Improved salmons swarm optimization algorithm

Abstract

The invention provides an improved salmons swarm optimization algorithm, and aims to solve the problems that the global exploration capability of the algorithm is weak, and the algorithm is easy to fall into local optimum in the later stage, and the method comprises the following steps: firstly, solving a population with more uniform distribution as an initial population through adding Cubic chaotic mapping, increasing the diversity of salmons population, and improving the convergence speed of the algorithm; secondly, a spiral search strategy is added in the prey search stage, so that the salmons have multiple search paths to better adjust the positions of the salmons, and the global search performance of the algorithm is improved; and finally, a sparrow early warning mechanism is introduced, so that the rate of convergence of the Karat swarm algorithm is increased more quickly. According to the method, 12 basic test functions of a CEC2017 test set are used for testing the improved algorithm, and compared with path planning experiments of PSO, GWO, AWOA, GA and SCSO algorithms under a complex map, the optimal values of the shortest paths of the improved algorithm are reduced by 9.46%, 14.83%, 14.32%, 5.76% and 1.06% respectively. It is verified that the search efficiency of the algorithm is improved, the capability of avoiding falling into local optimum is high, and the convergence speed, the convergence precision and the optimization time are all superior.

Description

Technical Field

[0001] The present invention belongs to the technical field of sand cat swarm optimization algorithms, and particularly relates to an improved sand cat swarm optimization algorithm. Technical Background

[0002] Swarm intelligence optimization algorithms originate from the behaviors of organisms in nature such as predation, fighting, and mating. Affected by different constraints such as population size, initial position, food quantity, and environment, the position is updated to search for the optimal position and obtain the best fitness value. During the iterative search process, it usually goes through two stages: global exploration and local exploitation. These two stages are sequential rather than synchronous processes, which may lead to the dilemma of being easily trapped in local optimal solutions. To enhance the global search ability and accelerate the convergence speed, researchers have proposed different thinking strategies for improving the algorithm, aiming to enhance the local search ability while accelerating the convergence speed to obtain the global optimal solution. Among them, the hybrid strategy has the most prominent improvement effect and can achieve higher accuracy in practical engineering applications.

[0003] Grey wolf optimization algorithm, dung beetle optimization algorithm, whale optimization algorithm, Harris hawk optimization algorithm, sparrow optimization algorithm, etc. as the original swarm intelligence algorithms have been continuously concerned by researchers. Common improvement strategies include using mapping strategies to initialize population diversity instead of random generation, integrating opposition learning strategies to expand the search ability, adding multiple strategies to increase the overall search range of iterations, etc. These strategies all help to accelerate the convergence speed. However, the current sand cat swarm optimization algorithm usually shows local optimality during the global optimization stage in the entire iterative process, resulting in weak global search ability of the algorithm. Summary of the Invention

[0004] The technical problem to be solved by the present invention is to overcome the existing defects and provide an improved sand cat swarm optimization problem to solve the problem of local optimality that occurs during the global optimization of the sand cat swarm in the entire iterative process, resulting in weak global search ability and slow convergence speed of the algorithm as mentioned in the above technical background.

[0005] To achieve the above object, the present invention provides the following technical solution: An improved sand cat swarm optimization algorithm, including the following steps:

[0006] Step 2: Exploration stage: The search ability of the SCSO algorithm depends on the auditory ability in the low-frequency detection of sand cats. According to the working mechanism proposed by the algorithm, as the iteration progresses, this sensitivity value r G decreases from 2 to 0, gradually approaching the prey. To find the prey, assume that the sensitivity range of the sand cat is from 2 kHz to 0, as shown in the following formula. Among them, s M The value is inspired by the auditory characteristics of the sand cat and is assumed to be 2.

[0007]

[0008] Let \(t\) be the current iteration number and \(T\) be the maximum iteration number. The values are selected according to different problems, which also proves the flexibility and generality of the proposed equation.

[0009] The transition from the exploration stage to the development stage is determined by the parameter \(R\), and the calculation formula of \(R\) is obtained as shown in Equation (2):

[0010] \(R = 2\times r\) G \(\times rand() - r\) G (2)

[0011] \(rand()\) represents a random number between 0 and 1. During the search process, the position of each current searcher is updated based on a random position. In this way, the searchers can explore new spaces in the search space. To avoid falling into local optima, the sensitivity range of each sand cat is different, which is achieved by Equation (3):

[0012] \(r_2 = r\) G \(\times rand()\) (3)

[0013] \(r\) G represents the sensitivity range that decreases from 2 to 0, and \(r_2\) represents the sensitivity range of each individual sand cat, which is used for updating the position of the sand cat group in the exploration or development stage. \(r\) G is used to guide the parameter \(R\) to achieve the transfer control between the two stages.

[0014] In the SCSO algorithm, the sand cat group updates the individual positions according to the optimal position of the current population, its own position, and the sensitivity range, generating new positions between the initial position and the prey position, and using randomness to ensure the convergence and complexity of the algorithm. The mathematical model is as follows:

[0015]

[0016] represents the position where the optimal individual of the current population is located, is the current position of the sand cat

[0017] Step 3: Development stage: The sand cat perceives the direction of the prey through hearing. By mathematically modeling the stage of attacking the prey, the distance between the current position and the best position is represented, and the update formula is as follows:

[0018]

[0019] In the formula, P represents the candidate position. It is set that the sensitivity range of the sand cat is a circle, and the moving direction can be defined by a random angle θ on the circle. θ takes a random value between 0° and 360°, and cosθ is limited between -1 and 1. In this way, each individual can ensure that it approaches the prey along the circumferential direction in the search space.

[0020] Step 4: The SSCSO algorithm is tested by the cec2017 test function set, and the convergence speed effect is not obvious. The reasons are as follows: First, in the initialization stage, the formula uses random numbers generated in the [lb, ub] interval to limit the initial position of the individual. Since the rand() function generates random numbers, the population is not randomly distributed. The reduction of population diversity makes it difficult for the algorithm to obtain the optimal solution in multiple iterations, resulting in a decrease in the convergence speed. Second, the traditional sand cat optimization algorithm may lack an effective global search strategy, resulting in the inability to quickly converge to the vicinity of the optimal solution during the search process;

[0021] Step 5: The chaotic mapping has good randomness, regularity, and ergodicity, which can not only make the population diverse but also improve the algorithm in terms of global search ability, convergence speed, and convergence accuracy. The SSCSO algorithm uses the Cubic chaotic mapping function in the initial stage, changing the way of finding random positions between the upper and lower bounds, getting rid of the problem of local path search, and expanding the search space to a certain extent. The standard Cubic chaotic mapping function is shown in Equation (6):

[0022]

[0023] where a and b are chaotic influence factors, and their values directly affect the range of the Cubic mapping. When a = 1, x n ∈(2, 2); when a = 4, x n ∈(-1, 1). When a ∈ (2.3, 3), the sequence generated by the Cubic mapping is in a chaotic state;

[0024] The relevant analysis of the maximum Lyapunov exponent of the commonly used one-dimensional chaotic mapping is carried out. The expression of the Cubic chaotic mapping function is shown in Equation (7):

[0025]

[0026] In the formula, ρ is the control parameter, and its value affects the chaos of the Cubic mapping. x n ∈(0, 1). The chaos of the Cubic mapping is similar to the maximum Lyapunov exponents of the Logistic mapping and Tent mapping, and is better than one-dimensional mappings such as the Sine mapping and Circle mapping. In the present invention, (0, 1) is x nFor the initial value range, when \(x_0 = 0.3\) and \(\rho = 2.595\), the Cubic chaotic map has good ergodicity;

[0027] Step Six: Spiral Search Strategy: It is a heuristic search algorithm mainly used to solve optimization problems and can be adjusted and optimized according to the characteristics of specific problems, having a certain degree of flexibility and adaptability in practical applications. When the sand cat swarm optimization algorithm attacks prey, it will randomly select an angle through the roulette method to approach the prey for attack. However, such an attack method has a large degree of randomness and is also prone to falling into a local optimum. During the stage of searching for prey, each sand cat individual updates its position at the next moment according to the best candidate position and the current position and its sensitivity range \(r_2\). Therefore, each sand cat individual can search for other feasible best prey positions. The update formula is shown in Equation (8):

[0028]

[0029] To enable the sand cat swarm to search for more achievable paths for position updates, the spiral search strategy is introduced. The defined parameter \(k\) in this strategy affects the decay rate of the algorithm. A larger value of \(k\) will cause the algorithm to fall into a local optimum, while a smaller value of \(k\) will result in a slower convergence rate of the algorithm. To solve this problem, based on the spiral search strategy, the strategy factor \(Z\) is designed according to the mathematical model of the spiral. On the basis of the original spiral model, the number of iterations is introduced to dynamically adjust the shape of the spiral. As the number of iterations increases, the parameter \(Z\) causes the spiral shape to change from large to small. The improved formula is shown in Equation (9):

[0030]

[0031] In the formula, \(Z\) is the spiral search factor and \(k\) is the parameter. By adding the spiral factor, the global search performance of the algorithm is effectively improved. The new spiral position update formula is as follows:

[0032]

[0033] \(l\) is the difference between the position at the next moment and the best position. The sand cat swarm uses the spiral search strategy to explore in the search space in a spiral form, thereby enhancing the exploration ability of unknown regions, avoiding an increased possibility of falling into a local optimum solution, and improving the global search performance of the algorithm;

[0034] Step 7: Sparrow Early Warning Mechanism: The sparrow early warning mechanism is mainly used to simulate the information transmission and collaborative behavior in a sparrow group. Through the early warning mechanism, sparrow individuals can transmit and receive information from each other to adjust their own behaviors and search directions, which helps to improve the convergence speed and global search ability of the algorithm and better solve various optimization problems. After the sand cat group searches for prey, the sparrow warning mechanism is added, which can make the sand cat group algorithm improve the convergence speed faster. When the sand cat group is in danger, in order to obtain a better position, the sand cats at the edge will quickly move to the safe area, while the sand cats in the middle of the population will move randomly to approach other sand cats. The mathematical expression is shown in Equation (11):

[0035]

[0036] In the formula, is the globally optimal position of the current search. β is the step size control parameter, which is a random number obeying the normal distribution with a mean of 0 and a variance of 1. K is a random number, and its value range is [-1, 1], which represents the moving direction of the sparrow and is also the control parameter of the step size. fi is the fitness value of the current sand cat individual, f g and f w are the best fitness value and the worst fitness value of the current global position respectively. ε is a constant to avoid the situation where the fraction is 0. f i >f g means that this sand cat is in a vulnerable position at the edge of the group. f i =f g means that the sand cat located inside the population realizes the danger and needs to get closer to other sand cats to avoid being preyed on;

[0037] Step 8: Algorithm Process: (1) Initialization: Initialize the population size, dimension variables, number of iterations, upper and lower boundary limits, fitness function, etc.; If R ≤ 1, update the position of the sand cat individual and calculate the fitness value according to Equation (5). If R > 1, update the position of the sand cat individual and calculate the fitness value according to Equation (11). After the exploration stage, add the sparrow early warning mechanism, use Equation (12) to update the position of the sand cat individual and calculate the optimal fitness value, compare the obtained fitness values, and select the position with the optimal fitness value to update the position of the sand cat individual;

[0038] Step 9: Build a Simulation Experiment Environment: The experimental hardware is a computer with 12th Gen Intel(R) Core(TM) i7-12650H, 32GB of memory, and 2.30GHz. The software is MATLAB R2022a;

[0039] Step 10: Performance Comparison of Benchmark Test Functions: Parameter Settings: The initial population size is 30, the number of iterations is 500, the dimension is 100. To compare the performance of the SSCSO algorithm, 12 benchmark test functions in the CEC2017 test function set were selected for testing this time;

[0040] Step 11: The comparison algorithms are selected as PSO, AWOA, GA, GWO, and SCSO. To ensure the rationality of the algorithm, each benchmark test function was experimented 30 times and the average value was taken;

[0041] Step 12: It can be seen from the experimental data that in high-dimensional unimodal functions, multimodal functions, hybrid functions, and composite functions, the SSCSO algorithm has a faster convergence speed, a smaller optimal value, and less optimization time than the other five algorithms;

[0042] Step 13: Compare the performance of the improved algorithm;

[0043] Step 14: To make the algorithm fall into the local optimal value and improve the convergence speed, a hybrid scheme based on the Cubic chaotic mapping function, spiral search strategy, and sparrow early warning mechanism is proposed to optimize the algorithm, and its performance is compared with the five original algorithms on the benchmark test functions;

[0044] In the above Step 1, the initialization process: swarm intelligence optimization algorithms often generate randomly. The population initialization of the SCSO algorithm is to randomly generate some point spaces in the search space of the problem, and these points are individuals. Assuming the population size is M and the problem dimension is N, the initialization formula is shown as follows:

[0045] x ij = lb j + r1 × (ub j - lb j )

[0046] i = 1, 2, …, M; j = 1, 2, …, N, x ij represents the j-th dimension variable of individual i, lb j and ub j represent the lower bound and upper bound of the j-th dimension variable respectively, and r1 is a random number between [0, 1];

[0047] In the above Step 5, the Cubic chaotic mapping function makes the population distribution more uniform, enhances the global search ability, shortens the time to jump out of the local area, and improves the convergence speed;

[0048] In step six, for the spiral search strategy, in order to enable the sand cat swarm to search for more achievable paths for position update, the spiral search strategy is introduced. The defined parameter k in this strategy affects the decay rate of the algorithm. A larger k value will cause the algorithm to fall into a local optimum, while a smaller k value will result in a slower convergence rate of the algorithm. To solve this problem, based on the spiral search strategy, a strategy factor Z is designed according to the mathematical model of the spiral. On the basis of the original spiral model, the number of iterations is introduced to dynamically adjust the shape of the spiral. As the number of iterations increases, the parameter Z will cause the shape of the spiral to change from large to small;

[0049] In step seven, for the sparrow early warning mechanism, after the sand cat swarm finishes searching for prey, a sparrow warning mechanism is added. This can enable the sand cat swarm algorithm to improve the convergence rate faster. When the sand cat swarm is in danger, in order to obtain a better position, the sand cats at the edge will quickly move to the safe area, while the sand cats in the middle of the population will move randomly to approach other sand cats;

[0050] In step eleven, through the data, it can be seen that the SSCSO algorithm has better optimization ability, that is, a faster convergence rate, on the unimodal function F1, the simple multimodal functions F4 and F9, the hybrid functions F13 - F15, F18, and F19, and the composite functions F25, F27, F28, and F30. The stability of the algorithm on the functions F1, F9, F15, F27, and F28 is slightly lower than that of the comparative algorithm. From this, it can be known that methods such as the Cubic chaotic mapping function, the spiral search strategy, and the sparrow early warning mechanism adopted for the algorithm can improve the optimization ability of the algorithm;

[0051] In step twelve, after comparing the test performance of the five original algorithms on the benchmark functions, except that the optimal fitness value is close on the high-dimensional simple multimodal function F9, the SSCSO algorithm shows significant optimization ability and a faster convergence rate. From this, it can be known that the improved strategy has the advantage of making up for the original algorithm's tendency to fall into local optima, and the global search ability is enhanced;

[0052] Compared with the prior art, the present invention provides an improved sand cat swarm optimization algorithm, which has the following

[0053] Beneficial effects:

[0054] The present invention optimizes it by proposing a hybrid scheme based on the Cubic chaotic mapping function, the spiral search strategy, and the sparrow early warning mechanism. After comparing the test performance of the five original algorithms on the benchmark functions, except that the optimal fitness value is close on the high-dimensional simple multimodal function F9, the SSCSO algorithm shows significant optimization ability and a faster convergence rate. From this, it can be known that the improved strategy has the advantage of making up for the original algorithm's tendency to fall into local optima, and the global search ability is enhanced.

[0055] From the data, it can be seen that the SSCSO algorithm has better optimization ability, that is, faster convergence speed, on the unimodal function F1, simple multimodal functions F4 and F9, hybrid functions F13 - F15, F18, F19, and composite functions F25, F27, F28, F30. On functions F1, F9, F15, F27, F28, the stability of the algorithm is slightly lower than that of the comparative algorithm. Thus, it can be known that methods such as the Cubic chaotic mapping function, spiral search strategy, and sparrow early warning mechanism adopted for the algorithm can improve the optimization ability of the algorithm; Description of the Drawings

[0056] The drawings are used to provide a further understanding of the present invention and constitute a part of the specification. Together with the embodiments of the present invention, they are used to explain the present invention and do not constitute a limitation to the present invention. In the drawings:

[0057] Figure 1 is the flowchart of the improved sand cat optimization algorithm proposed by the present invention;

[0058] Figure 2 is the fitness value curve of the improved sand cat optimization algorithm proposed by the present invention on the unimodal function;

[0059] Figure 3 is the fitness value curve of the improved sand cat optimization algorithm proposed by the present invention on the multimodal function;

[0060] Figure 4 is the fitness value curve of the improved sand cat optimization algorithm proposed by the present invention on the composite function;

[0061] Figure 5 is the fitness value curve of the improved sand cat optimization algorithm proposed by the present invention on the hybrid function;

[0062] Figure 6 is the application of the improved sand cat optimization algorithm proposed by the present invention in robot path planning; Detailed Embodiments

[0063] The following further describes the present invention in detail with reference to the drawings and examples. It should be noted that: the specific implementation examples described here are only used to explain the present invention and are not used to limit the present invention. In addition, for the convenience of description, only parts related to the present invention rather than all structures are shown in the drawings

[0064] Please refer to Figures 1-6 , the present invention provides a technical solution: an improved sand cat swarm optimization algorithm, including the following steps:

[0065] Step 1: In a d-dimensional optimization problem, a sand cat is a 1×d array representing the solution to the problem, and each variable value (x1, x2,..., x d) are all floating-point numbers and must be between the lower and upper bounds.

[0066] Step 2: Exploration stage: The search ability of the SCSO algorithm depends on the auditory ability in the low-frequency detection of the sand cat. According to the working mechanism proposed by the algorithm, as the iteration progresses, this sensitivity value will decrease from 2 to 0, gradually approaching the prey. To search for the prey, assume that the sensitivity range of the sand cat is from 2 kHz to 0, as shown in the following equation. Among them, the s G value is inspired by the auditory characteristics of the sand cat and is assumed to be 2; M where t is the current iteration number, T is the maximum iteration number, and the value selection is reset according to different problems, which also proves the flexibility and generality of the proposed equation.

[0067]

[0068] The transition from the exploration to the exploitation stage is determined by the parameter R, and the calculation formula of R is obtained as shown in equation (2):

[0069] R = 2 × r

[0070] × rand() - r G × rand() (2) G (2)

[0071] rand() represents a random number between 0 and 1. During the search process, the position update of each current searcher is based on a random position. In this way, the searchers can explore new spaces in the search space. To avoid being trapped in local optima, the sensitivity range of each sand dune cat is different, which is achieved by equation (3):

[0072] r2 = r G × rand() (3)

[0073] r G represents the sensitivity range that decreases from 2 to 0, r2 represents the sensitivity range of each individual sand cat, and is used for the position update of the sand cat group in the exploration or exploitation stage. r G is used to guide the parameter R to achieve the transfer control between the two stages.

[0074] In the SCSO algorithm, the sand cat group updates the individual position according to the optimal position of the current population, its own position, and the sensitivity range, generating a new position between the initial position and the prey position, and using randomness to ensure the convergence and complexity of the algorithm. The mathematical model is as follows:

[0075]

[0076] represents the position where the optimal individual of the current population is located, is the current position of the sand cat;

[0077] Step 3: Development stage: The sand cat perceives the orientation of the prey through hearing. By mathematically modeling the stage of attacking the prey, the distance between the current position and the optimal position is represented, and the update formula is as follows:

[0078]

[0079] In the formula, P represents the candidate position. The sensitivity range of the sand cat is set as a circle, and the moving direction can be defined by a random angle θ on the circle. θ takes a random value between 0° and 360°, and cosθ is limited between -1 and 1. In this way, each individual can ensure to approach the prey along the circumferential direction in the search space;

[0080] Step 4: The SSCSO algorithm is tested on the cec2017 test function set, and the convergence speed effect is not obvious. The reasons are as follows: First, in the initialization stage, the formula uses random numbers generated in the [lb, ub] interval to limit the initial position of the individual. Since the rand() function generates random numbers, the population is not randomly distributed, and the reduction of population diversity makes it difficult for the algorithm to obtain the optimal solution in multiple iterations, resulting in a decrease in the convergence speed. Second, the traditional sand cat optimization algorithm may lack an effective global search strategy, resulting in the inability to quickly converge to the vicinity of the optimal solution during the search process;

[0081] Step 5: The chaotic mapping has good randomness, regularity, and ergodicity, which can not only make the population diverse but also improve the algorithm in terms of global search ability, convergence speed, and convergence accuracy. The SSCSO algorithm uses the Cubic chaotic mapping function in the initial stage, changing the way of finding random positions between the upper and lower bounds, getting rid of the problem of local path search, and expanding the search space to a certain extent. The standard Cubic chaotic mapping function is shown in Equation (6):

[0082]

[0083] Among them, a and b are chaotic influence factors, and their values directly affect the range of the Cubic mapping. When a = 1, x n ∈(-2, 2); when a = 4, x n ∈(-1, 1). When a ∈ (2.3, 3), the sequence generated by the Cubic mapping is in a chaotic state.

[0084] The maximum Lyapunov exponent of the commonly used one-dimensional chaotic mapping is analyzed, and the expression of the Cubic chaotic mapping function is shown in Equation (7):

[0085]

[0086] where ρ is a control parameter, and the selection of its value affects the chaos of the Cubic mapping, and x n ∈(0,1). The chaos of the Cubic mapping is similar to the maximum Lyapunov exponents of the Logistic mapping and the Tent mapping, and is better than one-dimensional mappings such as the Sine mapping and the Circle mapping. In the present invention, (0, 1) is the initial value range of x n . When x0 = 0.3 and ρ = 2.595 are selected, the Cubic chaotic mapping has good ergodicity;

[0087] Step Six: Spiral Search Strategy: A heuristic search algorithm mainly used to solve optimization problems and can be adjusted and optimized according to the characteristics of specific problems, having a certain degree of flexibility and adaptability in practical applications. When attacking prey, the sand cat swarm optimization algorithm will randomly select an angle through the roulette method to approach the prey for attack, but such an attack method has a large degree of randomness and is also prone to falling into local optima. During the stage of searching for prey, an individual sand cat updates its position at the next moment according to the best candidate position and the current position and its sensitivity range r2. Therefore, an individual sand cat can search for other feasible best prey positions. The update formula is shown in Equation (8):

[0088]

[0089] To enable the sand cat swarm to search for more achievable paths for position updates, a spiral search strategy is introduced. The defined parameter k in this strategy affects the decay rate of the algorithm. A larger value of k will cause the algorithm to fall into local optima, while a smaller value of k will cause the algorithm to have a slower convergence rate. To solve this problem, based on the spiral search strategy, a strategy factor Z is designed according to the mathematical model of the spiral. On the basis of the original spiral model, the number of iterations is introduced to dynamically adjust the shape of the spiral. As the number of iterations increases, the parameter Z will cause the spiral shape to change from large to small. The improved formula is shown in Equation (9):

[0090]

[0091] In the formula, Z is the spiral search factor, and k is the parameter. By adding the spiral factor, the global search performance of the algorithm is effectively improved. The new spiral position update formula is as follows:

[0092]

[0093] l is the difference between the position at the next moment and the best position. The sand cat swarm uses the spiral search strategy to explore in the search space in a spiral form, thereby enhancing the exploration ability for unknown regions, reducing the possibility of falling into local optimal solutions, and improving the global search performance of the algorithm;

[0094] Step 7: Sparrow Early Warning Mechanism: The sparrow early warning mechanism is mainly used to simulate the information transmission and cooperative behavior in a sparrow group. Through the early warning mechanism, sparrow individuals can transmit and receive information from each other to adjust their own behaviors and search directions, which helps to improve the convergence speed and global search ability of the algorithm and better solve various optimization problems. After the sand cat group searches for prey, adding the sparrow warning mechanism can make the sand cat group algorithm improve the convergence speed faster. When the sand cat group is in danger, in order to obtain a better position, the sand cats on the edge will quickly move to the safe area, while the sand cats in the middle of the population will move randomly to approach other sand cats. The mathematical expression is shown in Equation (11):

[0095]

[0096] In the formula, is the globally optimal position of the current search, β is the step size control parameter, a random number that follows a normal distribution with a mean of 0 and a variance of 1. K is a random number, with a value range of [-1, 1], representing the moving direction of the sparrow and also the control parameter of the step size. f i is the fitness value of the current sand cat individual, f g and f w are the best fitness value and the worst fitness value of the current global position respectively, and ε is a constant to avoid the situation where the fraction is 0. f i > f g indicates that this sand cat is in a vulnerable position on the edge of the group, and f i = f g means that the sand cat located inside the population realizes the danger and needs to get closer to other sand cats to avoid being preyed upon;

[0097] Step 8: Algorithm Process: (1) Initialization: Initialize the population size, dimension variables, number of iterations, upper and lower boundary limits, fitness function, etc.; If R ≤ 1, update the sand cat individual position using Equation (5) and calculate the fitness value. If R > 1, update the sand cat individual position using Equation (11) and calculate the fitness value. After the exploration stage, add the sparrow early warning mechanism, update the sand cat individual position using Equation (12) and calculate the optimal fitness value, compare the obtained fitness values, and select the position with the optimal fitness value to update the sand cat individual position;

[0098] Step 9: Build a simulation experiment environment: The experimental hardware is a computer with 12th Gen Intel(R) Core(TM) i7-12650H, 32GB of memory, and 2.30GHz, and the software is MATLAB R2022a;

[0099] Step 10: Performance Comparison of Benchmark Test Functions: Parameter Settings: The initial population size is 30, the number of iterations is 500, and the dimension is 100. To compare the performance of the SSCSO algorithm, 12 benchmark test functions from the CEC2017 test function set were selected for testing this time;

[0100]

[0101] Step 11: The comparison algorithms are selected as PSO, AWOA, GA, GWO, and SCSO. To ensure the rationality of the algorithms, each benchmark test function was experimented 30 times and the average value was taken;

[0102]

[0103] Step 12: It can be seen from the experimental data that in high-dimensional unimodal functions, multimodal functions, hybrid functions, and composite functions, the SSCSO algorithm has a faster convergence speed, a smaller optimal value, and less optimization time than the other five algorithms;

[0104] Step 13: Compare the performance of the improved algorithm;

[0105] To make the algorithm fall into the local optimal value and improve the convergence speed, a hybrid scheme based on the Cubic chaotic mapping function, spiral search strategy, and sparrow early warning mechanism is proposed to optimize the algorithm, and its performance is compared with the five original algorithms on the benchmark test functions.

[0106] In the above Step 1, the initialization process: swarm intelligence optimization algorithms are often randomly generated. The population initialization of the SCSO algorithm is to randomly generate some points in the search space of the problem, and these points are individuals. Assuming the population size is M and the problem dimension is N, the initialization formula is shown as follows:

[0107] x ij = lb j + r1 × (ub j - lb j )

[0108] i = 1, 2, …, M; j = 1, 2, …, N, x ij represents the j-th dimension variable of individual i, lb j and ub j represent the lower bound and upper bound of the j-th dimension variable respectively, and r1 is a random number between [0, 1];

[0109] In the above Step 5, the Cubic chaotic mapping function makes the population distribution more uniform, enhances the global search ability, shortens the time to jump out of the local area, and improves the convergence speed;

[0110] In step six, for the spiral search strategy, in order to enable the sand cat swarm to search for more achievable paths for position update, the spiral search strategy is introduced. The defined parameter k in this strategy affects the decay rate of the algorithm. A larger k value will cause the algorithm to fall into a local optimum, while a smaller k value will result in a slower convergence rate of the algorithm. To solve this problem, based on the spiral search strategy, a strategy factor Z is designed according to the mathematical model of the spiral. On the basis of the original spiral model, the number of iterations is introduced to dynamically adjust the shape of the spiral. As the number of iterations increases, the parameter Z will cause the spiral shape to change from large to small;

[0111] In step seven, for the sparrow early warning mechanism, after the sand cat swarm finishes searching for prey, a sparrow warning mechanism is added. This can enable the sand cat swarm algorithm to improve the convergence rate faster. When the sand cat swarm is in danger, in order to obtain a better position, the sand cats at the edge will quickly move to the safe area, while the sand cats in the middle of the population will move randomly to approach other sand cats;

[0112] In step eleven, it can be seen from the data that the SSCSO algorithm has better optimization ability, that is, a faster convergence rate, on the unimodal function F1, the simple multimodal functions F4 and F9, the hybrid functions F13 - F15, F18, F19, and the composite functions F25, F27, F28, F30. The stability of the algorithm on the functions F1, F9, F15, F27, F28 is slightly lower than that of the comparison algorithm. It can be seen that methods such as the Cubic chaotic mapping function, the spiral search strategy, and the sparrow early warning mechanism adopted for the algorithm can improve the optimization ability of the algorithm;

[0113] In step twelve, after comparing the test performance with the five original algorithms on the benchmark functions, except that the optimal fitness value is close on the high-dimensional simple multimodal function F9, the SSCSO algorithm shows significant optimization ability and a faster convergence rate. It can be seen that the improved strategy has the advantage of making up for the original algorithm's tendency to fall into local optima, and the global search ability is enhanced.

[0114] The above examples are only used to illustrate the technical solutions of the present application, not to limit them.

Claims

1. An improved sand cat swarm optimization algorithm, characterized in that: It includes the following steps: Step 1: In a d-dimensional optimization problem, the fennec fox represents a 1×d array for problem solving, where each variable value (x1, x2, …, x d ) is a floating-point number and must lie between a lower bound and an upper bound; Step 2: Exploration stage: The search ability of the SCSO algorithm depends on the auditory ability in the low-frequency detection of sand cats. According to the working mechanism proposed by the algorithm, as the iteration progresses, this sensitivity value will decrease from r G from 2 to 0, gradually approaching the prey. To search for the prey, assume that the sensitivity range of the sand cat is from 2 kHz to 0, as shown in the following equation. Among them, s M value is inspired by the auditory characteristics of the sand cat, and its value is assumed to be 2. Let \(t\) be the current iteration number and \(T\) be the maximum iteration number. The values are re-set according to different problems, which also proves the flexibility and generality of the proposed equation. The transition from the exploration stage to the development stage is determined by the parameter \(R\), and the calculation formula of \(R\) is obtained as shown in Equation (2): R = 2 × r G ×rand() - r G (2) \(rand()\) represents a random number between 0 and 1. During the search process, the position update of each current searcher is based on a random position. In this way, the searchers can explore new spaces in the search space. To avoid being trapped in local optima, the sensitivity range of each sand cat is different, which is achieved by Equation (3): r2 = r G × rand() (3) r G represents the sensitivity range that decreases from 2 to 0, and r2 represents the sensitivity range of each individual sand cat, which is used for the position update of the sand cat group during the exploration or development stage, r G is used for the guiding parameter R to achieve the transfer control of the two stages. In the SCSO algorithm, the sand cat swarm updates the positions of individuals according to the optimal position of the current population, its own position, and the sensitivity range, generating new positions between the initial position and the prey position, and using randomness to ensure the convergence and complexity of the algorithm. The mathematical model is as follows: Represents the location of the optimal individual in the current population, which is the current location of the sand cat; Step 3: Development stage: The sand cat perceives the direction of the prey through hearing. By mathematically modeling the stage of attacking the prey, the distance between the current position and the best position is represented, and the update formula is as follows: In the formula, \(P\) represents the candidate position. It is set that the sensitivity range of the sand cat is a circle, and the moving direction can be defined by a random angle \(\theta\) on the circle. \(\theta\) takes a random value between \(0^{\circ}\) and \(360^{\circ}\), and \(\cos\theta\) is limited between -1 and 1, so that each individual can ensure to approach the prey along the circumferential direction in the search space; Step 4: The SSCSO algorithm is tested on the cec2017 test function set, and the convergence speed effect is not obvious. The reasons are as follows: First, in the initialization stage, the formula uses random numbers generated in the interval \([lb, ub]\) to limit the initial positions of individuals. Because the \(rand()\) function generates random numbers, the population is not randomly distributed, and the reduction of population diversity makes it difficult for the algorithm to obtain the optimal solution in multiple iterations, resulting in a decrease in the convergence speed. Second, the traditional sand cat optimization algorithm may lack an effective global search strategy, resulting in the inability to quickly converge to the vicinity of the optimal solution during the search process; Step 5: The chaotic mapping has good randomness, regularity, and ergodicity, which can not only make the population diverse but also improve the algorithm in terms of global search ability, convergence speed, and convergence accuracy. The SSCSO algorithm uses the Cubic chaotic mapping function in the initial stage, changing the way of finding random positions between the upper and lower bounds, getting rid of the problem of local path search, expanding the search space to a certain extent. The standard Cubic chaotic mapping function is shown in Equation (6): Among them, a and b are chaotic influence factors, and their values directly affect the range of the Cubic mapping. When a = 1, x n ∈ (-2, 2); when a = 4, x n ∈ (-1, 1). When a ∈ (2.3, 3), the sequence generated by the Cubic mapping is in a chaotic state. The relevant analysis of the maximum Lyapunov exponent of the commonly used one-dimensional chaotic mapping is carried out, and the expression of the Cubic chaotic mapping function is shown in Equation (7): where ρ is a control parameter, and the selection of its value affects the chaos of the Cubic mapping, and x n ∈(0, 1). The chaos of the Cubic mapping is similar to the maximum Lyapunov exponents of the Logistic mapping and the Tent mapping, and is superior to one-dimensional mappings such as the Sine mapping and the Circle mapping. In the present invention, (0, 1) is the initial value range of x n . When x0 = 0.3 and ρ = 2.595 are selected, the Cubic chaotic mapping has good ergodicity; Step 6: Spiral search strategy: It is a heuristic search algorithm mainly used to solve optimization problems and can be adjusted and optimized according to the characteristics of specific problems, having a certain degree of flexibility and adaptability in practical applications. When the sand cat swarm optimization algorithm attacks prey, it will randomly select an angle through the roulette method to approach the prey for attack. However, such an attack method has a large degree of randomness and is also prone to falling into local optima. During the stage of searching for prey, the sand cat individuals update the position at the next moment according to the best candidate position and the current position and its sensitivity range r2. Therefore, the sand cat individuals can search for other feasible best prey positions. The update formula is shown in Equation (8): To enable the sand cat swarm to search for more achievable paths for position update, a spiral search strategy is introduced. The defined parameter k in this strategy affects the decay rate of the algorithm. A larger k value will cause the algorithm to fall into a local optimum, while a smaller k value will result in a slower convergence rate of the algorithm. To solve this problem, based on the spiral search strategy, a strategy factor Z is designed according to the mathematical model of the spiral. On the basis of the original spiral model, the number of iterations is introduced to dynamically adjust the shape of the spiral. As the number of iterations increases, the parameter Z will cause the spiral shape to change from large to small. The improved formula is shown in Equation (9): In the formula, Z is the spiral search factor, k is the parameter. Adding the spiral factor effectively improves the global search performance of the algorithm. The new spiral position update formula is as follows: l is the difference between the position at the next moment and the best position. The sand cat swarm uses the spiral search strategy to explore in the search space in a spiral form, thus enhancing the exploration ability of the unknown area, avoiding the possibility of falling into a local optimum, and improving the global search performance of the algorithm; Step 7: Sparrow early warning mechanism: The sparrow early warning mechanism is mainly used to simulate the information transmission and collaborative behavior in the sparrow group. Through the early warning mechanism, sparrow individuals can transmit and receive information from each other to adjust their own behaviors and search directions, which helps to improve the convergence rate and global search ability of the algorithm and better solve various optimization problems. After the sand cat swarm finishes searching for prey, adding the sparrow warning mechanism can make the sand cat swarm algorithm improve the convergence rate faster. When the sand cat swarm is in danger, in order to obtain a better position, the sand cats at the edge will quickly move to the safe area, while the sand cats in the middle of the population will move randomly to approach other sand cats. The mathematical expression is shown in Equation (11): In the formula, is the globally optimal position of the current search. β is the step size control parameter, which is a random number following a normal distribution with a mean of 0 and a variance of 1. K is a random number with a value range of [-1, 1], representing the movement direction of the sparrow and also the control parameter of the step size. fi is the fitness value of the current sand cat individual, f g and fw are the best fitness value and the worst fitness value of the current global position respectively. ε is a constant to avoid the situation where the fraction is 0. fi > f g indicates that this sand cat is in a vulnerable position at the edge of the group. f i = f g means that the sand cat located inside the population realizes the danger and needs to get closer to other sand cats to avoid being preyed on; Step 8: Algorithm process: (1) Initialization: Initial population size, dimension variable, number of iterations, upper and lower boundary limits, fitness function, etc.; If R ≤ 1, update the position of the sand cat individual and calculate the fitness value according to Equation (5). If R > 1, update the position of the sand cat individual and calculate the fitness value according to Equation (11). After the exploration stage, add the sparrow early warning mechanism, use Equation (12) to update the position of the sand cat individual and calculate the optimal fitness value, compare the obtained fitness values, and select the position with the optimal fitness value to update the position of the sand cat individual; Step 9: Build a simulation experiment environment: The experimental hardware is a computer with 12th Gen Intel(R) Core(TM) i7-12650H, 32GB of memory, and 2.30GHz. The software is MATLAB R2022a; Step 10: Performance comparison of benchmark test functions: Parameter settings: The initial population size is 30, the number of iterations is 500, and the dimension is 100. To compare the performance of the SSCSO algorithm, 12 benchmark test functions in the CEC2017 test function set are selected for testing this time; Step 11: The comparison algorithms are selected as PSO, AWOA, GA, GWO, and SCSO. To ensure the rationality of the algorithm, each benchmark test function is experimented 30 times and the average value is taken; Step 12: It can be seen from the experimental data that in high-dimensional unimodal functions, multimodal functions, hybrid functions, and composite functions, the SSCSO algorithm has a faster convergence speed, a smaller optimal value, and less optimization time than the other five algorithms; Step 13: Compare the performance of the improved algorithm; Step 14: To make the algorithm fall into the local optimal value and improve the convergence speed, a hybrid scheme based on the Cubic chaotic mapping function, spiral search strategy, and sparrow early warning mechanism is proposed to optimize the algorithm, and its performance is compared with the five original algorithms on benchmark test functions.

2. An improved sand cat swarm optimization algorithm according to claim 1, characterized in that: In the above-mentioned Step 1, the initialization process: swarm intelligence optimization algorithms often generate randomly. The population initialization of the SCSO algorithm is to randomly generate some point spaces in the search space of the problem, and these points are individuals. Assuming the population size is M and the problem dimension is N, the initialization formula is shown as follows: x ij = lb j + r1×(ub j - lb j ) i = 1, 2, …, M; j = 1, 2, …, N, xi j represents the j-th dimensional variable of individual i, lb j and ub j represent the lower bound and upper bound of the j-th dimensional variable respectively, and r1 is a random number between [0, 1].

3. An improved sand cat swarm optimization algorithm according to claim 1, characterized in that: In the above-mentioned Step 5, the Cubic chaotic mapping function makes the population distribution more uniform, enhances the global search ability, reduces the time to jump out of the local area, and improves the convergence speed.

4. An improved sand cat swarm optimization algorithm according to claim 1, characterized in that: In the above-mentioned Step 6, the spiral search strategy is introduced to enable the sand cat swarm to search for more achievable paths for position update. The defined parameter k in this strategy affects the decay speed of the algorithm. A larger k value will cause the algorithm to fall into the local optimum, while a smaller k value will result in a slower convergence speed of the algorithm. To solve this problem, based on the spiral search strategy, a strategy factor Z is designed according to the mathematical model of the spiral. On the basis of the original spiral model, the number of iterations is introduced to dynamically adjust the shape of the spiral. As the number of iterations increases, the parameter z will cause the spiral shape to change from large to small.

5. An improved sand cat swarm optimization algorithm according to claim 1, characterized in that: In the above-mentioned Step 7, the sparrow early warning mechanism is added after the sand cat swarm finishes searching for prey. This can enable the sand cat swarm algorithm to improve the convergence speed more quickly. When the sand cat swarm is in danger, in order to obtain a better position, the sand cats on the edge will quickly move to the safe area, while the sand cats in the middle of the population will move randomly to approach other sand cats.

6. An improved sand cat swarm optimization algorithm according to claim 1, characterized in that: In the above-mentioned Step 11, it can be seen from the data that the SSCSO algorithm has better optimization ability, that is, a faster convergence speed, in the unimodal function F1, simple multimodal functions F4 and F9, hybrid functions F13 - F15, F18, F19, and composite functions F25, F27, F28, F30. The stability of the algorithm is slightly lower than that of the comparison algorithm in the functions F1, F9, F15, F27, F28. It can be seen that methods such as the Cubic chaotic mapping function, spiral search strategy, and sparrow early warning mechanism adopted for the algorithm can improve the optimization ability of the algorithm.

7. An improved sand cat swarm optimization algorithm according to claim 1, characterized in that: In the above-mentioned Step 12, after comparing the performance of the five original algorithms on the benchmark function test, except that the optimal fitness value of the SSCSO algorithm is close on the high-dimensional simple multimodal function F9, the SSCSO algorithm shows significant optimization ability and a faster convergence speed. It can be seen that the improvement strategy has the advantage of making up for the original algorithm's tendency to fall into the local optimum, and the global search ability is enhanced.

Images