes during the wandering process, the value of A(i) is incremented by 1. To intuitively display the characteristics of the position changes of the exploring wolves during the wandering process, taking the WPA to solve the Easom function as an example, the population sizes are set to 100, 150, 200, 250 respectively, and compare the number of position changes of the exploring wolves in the wolf pack after wandering 20 times. As Figure 2 shown as follows:

[0159] At the end of each wandering process, A will be updated to a non-zero row vector. At this time, the wolf with the maximum objective function value is selected as the alpha wolf. The Q wolves with the smallest A(i) values will be eliminated and new exploring wolves will be generated. Q is a random number with a value in (1, n / a), and its calculation formula is as

[0160] Q = floor(rand(1)*(n / a)) + 1 (6)

[0161] Among them, a is the exploring wolf elimination factor, usually taking values in [16, 20]. Generally, the larger the scale of the wolf pack, the more exploring wolves will be eliminated. By eliminating the "negatively working" artificial wolves during the wandering process, the searching ability of the wolf pack is improved, thereby accelerating the convergence speed of the algorithm and finding the global optimal solution faster. The implementation process is shown in the figure:

[0162] 2.2. Improvement of the summoning behavior

[0163] The lead wolf is the artificial wolf with the largest objective function value in the current wolf pack, coordinating the actions of the entire wolf pack. After the wandering ends, the lead wolf summons the wolf pack to rush towards it. During this process, if it is found that the objective function value of any other wolf is greater than that of the lead wolf, the lead wolf will be replaced and summoned again. The position update during the initial WPA rushing process is as shown in formula (7):

[0164]

[0165] Among them, and are the positions of the fierce wolf before and after the rush respectively, is the rush step size, is the position of the current lead wolf.

[0166] Its deficiencies are as follows: First, the fixed rush step size causes the fierce wolves far from the lead wolf to need several rushes to enter the siege range. Second, when the objective function is a multi-peak function, if the lead wolf falls into a local optimum, even if the exploring wolf is approaching the global optimum during the wandering, it still needs to abandon the wandering and transfer to the rush. For intuitive display, Figure 4 is the schematic diagram of the position of the exploring wolf at the end of the wandering when solving the Rastrigin function with a population size of 50 for WPA:

[0167] It is analyzed that after the wandering ends, the wolf pack tends to gather around the peak. Figure 4 In, the wolf pack is divided into several small groups and distributed around the global optimum and several local optimum values. At the same time, if an individual wolf is far from the lead wolf, it enters the siege distance slowly, affecting the running speed of the algorithm. Therefore, two judgment conditions are set for whether the exploring wolf conducts a rush: one is that the distance from the lead wolf reaches the threshold, and the other is that its A(i) value is large enough. The exploring wolf that meets one of the two conditions abandons the rush and continues to wander. During the wandering process, if Y(i) is greater than Y(toulang), it replaces the lead wolf and initiates a new summons. If there is no wolf whose distance from the lead wolf reaches the threshold, the wolf with the farthest distance is selected. To avoid the reduction of the population search accuracy, an equal number of new artificial wolves are randomly generated within the siege distance centered on the lead wolf. The implementation process is as Figure 5 shown:

[0168] 2.3. Improvement of the Siege Behavior

[0169] After the scout wolf finishes its wandering, the alpha wolf summons the wolf pack to rush towards it and regards the alpha wolf as the prey. In the initial siege formula of WPA, the value range of λ is [-1, 1], which leads to the situation of returning to the original position many times during the siege process (when the value of λ is negative, greater than when the value of λ is positive, less than ), ultimately reducing the search efficiency. At the same time, in extreme cases, if the siege step size of the fierce wolf is too large, it is easy to jump out of the siege range, making it difficult for the wolf pack to search for the optimal value. Before the fierce wolf's rushing action starts, first determine the positional relationship between it and the alpha wolf. Since the search space is an N×D solution space, the position of the artificial wolf will be a D-dimensional coordinate. In each dimension, the value of λ is divided into two cases. If that is, it means when the fierce wolf does not exceed the position of the alpha wolf, the value of λ is positive, and the fierce wolf can always move towards the alpha wolf; conversely, if and the fierce wolf does not exceed the position of the alpha wolf, the value of λ is negative, and the fierce wolf can also always move towards the alpha wolf. At the same time, when the siege step size is too large and the fierce wolf crosses the position of the alpha wolf, by judging the and size relationship, immediately adjust the advancing direction of the fierce wolf to continuously narrow the search range for the prey.

[0170] 2.4. Improvement of the Population Update Mechanism

[0171] In nature, in order to maintain competitiveness, the wolf pack distributes the captured prey from the weakest to the strongest. The initial WPA eliminates the R artificial wolves with the worst performance (i.e., the smallest objective function value), and at the same time randomly generates new R artificial wolves to complete the population update of the wolf pack. R takes a random integer between [N / (2β), N / β], and β is the update ratio factor.

[0172] In the figure, C1 and C2 are the adjacent ranges of two local optimal values. If the initial position of the updated wolf is within this range, it will repeatedly fall into the local optimal value, such as L1, L2, L3, L4, and L8. C3 is the adjacent range of the global optimal value. If the updated wolf is generated within the range, it will gradually approach the global optimal value, such as L9. L5, L6, and L7 are wolves with initial positions in other positions of the solution space. After their wandering behavior, they have the possibility of approaching C3, which depends on the values of the direction factor and step size during the wandering process. Obviously, L5, L6, L7, and L9 are effective updated wolves, and the convergence process of the algorithm towards the global optimal value is accelerated through these new artificial wolves. Other wolves are ineffective updated wolves and they cannot contribute to the convergence of the algorithm.

[0173] The deficiencies of the initial WPA are as follows: After wandering, raiding, and besieging, most of the artificial wolves gather around the leading wolf. If the leading wolf falls into a local optimum, it is highly likely that other wolves will also fall into the local optimum. The positions of the artificial wolves updated in the initial WPA are random and are likely to be generated around the leading wolf. Therefore, updating the artificial wolves near the leading wolf is of little significance, and the new artificial wolves are likely to fall into the same local optimum. At the same time, if the updated artificial wolf is close to the R wolves eliminated in the previous generation, it is highly likely to repeat the process of eliminating wolves, which is also an ineffective update. Therefore, a distance determination factor for population update is introduced. When updating the wolf pack, the new artificial wolves that meet the conditions are retained. The implementation process is as follows:

[0174] Step1: Record the original positions of the R wolves to be eliminated and the position of the current leading wolf.

[0175] Step2: Randomly generate an R*m position matrix to update the positions of the artificial wolves.

[0176] Step3: Judge two conditions: Calculate the distance from the old position. If it is greater than the distance determination factor, update the position; calculate the distance from the current leading wolf. If it is greater than the maximum distance of the current artificial wolf from the leading wolf, update the position. The wolves that do not meet the conditions are randomly updated again until all R wolves are updated.

[0177] Step4: After the wolf pack is updated, select the leading wolf in the new wolf pack.

[0178] 3. Theoretical Analysis of the Improved Wolf Pack Algorithm

[0179] 3.1. Convergence Analysis of the Algorithm

[0180] The Markov Chain is a stochastic process. Its characteristic is that the future state of the system depends only on the current state and is independent of the past states. This property is called memorylessness or lack of aftereffect and is often used to prove the convergence of algorithms. The wolf pack algorithm is an optimization algorithm based on swarm intelligence inspired by the hunting behavior of wolf packs. To analyze its convergence, the memoryless property of the Markov Chain can be utilized. Due to the memoryless property of the Markov Chain, the transition of the current state depends only on the current solution and is independent of the previous solutions. This property applies to the wolf pack algorithm because the decision-making in each iteration is only based on the current positions of the wolf pack.

[0181] Lemma 1 It can be known from the literature that to determine whether an intelligent algorithm converges to the global optimal solution of the problem with probability 1, the following conditions need to be met:

[0182] a) For any two solutions x 1 and x 2 in the solution space, x 2 is reachable from x 1 by various operators in the algorithm;

[0183] b) Population sequence Q 1 , Q 2 , …, Q N is monotonic.

[0184] According to Lemma 1 and related theories, to prove that the improved wolf pack algorithm can converge to the global optimal solution of the problem to be solved with probability 1, it is first necessary to prove that

[0185] a) The population sequence of the wolf pack algorithm is an ergodic Markov chain;

[0186] b) The sequence solution of the wolf pack algorithm is a finite homogeneous Markov chain.

[0187] The wolf pack algorithm regards the solution space of the algorithm as a state space (S), where each state represents a possible solution (i.e., the position of the wolf). Assume that each state (s ∈ S) represents the solution vector at a certain moment. Define the state transition probability (P(s_i, s_j)), which represents the probability of transferring from state (s_i) to state (s_j). In the wolf pack algorithm, the transfer mechanism can be described by the movement strategies of the wolves (wandering, summoning, besieging, population update, etc.). Construct the transition probability matrix P, where each element (P(ij)) represents the probability of transferring from state (s_i) to state (s_j). Let the solution space be Ω, and the state transition matrices of wandering, summoning, besieging, and wolf pack update behaviors be Y, Z, W, and G respectively. Then its probability matrix P is

[0188] P = Y × Z × W × G (8)

[0189] Ensure that the sum of the probabilities of all rows of the probability matrix is 1, that is

[0190]

[0191] First, give the following definitions

[0192] Definition 1: For a Markov chain, starting from state (s_i) at time 0, if there exists a time t when the probability of transferring to state (s_j) is greater than zero, this Markov chain is called irreducible. As shown in the following formula

[0193]

[0194] Definition 2: For a Markov chain with state space Ω, if starting from state i at time 0, the greatest common divisor of the non-empty set of times t when it returns to state i is 1, then this Markov chain is called aperiodic.

[0195] Definition 3: For a Markov chain (X 1 , X 2...X t ,...), starting from state i at time 0, if the probability of first transferring to state j at time t is denoted as If for all states i, j satisfy:

[0196]

[0197] Then this Markov chain is said to be positive recurrent. In particular, when the Markov chain is both aperiodic and positive recurrent, it is ergodic.

[0198] Next, it is proved that the Markov chain of the population sequence of the improved wolf pack algorithm is ergodic.

[0199] In the wolf pack algorithm, the position of each wolf represents a solution, and the state of the entire population can be regarded as a combination of multiple solutions. The state space consists of all possible solutions, corresponding to the positions of each artificial wolf in the wolf pack. Let the k-th generation population of the algorithm be Q k ={X 1 , X 2 , …, X N}, where Xi is the state of the i-th artificial wolf. The artificial wolves move around, summon, rush, besiege, and update the population in the solution space. According to the basic principle of WPA, the position changes brought about by the above behaviors are random, ensuring global exploration to a certain extent, thus making it possible to transfer between different states. Therefore, it is possible for each artificial wolf in the solution space to transfer from position i to position j, that is Since the probability matrix P only depends on the starting and ending positions, P is a positive definite matrix. According to Definition 1 again, it can be obtained that the population sequence of the improved wolf pack algorithm is irreducible.

[0200] Obviously, for an irreducible Markov chain, after a long enough time, it can transfer from a certain state to any state. And for any k > 0, if the Markov chain is irreducible, it is inevitable that Pij > 0 holds. Combining with Definition 2, it can be known that k = 1. Therefore, the greatest common divisor of Q is 1, so the Markov chain of the improved wolf pack algorithm population is aperiodic.

[0201] Next, it is proved that the Markov chain of the population sequence of the improved wolf pack algorithm is finitely homogeneous.

[0202] In the wolf pack algorithm, the position of each wolf represents a solution, and the state of the entire population can be regarded as a combination of multiple solutions. Let the state space (S) contain all possible solutions. This is a finite state space, so the representation of the solution (such as position) is finite-dimensional, and the search space is usually bounded. The update of the wolf's position can be described by the transition probability. Each wolf updates its own position according to the current position, the positions of other wolves, and the fitness. The position update rule of the wolf is the same, that is, the position of the next generation is only related to the position of the current generation, and has nothing to do with the position of the previous generation, the number of iterations, and the probability matrix. Therefore, after the algorithm iteration ends, the obtained solution is a finite homogeneous Markov chain.

[0203] In summary, it can be proved that the Markov chain of the population sequence of the improved wolf pack algorithm is ergodic and finitely homogeneous, and the global optimal solution obtained at the end of the iteration is a finitely homogeneous Markov chain. Furthermore, it can be proved that the improved wolf pack algorithm converges to the global optimal solution of the problem to be solved with probability 1.

[0204] 4. Simulation Experiments and Algorithm Effectiveness Testing

[0205] 4.1. Test Functions

[0206] In the IEEE Congress on Evolutionary Computation (CEC), standard test functions are usually used to evaluate the performance of evolutionary algorithms and other optimization algorithms. To verify that the improved wolf pack algorithm has better performance, first, 12 standard test functions are selected from the CEC2014, CEC2017, and CEC2022 test function sets, and a comparative experiment is conducted between the improved wolf pack algorithm and the standard wolf pack algorithm WPA to test the performance of the improved algorithm. The selected test functions include unimodal and multimodal, separable and inseparable functions. The specific characteristics are shown in Table 1, where "U" represents unimodal functions, "M" represents multimodal functions, "S" represents separable functions, and "N" represents inseparable functions. F1 to F5 are unimodal standard test functions, F6 to F10 are multimodal standard test functions, and F11 to F12 are fixed-dimensional multimodal standard test functions. Unimodal functions are relatively simple with only one global optimum, suitable for testing the precise search ability of the improved algorithm. Multimodal functions have multiple local extrema, which can better reflect the global search ability of the improved algorithm. High-dimensional inseparable functions reflect the ability of the improved algorithm to solve complex functions.

[0207] Table 1 Standard Test Functions

[0208]

[0209]

[0210] Figure 9 Vividly demonstrates the search space of functions F1 to F12, with clear function characteristics.

[0211] To further verify the feasibility and better performance of the improved algorithm, a comparative test was conducted on 14 swarm intelligence optimization algorithms: Genetic Algorithm (GA), Particle Swarm Optimization (PSO), Differential Evolution Algorithm (DE), Grey Wolf Optimization Algorithm (GWO), Whale Optimization Algorithm (WOA), IBI Logic Optimization Algorithm (IBL), Snake Optimization Algorithm (SO), Snow Ablation Optimizer (SAO), Squirrel Search Algorithm (SSA), Coati Optimization Algorithm (COA), Fishing Optimization Algorithm (CFOA), Secretary Bird Optimization Algorithm (SBOA), Red-billed Blue Magpie Optimization Algorithm (RBMO), Black-winged Kite Algorithm (BWA) under the test conditions of 10 functions in two dimensions (10, 20) of the CEC2020 test set and 12 functions in two dimensions (10, 20) of the CEC2022 test set. In the above test functions, the general parameters were set to the same values to ensure strict control of variables, and other parameters were set to the default values of the algorithm. The CEC-2020 and CEC-2022 test functions can also be divided into: unimodal functions, multimodal functions, hybrid functions, and composite functions. The characteristics of unimodal functions and multimodal functions are suitable for testing the precise search ability and global search ability of the algorithm. Hybrid functions and composite functions have complex structures and are designed to evaluate the comprehensive performance of the algorithm. In the present invention, the maximum number of iterations was set to 500, the population size was set to 100, and the remaining parameters were set as shown in Table 2.

[0212] To ensure the effectiveness of the verification experiment, the above swarm intelligence optimization algorithms were each conducted 20 times of experiments, and the experimental results were statistically analyzed to evaluate the effectiveness of the improved algorithm in terms of the optimal value, average value, standard deviation, worst value, and the average value of the success rate of 20 experiments. The optimal value is the final result obtained by the algorithm. Comparing with the standard value of the function, the closer the optimal value is to the standard value, the stronger the solving ability of the algorithm. The average value represents the accuracy of the algorithm's solution result. In the experiments of the present invention, the smaller the value, the closer it is to the function standard value, which means the higher the solving accuracy. The standard deviation is used to evaluate the stability of the algorithm's ability to solve the function problem. The smaller the value, the smaller the fluctuation of the solution result around the function standard value, indicating the better stability of the algorithm. The worst value reflects the worst result or performance of the algorithm's solution result, which can comprehensively evaluate the stability and performance lower limit of the algorithm, and can also expand the progress space of the algorithm. The success rate is the percentage of the number of times the theoretical optimal value of the test function is found during the algorithm optimization process. The higher the success rate, the better the global optimization performance of the algorithm.

[0213] Table 2 Parameter Settings of Swarm Intelligence Optimization Algorithms

[0214]

[0215] The above intelligent algorithm was used to conduct 20 independent experiments on the CEC2020 and CEC2022 typical test function sets respectively. The simulation experiment environment was: HONOR Magic Book Pro, windows10, Intel core i5-10210U, and the program was implemented in Matlab R2019b with m language.

[0216] 4.2. Results of the control experiment

[0217] To evaluate the superior performance of DDWPA compared to WPA, a series of experiments were conducted under the conditions of the standard test functions given in Table 1 to evaluate the convergence of the algorithm and its exploration and exploitation capabilities. Table 3 presents the optimization results of WPA and DDWPA for the 12 test functions in Table 1 under the same parameter conditions, including the optimal value, mean, worst value, standard deviation, and success rate.

[0218] Table 3 Results of the control experiment

[0219]

[0220]

[0221] From the experimental results, DDWPA demonstrated higher solution accuracy and stability in multiple dimensions and functions. Under the three evaluation criteria of the best value, average value, and worst value, DDWPA achieved better results than WPA in multiple dimensions. This indicates that DDWPA has stronger global search capabilities and more stable solution performance during the solution process. Specifically, when solving unimodal functions such as F1, F2, and F3, DDWPA can converge to the global optimal solution faster, which benefits from its improved search strategy and stronger global search capabilities. Especially for function F5, DDWPA showed far better performance than WPA, and all indicators demonstrated its superiority, generally leading the experimental results of WPA by 3 to 4 orders of magnitude. In contrast, WPA may fall into local optimal solutions when solving unimodal functions, resulting in a decrease in solution accuracy, as evidenced by its performance on F1. Its experimental results indicate that WPA may have fallen into a local optimum and was unable to escape during the solution process.

[0222] When solving multi-modal functions, DDWPA also shows high solution accuracy and stability. Multi-modal functions usually have multiple local optimal solutions, so they are more difficult to solve. However, DDWPA has a stronger global search ability through an improved search strategy, which can effectively jump out of local optimal solutions and find the global optimal solution. When solving multi-modal functions such as F6, F7, and F8, DDWPA shows strong global search ability in both low-dimensional and multi-dimensional spaces. Taking F6 as an example, all evaluation indicators of its experimental results are one order of magnitude better than those of the WPA optimization results, and its success rate has increased by 90%, demonstrating its good search ability. In contrast, WPA may fall into local oscillations when solving multi-modal functions, resulting in a decrease in solution accuracy and stability.

[0223] In the process of solving the fixed multi-modal functions F11 and F12, in the solution of function F11, DDWPA still maintains its advantages in the results of the optimal value, average value, worst value, and variance, and achieves a high success rate, indicating that the theoretical optimal value is obtained in multiple solution processes, demonstrating good search ability and the ability to jump out of local optimal solutions. In the solution of F12, all evaluation indicators have made progress, and the average value, worst value, and variance are one order of magnitude better than those of WPA, which also fully proves the effectiveness of the improved algorithm.

[0224] Generally speaking, DDWPA shows better results than WPA in all indicators. This indicates that DDWPA has higher stability and reliability during the solution process and can better adapt to different test functions and dimensions. To visually show the significant gap, the optimization results of functions F1 to F12 are presented in a line chart, as Figure 10 .

[0225] Figure 10 shown. In figure (a), it is the line chart of the comparison of the optimal values of the results before and after the improvement; in figure (b), it is the comparison chart of the average values of the results before and after the improvement; in figure (c), it is the line chart of the comparison of the worst values of the results before and after the improvement; in figure (d), it is the line chart of the comparison of the variances of the results before and after the improvement. In function F4, DDWPA does not show an advantage, and its optimal value and variance are both greater than the WPA result values, which well illustrates that no algorithm is perfect. Except for F4, DDWPA shows its superiority in the optimization of most other functions, and its optimal value and variance values are both smaller than those of WPA, proving the good global optimization and stability of DDWPA.

[0226] From the perspective of function dimension, in the solution of unimodal low-dimensional functions F1, F2, and F3, DDWPA also performs better than WPA. Especially for the F1 function, its final value result is much smaller than the final result of WPA, demonstrating the superiority of DDWPA in dealing with the optimal value problem of unimodal low-dimensional functions. In the results of dealing with multimodal low-dimensional functions, DDWPA shows stronger stability when solving the F6 function, and the results of solving the F8 function are better than those of WPA, showing good computational accuracy. In solving fixed-dimensional multimodal standard test functions, DDWPA still shows good performance. Although there is no significant difference in the optimal value, average value, and worst value in the results of solving F11 and F12, it shows a leading advantage in terms of standard deviation.

[0227] 5.3. Quantitative analysis of experimental results

[0228] To more comprehensively evaluate the performance of DDWPA, a comprehensive comparative experiment is conducted between it and 14 current advanced swarm intelligence optimization algorithms on the CEC2020 and CEC2022 test sets.

[0229] 5.3.1 CEC2022 test results

[0230] When using the CEC2022 test function for verification, DDWPA is compared with 14 algorithms in dimensions 10 and 20. The results of different dimensions are presented in Tables 4 and 5 respectively, and the statistical graphs of the optimal value comparison are as shown in Figure 11 、 Figure 12 shown, the convergence curves are as shown in Figure 13 、 14 shown, and the box plots are as shown in Figure 15 、 16 shown.

[0231] Table 4 Experimental results of 15 algorithms in CEC-2022 test functions (dimension = 10)

[0232]

[0233]

[0234] Table 5 Experimental results of 15 algorithms in CEC-2022 (dimension = 20)

[0235]

[0236]

[0237] Figure 11 、 Figure 12The comparison of the optimal value results obtained by different algorithms in each dimension is shown. It can be seen intuitively from the height of the bar graph that DDWPA has achieved lower results of the optimal value (or close to the optimal value) on multiple test functions, which means that it can find higher quality solutions when solving these optimization problems. Especially in the solution of function F12, the height of the bar graph intuitively reflects that the results obtained by DDWPA in both dimensions are better than those of other algorithms. In the 10-dimensional test, DDWPA obtained the best value on 7 functions compared with other algorithms, showing good robustness. Although DDWPA's performance on some functions in the 20-dimensional test was not ideal, it still achieved good results and significantly outperformed most other algorithms in these cases. For example, for functions F2 and F10, although DDWPA did not achieve the best results, it was basically at the same level as other algorithms. It is worth noting that DDWPA did not obtain the worst fitness in the test results of 10 and 20 dimensions. Although DDWPA does not perform as well as expected on the F2, F6, and F11 functions in the 20-dimensional test, it still achieves decent results, significantly outperforming most other algorithms in these cases.

[0238] Figure 13 , 14 The convergence curves of DDWPA and 14 benchmark algorithms are shown. By observing the convergence speed of each algorithm, it can be found that DDWPA shows a faster convergence speed in the early stage and quickly approaches the optimal solution. This shows that DDWPA may have a more efficient search mechanism and faster convergence speed in algorithm design. Especially on some test functions (such as F2 and F5), the convergence curve of the DDWPA algorithm is significantly better than other algorithms, and the error level can be maintained at a low level as the number of iterations gradually increases. DDWPA showed high stability throughout the optimization process, and its convergence curve was relatively smooth without large fluctuations. The search strategy and population update mechanism adopted by DDWPA enable it to better avoid falling into local optimal solutions. On the CEC-2022 test set (D=20), as the number of iterations increases, the convergence curve of the DDWPA algorithm drops rapidly and stabilizes at a low error value, indicating that it has strong global search capabilities and a faster convergence speed.

[0239] Figure 15 and Figure 16The performance and results of DDWPA and 14 algorithms in two dimensions of the CEC2022 test set are presented in the form of box plots. These two figures clearly reveal that when facing the challenges of complex and variable test functions, each algorithm exhibits a completely different efficiency spectrum. The box plot of DDWPA is at a relatively low level in the lower quartile, median, and upper quartile, indicating that DDWPA can obtain relatively stable and low objective function values in multiple runs. At the same time, the box plot of DDWPA is relatively compact without too many outliers, while the box plots of some algorithms are stretched longer, possibly due to getting trapped in local optima during the search process and resulting in outliers, which further verifies its stability and reliability in the optimization process. The box plot of the DDWPA algorithm shows a small degree of dispersion and generally has short whiskers, indicating that the performance of the algorithm fluctuates less in multiple runs and has relatively good stability, further indicating that the algorithm can avoid or jump out of local optimal solutions during the search process.

[0240] 5.3.1 CEC2020 Test Results

[0241] When using the CEC2020 test function for verification, it is compared with 14 algorithms in 10 dimensions and 20 dimensions. The results in different dimensions are presented in Tables 6 and 7 respectively, and the bar chart of the optimal value statistics is as Figure 17 , 18 shown, and the convergence curves are as Figure 19 , 20 shown, and the box plots are as Figure 21 , 22 shown.

[0242] Table 6 Experimental Results of 15 Algorithms in CEC-2020 (Dimension = 10)

[0243]

[0244] Table 7 Experimental Results of 15 Algorithms in CEC-2020 (Dimension = 20)

[0245]

[0246]

[0247] Figure 17 , Figure 18Shows the comparison of the optimal value results obtained by different algorithms for the CEC2020 test set in each dimension. On multiple functions, DDWPA has demonstrated excellent performance. Taking F7 as an example, the optimal value of DDWPA on this function is significantly lower than that of some other algorithms. This indicates that when solving this function, the search strategy and solution method of DDWPA are more efficient and can converge to the global optimal solution or a solution close to the global optimal solution faster. It can be directly seen from the height of the bar chart that DDWPA has obtained good optimal value results on multiple test functions, which means that it can find solutions with higher quality when solving these optimization problems. Especially in the solution of functions F4, F5, and F7, it can be intuitively reflected from the height of the bar chart that the results obtained by DDWPA in the 10-dimensional test are better than those of other algorithms. Although in the 20-dimensional test, the performance of DDWPA on some functions is not ideal, it still achieved good results and showed superiority over most other algorithms in these cases. It is worth noting that the test results of DDWPA in the 10-dimensional and 20-dimensional tests still did not show the situation of obtaining the worst fitness value.

[0248] Figure 19 , 20 Shows the convergence curves of DDWPA and 14 benchmark algorithms. By observing the convergence speed of each algorithm, it can be found that DDWPA shows a relatively fast convergence speed in the initial stage and quickly approaches the optimal solution. For example, in the test function F1, the convergence curve of the DDWPA algorithm is better than that of most other algorithms, and the curve drops relatively fast, proving that DDWPA has a relatively fast convergence speed and can still maintain a low error level as the number of iterations increases. DDWPA shows high stability throughout the optimization process, and its convergence curve is relatively smooth without large fluctuations, which indicates that it can better avoid local optimal solutions during the search process and has the ability to jump out of local optimal values. This is due to the search strategy and population update mechanism adopted by DDWPA, which enables it to better avoid falling into local optimal solutions.

[0249] Figure 21 and Figure 22The performance and results of DDWPA and 14 algorithms in two dimensions of the CEC2020 test set are shown in the form of box plots. When facing the challenges of complex and variable test functions, the 15 algorithms tested showed very different performance. The box plot of DDWPA is relatively stable, clearly showing that its lower quartile, median, and upper quartile are all at relatively low levels, indicating that DDWPA can obtain relatively stable and low objective function values in multiple runs. At the same time, the box plot of DDWPA is relatively compact, indicating a small degree of dispersion, generally having short whiskers and not too many outliers, which further verifies its good stability and reliability, with small performance fluctuations in multiple runs and relatively good stability, further indicating that the algorithm can avoid or jump out of local optimal solutions during the search process.

[0250] 4. Conclusion

[0251] The present invention proposes an improved wolf pack algorithm, aiming to solve the problems of low convergence accuracy, long running time, and easy entrapment in local optima existing in the traditional wolf pack algorithm when solving multi-peak high-dimensional functions. By introducing strategies such as position change record values, distance determination factors, replicating individual wolves, and optimizing the population update mechanism, the improved algorithm of the present invention has achieved significant optimization in aspects such as wandering behavior, summoning behavior, besieging behavior, and population update. The experimental results show that the improved wolf pack algorithm has significantly improved the solution accuracy compared with the original algorithm in low-dimensional unimodal functions, and at the same time further improved the solution accuracy of high-dimensional multi-peak functions. In addition, the algorithm also has characteristics such as good global convergence and fast running speed, and performs excellently on multiple standard test functions. At the theoretical level, the present invention deeply analyzes the convergence of the algorithm by using the memoryless property of the Markov chain, and proves through mathematical derivation that the population sequence of the improved wolf pack algorithm is ergodic, thus ensuring that the algorithm can converge to the global optimal solution of the problem to be solved with probability 1. However, although the improved wolf pack algorithm has achieved remarkable results in many aspects, there are still some deficiencies. For example, when dealing with extremely complex or ultra-high-dimensional functions, its convergence speed and accuracy still need to be further improved. In addition, the parameter settings in the algorithm (such as wandering step size, raiding step size, besieging step size, and update ratio factor, etc.) have a great impact on the algorithm performance. How to adaptively adjust these parameters to adapt to different problem scales and solution space characteristics is an important direction for future research.

[0252] The application embodiment of the present invention provides a computer device, which includes a memory and a processor. The memory stores a computer program, and when the computer program is executed by the processor, the processor executes the steps of the high-dimensional function solving method based on the distance determination wolf pack algorithm.

[0253] An application embodiment of the present invention provides a computer-readable storage medium storing a computer program. When the computer program is executed by a processor, the processor is caused to execute the steps of a method for solving a high-dimensional function based on a distance-determined wolf pack algorithm.

[0254] An application embodiment of the present invention provides an information data processing terminal, and the information data processing terminal includes a high-dimensional function solving system based on a distance-determined wolf pack algorithm.

[0255] It should be noted that the embodiments of the present invention can be implemented by hardware, software, or a combination of software and hardware. The hardware part can be implemented using dedicated logic; the software part can be stored in a memory and executed by an appropriate instruction execution system, such as a microprocessor or dedicated designed hardware. Those of ordinary skill in the art can understand that the above devices and methods can be implemented using computer-executable instructions and / or included in processor control code, for example, such code is provided on a carrier medium such as a disk, CD, or DVD-ROM, a programmable memory such as a read-only memory (firmware), or a data carrier such as an optical or electronic signal carrier. The devices and modules of the present invention can be implemented by hardware circuits of programmable hardware devices such as very large scale integrated circuits or gate arrays, semiconductors such as logic chips and transistors, or field programmable gate arrays and programmable logic devices, can also be implemented by software executed by various types of processors, or can be implemented by a combination of the above hardware circuits and software such as firmware.

[0256] As described above, the above are only specific embodiments of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art within the technical scope disclosed by the present invention, any modifications, equivalent replacements, and improvements made within the spirit and principle of the present invention should be covered within the protection scope of the present invention.

Claims

1. A method for solving high-dimensional functions based on a distance-determined wolf pack algorithm, characterized in that: include: Step 1: First, in the N×D solution space, set the size of the wolf pack and then randomly generate artificial wolves to initialize the wolf pack; at this time, the spatial position coordinates X of each artificial wolf i They are all randomly generated, and the objective function value Y of each artificial wolf's location is calculated i , select the artificial wolf with the largest function value as the leader, and set the objective function value Y i The next best R wolf is selected as the scout wolf, and the rest are fierce wolves; Step 2: The scout wolf performs wandering behavior to greedily search the solution space; each scout wolf will move forward in each of the random h directions for a wandering step length while comparing the objective function values ​​corresponding to the previous and next positions, and finally move forward in the direction with the largest objective function value. If the objective function value of the original position is the largest, it will return to the original position; after the scout wolf finishes wandering, the objective function values ​​of all artificial wolves are compared, and the wolf with the largest objective function value becomes the new leader; the position of the scout wolf that ends wandering is calculated by the wandering formula, as shown in formula (1) In the formula is the walking step length; d = 1, 2, ..., D; P = 1, 2, ..., h. Since wolves in nature have different search and walking abilities, the value of h is random. h is a random integer. Generally, the larger h is, the more directions the wolf can search, and the better the global optimization performance is, but it will also increase the search time. Step 3: The alpha wolf howls to summon the wolves to come closer; all the wolves advance one galloping step toward the alpha wolf. Charge towards the alpha wolf until the distance between the alpha wolf and the alpha wolf is less than the siege distance; during the charge process, if the alpha wolf finds a position with a larger target function value than the alpha wolf, it will replace it as the new alpha wolf and initiate the summoning behavior again; when all wolves reach the siege distance, the wolf pack switches to siege behavior; the position change of the alpha wolf during the charge process is as follows: In the formula This is the position of the wolf before the attack. is the current position of the alpha wolf, The position of the k+1th generation of fierce wolves after the raid; Step 4: The wolf pack launches a siege on the prey, and regards the current leader as the prey. The fierce wolf and the scout wolf launch a siege on the prey and move forward one siege step in the direction of the prey. During the siege, if the objective function value corresponding to the current prey position is found to be better than that of the current prey position, the prey position is updated and the siege is launched again until the termination condition is reached. The position change of the artificial wolf during the siege is as follows: λ is a random number with a value between (0, 1), Indicates the siege step length; Step 5: The wolf pack eliminates the R artificial wolves with the smallest objective function value according to the set population update scale factor, and randomly generates R new artificial wolves in the solution space to complete the population update; R is a random number whose value range is [N / (2β), N / β], and β is the update scale factor; Step 6: Determine whether the accuracy requirement of the solution or the termination condition of the maximum number of iterations is met. If the termination condition is met, output the spatial position of the leader wolf, which is the optimal solution to the problem. Otherwise, continue to iterate until the termination condition is met.

2. The high-dimensional function solving method based on the distance determination wolf pack algorithm according to claim 1 is characterized in that: WPA has a walking step length Stride length and attack step length Three step sizes; suppose the value range of the variable in the d-th dimension space is [min d ,max d ], then the relationship between the three step sizes in the d-th dimension space is as shown in formula (4): In the above formula, S is the step size factor, which indicates the precision of the search.

3. The high-dimensional function solving method based on the distance determination wolf pack algorithm according to claim 1 is characterized in that: After each wandering process, the poorly performing wolves are eliminated and an equal number of new artificial wolves are generated to ensure the search accuracy, which can speed up the operation of WPA and improve the global convergence. At the same time, in order to improve the efficiency of wandering, the alpha wolf is selected after the population initialization is improved to the alpha wolf after the first wandering, so as to avoid the alpha wolf usually updated in the early stage of the algorithm is only a local optimal solution and wastes computing time; after the population initialization is completed, the explorer wolf starts to wander around itself to find the optimal solution, and the basic wandering formula is in, is the current wolf detection position, To explore the wolf's position after wandering, is the walking step length, h is the walking direction, p=1,2,.....,h; after the explorer walks in h directions respectively, it returns to the original position, and finally chooses to walk in the direction with the maximum objective function value after moving forward. If the original position is optimal, it does not change the position; At the end of each walk, A will be updated to a non-zero row vector. At this time, the wolf with the largest objective function value will be selected as the leader. The Q wolves with the smallest A(i) value will be eliminated and a new scout will be generated. Q is a random number with a value between (1, n / a), and its calculation formula is as follows: Q=floor(rand(1)*(n / a))+1 (6) Among them, a is the scout wolf elimination factor, which is usually taken as [16,20]. Generally, the larger the wolf pack is, the more scout wolves are eliminated. By eliminating the artificial wolves that "work passively" during the wandering process, the search ability of the wolf pack is improved, thereby accelerating the convergence speed of the algorithm and finding the global optimal solution faster.

4. The high-dimensional function solving method based on the distance determination wolf pack algorithm according to claim 1 is characterized in that: After the wandering is over, the alpha wolf summons the wolves to charge towards it. During this process, if any other wolf's objective function value is found to be greater than that of the alpha wolf, the alpha wolf is replaced and summoned again. The position update during the initial WPA charge is as shown in formula (7): in, and These are the positions of the wolves before and after the attack. For the running stride length, The current position of the alpha wolf.

5. The high-dimensional function solving method based on the distance determination wolf pack algorithm according to claim 1 is characterized in that: After the scout wolf finishes wandering, the alpha wolf summons the wolf pack to charge at it, treating the alpha wolf as prey. In the initial WPA siege formula, the λ value range is [-1,1], which leads to multiple returns to the original position during the siege. When the λ value is negative, Greater than When λ is a positive number, Less than Ultimately, the search efficiency is reduced. At the same time, in extreme cases, if the wolf's attack step length is too large, it is easy to jump out of the siege range, making it difficult for the wolf pack to search for the optimal value. Before the wolf starts its attack, first determine its position relationship with the leader wolf. Since the search space is an N×D solution space, the position of the artificial wolf will be a D-dimensional coordinate. In each dimension, the value of λ is divided into two cases. If This means that when the fierce wolf has not surpassed the leader, the value of λ is positive, and the fierce wolf can always move towards the leader. When the fierce wolf has not surpassed the position of the alpha wolf, λ is negative, and the fierce wolf can always move towards the alpha wolf; at the same time, when the siege step length is too large and the fierce wolf surpasses the position of the alpha wolf, and The wolf can adjust its direction of advance based on the size of its prey and continuously narrow its search range for prey.

6. The high-dimensional function solving method based on the distance determination wolf pack algorithm according to claim 1 is characterized in that: Introduce a distance determination factor for population update. When updating the wolf pack, new artificial wolves that meet the conditions will be retained. The implementation process is as follows: Step 1: Record the original positions of the R wolves to be eliminated and the position of the current leader wolf; Step 2: Randomly generate an R*m position matrix to update the position of the artificial wolf; Step 3: Determine two conditions: calculate the distance to the old position, and if it is greater than the distance determination factor, update the position; Calculate the distance to the current leader wolf. If the distance is greater than the maximum distance between the current artificial wolf and the leader wolf, update the position; Wolves that do not meet the conditions are randomly updated again until all R wolves are updated; Step 4: The wolf pack is updated and the leader of the new wolf pack is selected.

7. A high-dimensional function solving system based on a distance-determined wolf pack algorithm that implements the high-dimensional function solving method based on a distance-determined wolf pack algorithm as claimed in any one of claims 1 to 6, characterized in that: include: Wolf pack initialization module: First, in the N×D solution space, the wolf pack size is set and then artificial wolves are randomly generated to initialize the wolf pack; at this time, the spatial position coordinates X of each artificial wolf are i They are all randomly generated, and the objective function value Y of each artificial wolf's location is calculated i , select the artificial wolf with the largest function value as the leader, and set the objective function value Y i The next best R wolf is selected as the scout wolf, and the rest are fierce wolves; Solution space search module: The scout wolf performs wandering behavior to greedily search the solution space; each scout wolf will move forward in each of the random h directions for a wandering step length while comparing the objective function values ​​corresponding to the previous and next positions, and finally move forward in the direction with the largest objective function value. If the objective function value of the original position is the largest, it will return to the original position; after the scout wolf finishes wandering, it compares the objective function values ​​of all artificial wolves, and the wolf with the largest objective function value becomes the new leader; the position of the scout wolf that ends wandering is calculated by the wandering formula, as shown in formula (1) In the formula is the walking step length; d = 1, 2, ..., D; P = 1, 2, ..., h. Since wolves in nature have different search and walking abilities, the value of h is random. h is a random integer. Generally, the larger h is, the more directions the wolf can search, and the better the global optimization performance is, but it will also increase the search time. Wolf Summoning Module: The alpha wolf howls to summon the wolves to come closer; all wolves advance one galloping step toward the alpha wolf. Charge towards the alpha wolf until the distance between the alpha wolf and the alpha wolf is less than the siege distance; during the charge process, if the alpha wolf finds a position with a larger target function value than the alpha wolf, it will replace it as the new alpha wolf and initiate the summoning behavior again; when all wolves reach the siege distance, the wolf pack switches to siege behavior; the position change of the alpha wolf during the charge process is as follows: In the formula This is the position of the wolf before the attack. is the current position of the alpha wolf, After the raid k+1 Replace the wolf's position; Wolf pack siege module: The wolf pack launches a siege on the prey, and regards the current leader as the prey. The fierce wolf and the scout wolf initiate a siege on the prey and move forward one siege step in the direction of the prey. During the siege, if the objective function value corresponding to the current prey position is found to be better than that of the current prey position, the prey position is updated and the siege is initiated again until the termination condition is reached. The position change of the artificial wolf during the siege is as follows: λ is a random number with a value between (0, 1), Indicates the siege step length; Proportional factor update module: The wolf pack updates the proportional factor according to the set group, eliminates the R artificial wolves with the smallest objective function value, and randomly generates R new artificial wolves in the solution space to complete the population update; R is a random number, and its value range is [N / (2β), N / β], β is the update proportional factor; Termination condition judgment module: judge whether the accuracy requirement of the solution or the termination condition of the maximum number of iterations is met. If the termination condition is met, the spatial position of the leader wolf is output, which is the optimal solution to the problem. Otherwise, continue to iterate until the termination condition is met.

8. A computer device, comprising a memory and a processor as claimed in any one of claims 1 to 6, wherein the memory stores a computer program, and when the computer program is executed by the processor, the processor executes the steps of the high-dimensional function solving method based on the distance judgment wolf pack algorithm.

9. A computer-readable storage medium storing a computer program, which, when executed by a processor, enables the processor to execute the steps of the high-dimensional function solving method based on the distance determination wolf pack algorithm as described in any one of claims 1 to 6.

10. An information data processing terminal, comprising the high-dimensional function solving system based on the distance determination wolf pack algorithm as claimed in claim 7.

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