Mobile robot path planning method based on multi-strategy fusion whale optimization algorithm

Abstract

The invention provides a multi-strategy fusion white whale optimization algorithm, and aims to solve the problems that the global exploration capability of the algorithm is weak, the algorithm is easy to fall into local optimum in the later stage and the like, and the algorithm comprises the following steps: firstly, dynamically adjusting a balance factor by adopting a nonlinear balance factor based on a Softplus activation function, effectively balancing the global search and local development capability, and solving the problem that the global exploration capability is poor; the convergence speed is improved; secondly, a search strategy based on dimension learning is fused, the local search capability of the whale optimization algorithm in the later period is enhanced, and the convergence precision is improved; and finally, according to a sine and cosine optimization algorithm, optimizing a position updating formula, improving the global search capability of the algorithm and accelerating the search speed so as to enhance the robustness of the algorithm. According to the method, 29 basic test functions of a CEC2017 test set are used for testing the improved algorithm, and compared with path planning experiments of GWO, GA, WOA, HHO, ACO and original BWO algorithms under a complex map, the optimal values of the shortest paths of the improved algorithm are reduced by 16.01%, 4.34%, 9.56%, 10.01%, 6.24% and 13.7% 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 invention belongs to the technical field of White Whale optimization algorithms, and in particular relates to an improved White Whale optimization algorithm. Technical Background

[0002] The swarm intelligence optimization algorithm originates from the predation, fighting, mating and other behaviors of organisms in nature. It updates the position to search for the optimal position and obtain the best fitness value by being affected by constraints such as different population sizes, initial positions, food amounts, and environments. In the iterative search process, it usually goes through two stages: global exploration and local development. These two stages are sequential rather than synchronous processes, which may lead to the dilemma of falling into the local optimal solution. In order to enhance the global search capability and speed up the convergence speed, the researchers proposed different thinking strategies for improving the algorithm, aiming to speed up the convergence speed while enhancing the local search capability to obtain the global optimal solution. Among them, the hybrid strategy has the most outstanding improvement effect, which can achieve higher accuracy in practical engineering applications.

[0003] Gray wolf optimization algorithm, ant colony algorithm, whale optimization algorithm, Harris eagle optimization algorithm, etc., as the original algorithms of swarm intelligence, have been continuously concerned by researchers. Commonly used improvement strategies include using chaos mapping strategy to initialize population diversity instead of random generation, incorporating adversarial learning strategy to expand search capabilities, adding multiple strategies to improve the overall search range of iteration, etc. These strategies are helpful to speed up convergence. However, the current Beluga optimization algorithm usually has a local optimal situation in the global optimization stage of the entire iteration process, resulting in weak global search capabilities 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 Beluga optimization algorithm to solve the problem that the local optimal situation occurs when Beluga seeks global optimization in the entire iterative process proposed in the above technical background, resulting in weak global search ability and slow convergence speed of the algorithm.

[0005] To achieve the above object, the present invention provides the following technical solution: a multi-strategy fusion Beluga optimization algorithm, comprising the following steps:

[0006] Step 1: Initial stage: The BWO algorithm assumes each beluga whale as a candidate solution and continuously updates it during the optimization process. The initial position of the population is:

[0008] In the formula, pop represents the size of the beluga population, and dim represents the variable dimension. The fitness values ​​of all beluga whales are:

[0009]

[0010] Step 2: Exploration Phase: The exploration phase of BWO is established by considering the swimming behavior of beluga whales. Therefore, the position of the search agent is determined by the paired swimming of beluga whales. The position of beluga whales is updated as follows:

[0011]

[0012] In the formula, represents the position of the i-th beluga whale in the j-th dimension; p j (j=1,2,…,d) represents a random integer selected from the d dimension, T represents the current number of iterations; r1 and r2 represent random operators, both of which are random numbers in the range of (0,1); Indicates that the i-th beluga whale is in the p-th j Update position of the dimension; and Represent the current positions of the i-th and r-th beluga whales respectively; sin(2πr2) and cos(2πr2) indicate that the fin direction dimensions of the mirrored beluga whales are selected by odd and even numbers, and the updated positions can reflect the mirroring or synchronization behavior of the beluga whales when diving or swimming. The two random numbers r1 and r2 are used to enhance the random operator in the exploration phase.

[0013] Step 3: Development phase

[0014] In order to enhance the convergence of the algorithm, BWO introduced the Levy flight strategy

[17] in the development stage. It is assumed that the beluga whales use the above strategy to share each other's positions to capture prey at this stage, while considering other and best candidate solutions. In this process, the position update formula of the beluga whale is as follows:

[0015]

[0016] in, represents the updated position of the ith beluga whale; T represents the current iteration number; r3 and r4 represent random numbers in the range of (0,1); is the current position of the i-th beluga whale; represents the current position of the randomly selected rth beluga whale; represents the optimal position in the beluga population; C1 represents the random jump intensity, which is used to measure the Levy flight intensity. Its expression is as follows:

[0017] C1=2r4(1-T / T max ) (5)

[0018] L F represents the Levy flight function, which is expressed as:

[0019]

[0020]

[0021] Wherein, u and v represent normally distributed random numbers; β = 1.5.

[0022] Step 4: Whale Fall

[0023] During migration and foraging, beluga whales face threats from killer whales, polar bears, and humans. Most beluga whales successfully escape threats by sharing information, but a few fail to survive and sink to the bottom of the sea, forming a "whale fall" phenomenon. To simulate this phenomenon, we select individuals with a certain probability in each iteration to simulate the "whale fall" behavior, assuming that these beluga whales either migrate or sink to the bottom of the sea. In order to ensure that the population size remains unchanged, the position of the beluga whale and the step length of the whale fall are used to establish the updated position. The mathematical model is expressed as:

[0024]

[0025] Among them, r5, r6 and r7 are random numbers between (0, 1), X step is the step size of the whale's descent, as follows:

[0026] X step =(u b -l b )exp(-C2T / T max ) (9)

[0027] Among them, u b Indicates the upper limit of the variable; l b represents the lower limit of the variable; C2 represents the step size factor related to the population size and the probability of whale fall, and its calculation formula is:

[0028] C2=2W f ×n (10)

[0029] Among them, W f represents the probability of a whale falling, and its expression is as follows:

[0030]

[0031] The probability of a whale falling dropped from 0.1 in the initial iteration to 0.05 in the final iteration, indicating that during the optimization process, the beluga whales became less dangerous when they were closer to the food source.

[0032] Step 5: Nonlinear balancing factor: One advantage of the BWO algorithm over other metaheuristic algorithms is that it can use the balancing factor B f To adjust the optimization strategy. The traditional balance factor is slow in transition to the development stage. In order to improve the convergence speed and find the optimal solution, this study uses a nonlinear balance factor based on the Softplus activation function:

[0033] Softplus(x)=log(1+e x ) (12)

[0034] The Softplus activation function is a smooth variant of the Rectified Linear Unit (ReLU) function. In order to adapt to the selection of the balance factor in the BWO iteration process, the function value is changed in the interval [0,1] by changing the ordinate axis, then swapping the coordinates, and multiplying by a factor to enhance the smoothness of the convergence process. The improved Softplus activation function is expressed as:

[0035]

[0036] Therefore, the original balance factor formula has been improved to the formula:

[0037] Where T is the current iteration, T max is the maximum number of iterations.

[0038] Step 6: Dimensional learning strategy: In the dimensional learning strategy, the search agent updates its position by learning the optimal local values ​​of different neighborhoods and the information of randomly selected search members in the group.

[0039] First, the dimension learning search strategy is The new position generates two candidate solutions: one is the candidate position generated by the BWO algorithm The other is the candidate position generated by the dimension learning strategy

[0040] The neighborhood radius in the dimensional learning strategy is determined by the current position and candidate locations The Euclidean distance between by As the radius, The neighborhood of is represented as:

[0041]

[0042] Where D i express and Once the Euclidean distance between The neighborhood of , multi-neighborhood learning is performed through the following formula:

[0043]

[0044] in, The dth dimension of A randomly selected neighborhood solution and belugas randomly selected from the group In the selection and update phase, two candidate solutions are compared by the following formula: and The fitness value of:

[0045]

[0046] Select a better candidate solution. If the fitness value of the selected candidate solution is less than Then update with the candidate solution otherwise The population matrix Pop remains unchanged. After each iteration, the candidate position is built into the neighborhood to create another candidate solution. By comparing the fitness values ​​of the two candidate solutions, the better candidate solution is selected, thereby improving the local search ability.

[0047] Step 7: Sin-Cosine Optimization Algorithm: The Sin-Cosine Optimization Algorithm uses the mathematical properties of the sine and cosine functions to adaptively adjust their amplitudes, effectively balancing global exploration and local development capabilities during the search process. After exploring the area near a solution, the agent can run around another solution in the next cycle, making full use of the search space defined by the sine and cosine functions. SCA requires the search agent to make larger changes in the early stages of optimization and gradually reduce the amplitude of changes in the later stages of optimization. The geometric calculation characteristics of the sine-cosine algorithm provide more accurate path planning for the position update of the White Whale Optimization Algorithm (BWO), significantly enhance the global search capability of the algorithm, and accelerate the overall search process. Its position update formula is as follows:

[0048]

[0049] in, is the position of the individual at the Tth iteration; p T represents the position of the individual with the highest fitness at the Tth iteration; a2 is a random number belonging to (0,2π); a3 is a random number belonging to (-1,1); a4 is a random number belonging to (0,1); a1 is a linearly decreasing inertia factor, as shown below.

[0051] Among them, q is a constant with a fixed value of 2; T max Indicates the maximum number of iterations. Based on the sine-cosine algorithm, the exploration process of the improved BWO is adjusted as follows:

[0052]

[0053] This improved search method enhances Beluga's global search capability and local development capability, speeds up convergence, and increases the possibility of obtaining better solutions.

[0054] Step 8: Algorithm flow: (1) Initialization: initial population size, dimension variables, number of iterations, upper and lower boundary limits, fitness function, etc.; if B f > 0.5, update random dimension p j And calculate the fitness value according to formula (20), if B f ≤0.5, then update the jump intensity C1, use Levy flight, execute formula (17) to update the individual position of the beluga whale and calculate the fitness value; if B f <W f , then update the step factor C2, calculate the step length, use formula (8) to update the individual position of the beluga whale and calculate the optimal fitness value, compare the obtained fitness values, and select the position with the optimal fitness value to update the individual position of the beluga whale;

[0055] Step 9: Build a simulation experiment environment: The experimental hardware is an Intel Core i5-8300H, 32GB memory, a computer with a base frequency of 2.30GHz, and the software is MATLAB R2022a;

[0056] Step 10: Benchmark test function performance comparison: Parameter setting: the initial population size is 30, the number of iterations is 500, and the dimension is 50. In order to compare the performance of the MBWO algorithm, 29 benchmark test functions in the CEC2017 test function set were selected for testing;

[0057] Step 11: Six comparison algorithms are selected: GWO, GA, WOA, HHO, ACO, and BWO. To ensure the rationality of the algorithm, 30 experiments are conducted for each benchmark test function and the average value is taken;

[0058] Step 12: It can be seen from the experimental data that in high-dimensional unimodal functions, multimodal functions, mixed functions, and composite functions, the MBWO algorithm converges faster, has a smaller optimal value, and takes less time to find the optimal solution than the other five algorithms;

[0059] Step 13: Compare the performance of the improved algorithms;

[0060] Step 14: In order to prevent the algorithm from falling into the local optimal value and improve the convergence speed, a hybrid scheme based on nonlinear balance factor, dimensional learning strategy, and sine-cosine optimization algorithm is proposed to optimize the algorithm, and the performance is compared with the six original algorithms on the benchmark test function;

[0061] In the step 1, the initialization process: swarm intelligence optimization algorithms are often generated randomly. The population initialization of the MBWO algorithm is to randomly generate some point spaces in the search space of the problem. These points are individuals. Assuming that the population size is M and the problem dimension is N, the initialization formula is as follows:

[0062] x ij =lb j +r1×(ub j -lb j )

[0063] i=1,2,…,M; j=1,2,…,N,x ij represents the j-th dimension variable of individual i, lb j andub j They represent the lower and upper bounds of the j-th dimension variable, respectively, and r1 is a random number between [0, 1];

[0064] In step 5, the nonlinear balance factor enables the improved BWO algorithm to show better local development capabilities after an effective global exploration phase. The improved balance factor is more inclined to enter the development phase, thereby significantly improving the algorithm's solution capability.

[0065] In step 6, the dimension learning strategy, the search agent updates its position by learning the optimal local values ​​of different neighborhoods and the information of randomly selected search members in the group. This enhances the local search capability of the White Whale optimization algorithm in the later stage and speeds up the search speed.

[0066] In step seven, the sine-cosine optimization algorithm, by adaptively adjusting its amplitude, effectively balances global exploration and local development capabilities during the search process. After exploring the area near a solution, the agent can run around another solution in the next cycle, making full use of the search space defined by the sine and cosine functions. SCA requires the search agent to make larger changes in the early stages of optimization and gradually reduce the amplitude of changes in the later stages of optimization. The geometric calculation characteristics of the sine-cosine algorithm provide more accurate path planning for the position update of the White Whale Optimization Algorithm (BWO), significantly enhance the global search capability of the algorithm, and accelerate the overall search process.

[0067] In the step eleven, it can be seen from the data that the MBWO algorithm has better optimization ability, that is, faster convergence speed, on the single-peak function F1, simple multi-peak functions F4-F7, F9, F10, mixed functions F11-F17, F19, and composite functions F20, F22-F30. The optimal value of the algorithm on functions F3, F8, F18, and F21 is slightly lower than that of the comparison algorithm. It can be seen that the nonlinear balance factor, dimensional learning strategy, sine-cosine optimization algorithm and other methods adopted by the algorithm can improve the algorithm's optimization ability;

[0068] In step twelve, after comparing the performance of the six original algorithms in the benchmark function test, the MBWO algorithm showed significant optimization ability and faster convergence speed on 17 test functions. It can be seen that the improved strategy has the advantage of compensating for the original algorithm's easy falling into local optimality, and the global search ability is enhanced; the stability is enhanced.

[0069] Compared with the prior art, the present invention provides a multi-strategy fusion White Whale optimization algorithm, which has the following beneficial effects:

[0070] The present invention optimizes the algorithm by proposing a hybrid scheme based on nonlinear balance factor, dimensional learning strategy and sine-cosine optimization algorithm. After comparing the performance with the six original algorithms in the benchmark function test, the MBWO algorithms show significant optimization ability and faster convergence speed in addition to the close optimal fitness values ​​on high-dimensional simple multi-peak functions F3, F8, F18 and F21. It can be seen that the improved strategy has the advantage of making up for the original algorithm's easy falling into local optimality, and the global search capability is enhanced.

[0071] The data show that the MBWO algorithm has better optimization ability, that is, faster convergence speed, on the single-peak function F1, simple multi-peak functions F4-F7, F9, F10, mixed functions F11-F17, F19, and composite functions F20, F22-F30. The stability of the algorithm on the 17 test functions is higher than that of the comparison algorithm. It can be seen that the nonlinear balance factor, dimensional learning strategy, sine-cosine optimization algorithm and other methods used in the algorithm can improve the algorithm's optimization ability; BRIEF DESCRIPTION OF THE DRAWINGS

[0072] The accompanying 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 of the present invention. In the accompanying drawings:

[0073] Figure 1 This is a flow chart of the White Whale optimization algorithm of multi-strategy fusion proposed by the present invention;

[0074] Figure 2 The fitness value curve of the multi-strategy fusion Beluga optimization algorithm proposed in the present invention on a single-peak function;

[0075] Figure 3 The fitness value curve of the multi-strategy fusion Beluga optimization algorithm proposed in the present invention on the multi-peak function;

[0076] Figure 4 The fitness value curve of the White Whale optimization algorithm of multi-strategy fusion proposed by the present invention on the hybrid function;

[0077] Figure 5The fitness value curve of the White Whale optimization algorithm of multi-strategy fusion proposed by the present invention on the composite function;

[0078] Figure 6-7 The application of the multi-strategy fusion Beluga optimization algorithm proposed in the present invention in robot path planning; DETAILED DESCRIPTION

[0079] The present invention is further described in detail below with reference to the accompanying drawings and examples. It should be noted that the specific implementation examples described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, for ease of description, the accompanying drawings only show parts related to the present invention rather than all structures.

[0080] See also Figure 1-7 The present invention provides a technical solution: an improved sand cat group optimization algorithm, comprising the following steps:

[0081] Step 1: Initial stage: The BWO algorithm assumes each beluga whale as a candidate solution and continuously updates it during the optimization process. The initial position of the population is:

[0082]

[0083] In the formula, pop represents the size of the beluga population, and dim represents the variable dimension. The fitness values ​​of all beluga whales are:

[0084]

[0085] Step 2: Exploration Phase: The exploration phase of BWO is established by considering the swimming behavior of beluga whales. Therefore, the position of the search agent is determined by the paired swimming of beluga whales. The position of beluga whales is updated as follows:

[0086]

[0087] In the formula, represents the position of the i-th beluga whale in the j-th dimension; p j (j=1,2,…,d) represents a random integer selected from the d dimension, T represents the current number of iterations; r1 and r2 represent random operators, both of which are random numbers in the range of (0,1); Indicates that the i-th beluga whale is in the p-th j Update position of the dimension; and Represent the current positions of the i-th and r-th beluga whales respectively; sin(2πr2) and cos(2πr2) indicate that the fin direction dimensions of the mirrored beluga whales are selected by odd and even numbers, and the updated positions can reflect the mirroring or synchronization behavior of the beluga whales when diving or swimming. The two random numbers r1 and r2 are used to enhance the random operator in the exploration phase.

[0088] Step 3: Development phase

[0089] In order to enhance the convergence of the algorithm, BWO introduced the Levy flight strategy

[17] in the development stage. It is assumed that the beluga whales use the above strategy to share each other's positions to capture prey at this stage, while considering other and best candidate solutions. In this process, the position update formula of the beluga whale is as follows:

[0090]

[0091] in, represents the updated position of the ith beluga whale; T represents the current number of iterations; r3 and r4 represent random numbers in the range (0,1); is the current position of the i-th beluga whale; represents the current position of the randomly selected rth beluga whale; represents the optimal position in the beluga population; C1 represents the random jump intensity, which is used to measure the Levy flight intensity. Its expression is as follows:

[0092] C1=2r4(1-T / T max ) (25)

[0093] L F represents the Levy flight function, which is expressed as:

[0094]

[0095]

[0096] Wherein, u and v represent normally distributed random numbers; β = 1.5.

[0097] Step 4: Whale Fall

[0098] During migration and foraging, beluga whales face threats from killer whales, polar bears, and humans. Most beluga whales successfully escape threats by sharing information, but a few fail to survive and sink to the bottom of the sea, forming a "whale fall" phenomenon. To simulate this phenomenon, we select individuals with a certain probability in each iteration to simulate the "whale fall" behavior, assuming that these beluga whales either migrate or sink to the bottom of the sea. In order to ensure that the population size remains unchanged, the position of the beluga whale and the step length of the whale fall are used to establish the updated position. The mathematical model is expressed as:

[0099]

[0100] Among them, r5, r6 and r7 are random numbers between (0, 1), X step is the step size of the whale's descent, as follows:

[0101] X step =(ub -l b )exp(-C2T / T max ) (29)

[0102] Among them, u b Indicates the upper limit of the variable; l b represents the lower limit of the variable; C2 represents the step size factor related to the population size and the probability of whale fall, and its calculation formula is:

[0103] C2=2W f ×n (30)

[0104] Among them, W f represents the probability of a whale falling, and its expression is as follows:

[0105]

[0106] The probability of a whale falling dropped from 0.1 in the initial iteration to 0.05 in the final iteration, indicating that during the optimization process, the beluga whales became less dangerous when they were closer to the food source.

[0107] Step 5: Nonlinear balancing factor: One advantage of the BWO algorithm over other metaheuristic algorithms is that it can use the balancing factor B f To adjust the optimization strategy. The traditional balance factor is slow in transition to the development stage. In order to improve the convergence speed and find the optimal solution, this study uses a nonlinear balance factor based on the Softplus activation function:

[0108] Softplus(x)=log(1+e x ) (32)

[0109] The Softplus activation function is a smooth variant of the Rectified Linear Unit (ReLU) function. In order to adapt to the selection of the balance factor in the BWO iteration process, the function value is changed in the interval [0,1] by changing the ordinate axis, then swapping the coordinates, and multiplying by a factor to enhance the smoothness of the convergence process. The improved Softplus activation function is expressed as:

[0110]

[0111] Therefore, the original balance factor formula has been improved to the formula:

[0112] Where T is the current iteration, T max is the maximum number of iterations.

[0113] Step 6: Dimensional learning strategy: In the dimensional learning strategy, the search agent updates its position by learning the optimal local values ​​of different neighborhoods and the information of randomly selected search members in the group.

[0114] First, the dimension learning search strategy is The new position generates two candidate solutions: one is the candidate position generated by the BWO algorithm The other is the candidate position generated by the dimension learning strategy

[0115] The neighborhood radius in the dimensional learning strategy is determined by the current position and candidate locations The Euclidean distance between by As the radius, The neighborhood of is represented as:

[0116]

[0117] Where D i express and Once the Euclidean distance between The neighborhood of , multi-neighborhood learning is performed through the following formula:

[0118]

[0119] in, The dth dimension of A randomly selected neighborhood solution and belugas randomly selected from the group In the selection and update phase, two candidate solutions are compared by the following formula: and The fitness value of:

[0120]

[0121] Select a better candidate solution. If the fitness value of the selected candidate solution is less than Then update with the candidate solution otherwise The population matrix Pop remains unchanged. After each iteration, the candidate position is built into the neighborhood to create another candidate solution. By comparing the fitness values ​​of the two candidate solutions, the better candidate solution is selected, thereby improving the local search ability.

[0122] Step 7: Sin-Cosine Optimization Algorithm: The Sin-Cosine Optimization Algorithm uses the mathematical properties of the sine and cosine functions to adaptively adjust their amplitudes, effectively balancing global exploration and local development capabilities during the search process. After exploring the area near a solution, the agent can run around another solution in the next cycle, making full use of the search space defined by the sine and cosine functions. SCA requires the search agent to make larger changes in the early stages of optimization and gradually reduce the amplitude of changes in the later stages of optimization. The geometric calculation characteristics of the sine-cosine algorithm provide more accurate path planning for the position update of the White Whale Optimization Algorithm (BWO), significantly enhance the global search capability of the algorithm, and accelerate the overall search process. Its position update formula is as follows:

[0123]

[0124] in, is the position of the individual at the Tth iteration; p T represents the position of the individual with the highest fitness at the Tth iteration; a2 is a random number belonging to (0,2π); a3 is a random number belonging to (-1,1); a4 is a random number belonging to (0,1); a1 is a linearly decreasing inertia factor, as shown below.

[0125]

[0126] Among them, q is a constant with a fixed value of 2; T max Indicates the maximum number of iterations. Based on the sine-cosine algorithm, the exploration process of the improved BWO is adjusted as follows:

[0127]

[0128] This improved search method enhances Beluga's global search capability and local development capability, speeds up convergence, and increases the possibility of obtaining better solutions.

[0129] Step 8: Algorithm flow: (1) Initialization: initial population size, dimension variables, number of iterations, upper and lower boundary limits, fitness function, etc.; if B f > 0.5, update random dimension p j And calculate the fitness value according to formula (20), if B f ≤0.5, then update the jump intensity C1, use Levy flight, execute formula (17) to update the individual position of the beluga whale and calculate the fitness value; if B f <W f , then update the step factor C2, calculate the step length, use formula (8) to update the individual position of the beluga whale and calculate the optimal fitness value, compare the obtained fitness values, and select the position with the optimal fitness value to update the individual position of the beluga whale;

[0130] Step 9: Build a simulation experiment environment: The experimental hardware is an Intel Core i5-8300H, 32GB memory, a computer with a base frequency of 2.30GHz, and the software is MATLAB R2022a;

[0131] Step 10: Benchmark test function performance comparison: Parameter setting: the initial population size is 30, the number of iterations is 500, and the dimension is 50. In order to compare the performance of the MBWO algorithm, 29 benchmark test functions in the CEC2017 test function set were selected for testing;

[0132] Step 11: Six comparison algorithms are selected: GWO, GA, WOA, HHO, ACO, and BWO. To ensure the rationality of the algorithm, 30 experiments are conducted for each benchmark test function and the average value is taken;

[0133] Step 12: It can be seen from the experimental data that in high-dimensional unimodal functions, multimodal functions, mixed functions, and composite functions, the MBWO algorithm converges faster, has a smaller optimal value, and takes less time to find the optimal solution than the other five algorithms;

[0134] Step 13: Compare the performance of the improved algorithms;

[0135] Step 14: In order to prevent the algorithm from falling into the local optimal value and improve the convergence speed, a hybrid scheme based on nonlinear balance factor, dimensional learning strategy, and sine-cosine optimization algorithm is proposed to optimize the algorithm, and the performance is compared with the six original algorithms on the benchmark test function;

[0136] In the step 1, the initialization process: swarm intelligence optimization algorithms are often generated randomly. The population initialization of the MBWO algorithm is to randomly generate some point spaces in the search space of the problem. These points are individuals. Assuming that the population size is M and the problem dimension is N, the initialization formula is as follows:

[0137] x ij =lb j +r1×(ub j -lb j )

[0138] i=1,2,…,M; j=1,2,…,N,x ij represents the j-th dimension variable of individual i, lb j andub j They represent the lower and upper bounds of the j-th dimension variable, respectively, and r1 is a random number between [0, 1];

[0139] In step 5, the nonlinear balance factor enables the improved BWO algorithm to show better local development capabilities after an effective global exploration phase. The improved balance factor is more inclined to enter the development phase, thereby significantly improving the algorithm's solution capability.

[0140] In step 6, the dimension learning strategy, the search agent updates its position by learning the optimal local values ​​of different neighborhoods and the information of randomly selected search members in the group. This enhances the local search capability of the White Whale optimization algorithm in the later stage and speeds up the search speed.

[0141] In step seven, the sine-cosine optimization algorithm, by adaptively adjusting its amplitude, effectively balances global exploration and local development capabilities during the search process. After exploring the area near a solution, the agent can run around another solution in the next cycle, making full use of the search space defined by the sine and cosine functions. SCA requires the search agent to make larger changes in the early stages of optimization and gradually reduce the amplitude of changes in the later stages of optimization. The geometric calculation characteristics of the sine-cosine algorithm provide more accurate path planning for the position update of the White Whale Optimization Algorithm (BWO), significantly enhance the global search capability of the algorithm, and accelerate the overall search process.

[0142] In the step eleven, it can be seen from the data that the MBWO algorithm has better optimization ability, that is, faster convergence speed, on the single-peak function F1, simple multi-peak functions F4-F7, F9, F10, mixed functions F11-F17, F19, and composite functions F20, F22-F30. The optimal value of the algorithm on functions F3, F8, F18, and F21 is slightly lower than that of the comparison algorithm. It can be seen that the nonlinear balance factor, dimensional learning strategy, sine-cosine optimization algorithm and other methods adopted by the algorithm can improve the algorithm's optimization ability;

[0143] In step twelve, after comparing the performance of the six original algorithms in the benchmark function test, the MBWO algorithm showed significant optimization ability and faster convergence speed on 17 test functions. It can be seen that the improved strategy has the advantage of compensating for the original algorithm's easy falling into local optimality, and the global search ability is enhanced; the stability is enhanced.

[0144] The above examples are only used to illustrate the technical solution of the present application, but not to limit it.

Claims

1. A multi-strategy fusion White Whale optimization algorithm, Features: The following steps are involved: Step 1: Initial stage: The BWO algorithm assumes each beluga whale as a candidate solution and continuously updates it during the optimization process. The initial position of the population is: In the formula, pop represents the size of the beluga population, and dim represents the variable dimension. The fitness values ​​of all beluga whales are: Step 2: Exploration Phase: The exploration phase of BWO is established by considering the swimming behavior of beluga whales. Therefore, the position of the search agent is determined by the paired swimming of beluga whales. The position of beluga whales is updated as follows: In the formula, represents the position of the i-th beluga whale in the j-th dimension; p j (j=1,2,…,d) represents a random integer selected from the d dimension, T represents the current number of iterations; r 1 and r 2 Represents random operators, which are all random numbers in the range of (0,1); Indicates that the i-th beluga whale is in the p-th j Update position of the dimension; and Respectively represent the current positions of the i-th and r-th beluga whales; sin(2πr 2 ) and cos(2πr 2 ) indicates that the fin direction dimension of the mirrored beluga whale is selected by odd and even numbers, and the updated position can reflect the mirroring or synchronization behavior of the beluga whale when diving or swimming. 1 and r 2 Random operators used to enhance the exploration phase; Step 3: Development phase In order to enhance the convergence of the algorithm, BWO introduced the Levy flight strategy [17] in the development stage. It is assumed that the beluga whales use the above strategy to share each other's positions to capture prey at this stage, while considering other and best candidate solutions. In this process, the position update formula of the beluga whale is as follows: in, represents the updated position of the i-th beluga whale; T represents the current iteration number; r 3 and r 4 Represents a random number in the range (0,1); is the current position of the i-th beluga whale; represents the current position of the randomly selected rth beluga whale; represents the optimal position in the beluga population; C 1 Represents the random jump intensity, which is used to measure the Levy flight intensity. Its expression is as follows: C 1 =2r 4 (1-T / T max ) (5); L F represents the Levy flight function, which is expressed as: Where u and v represent normally distributed random numbers; β = 1.5; Step 4: Whale Fall During migration and foraging, beluga whales face threats from killer whales, polar bears, and humans. Most beluga whales successfully escape threats by sharing information, but a few fail to survive and sink to the bottom of the sea, forming a "whale fall" phenomenon. To simulate this phenomenon, we select individuals with a certain probability in each iteration to simulate the "whale fall" behavior, assuming that these beluga whales either migrate or sink to the bottom of the sea. In order to ensure that the population remains unchanged, the position of the beluga whale and the step length of the whale fall are used to establish the updated position. The mathematical model is expressed as: Among them, r 5 、r 6 and r 7 is a random number between (0, 1), X step is the step size of the whale's descent, as follows: X step =(u b -l b )exp(-C 2 T / T max ) (9); Among them, u b Indicates the upper limit of the variable; l b Indicates the lower limit of the variable; C 2 It represents the step size factor related to population size and whale fall probability, and its calculation formula is: C 2 =2W f ×n (10); Among them, W f represents the probability of a whale falling, and its expression is as follows: The probability of a whale falling dropped from 0.1 in the initial iteration to 0.05 in the final iteration, indicating that during the optimization process, the beluga whale became less dangerous when it was closer to the food source; Step 5: Nonlinear balancing factor: One advantage of the BWO algorithm over other metaheuristic algorithms is that it can use the balancing factor B f To adjust the optimization strategy. The traditional balance factor is slow in transition to the development stage. In order to improve the convergence speed and find the optimal solution, this study uses a nonlinear balance factor based on the Softplus activation function: Softplus(x)=log(1+e x ) (12); The Softplus activation function is a smooth variant of the Rectified Linear Unit (ReLU) function. In order to adapt to the selection of the balance factor in the BWO iteration process, the function value is changed in the interval [0,1] by changing the ordinate axis, then swapping the coordinates, and multiplying by a factor to enhance the smoothness of the convergence process. The improved Softplus activation function is expressed as: Therefore, the original balance factor formula has been improved to the formula: Where T is the current iteration, T max is the maximum number of iterations; Step 6: Dimensional learning strategy: In the dimensional learning strategy, the search agent updates its position by learning the optimal local values ​​of different neighborhoods and the information of randomly selected search members in the group; First, the dimension learning search strategy is The new position generates two candidate solutions: one is the candidate position generated by the BWO algorithm The other is the candidate position generated by the dimension learning strategy The neighborhood radius in the dimensional learning strategy is determined by the current position and candidate locations The Euclidean distance between by As the radius, The neighborhood of is represented as: Where D i express and Once the Euclidean distance between The neighborhood of , multi-neighborhood learning is performed through the following formula: in, The dth dimension of A randomly selected neighborhood solution and belugas randomly selected from the group In the selection and update phase, two candidate solutions are compared by the following formula: and The fitness value of: Select a better candidate solution. If the fitness value of the selected candidate solution is less than Then update with the candidate solution otherwise The population matrix Pop remains unchanged. After each iteration, the candidate position is built into the neighborhood to create another candidate solution. By comparing the fitness values ​​of the two candidate solutions, the better candidate solution is selected, thereby improving the local search capability; Step 7: Sin-Cosine Optimization Algorithm: The Sin-Cosine Optimization Algorithm uses the mathematical properties of the sine and cosine functions to adaptively adjust their amplitudes, effectively balancing global exploration and local development capabilities during the search process. After exploring the area near a solution, the agent can run around another solution in the next cycle, making full use of the search space defined by the sine and cosine functions. SCA requires the search agent to make larger changes in the early stages of optimization and gradually reduce the amplitude of changes in the later stages of optimization. The geometric calculation characteristics of the sine-cosine algorithm provide more accurate path planning for the position update of the White Whale Optimization Algorithm (BWO), significantly enhance the global search capability of the algorithm, and accelerate the overall search process. Its position update formula is as follows: in, is the position of the individual at the Tth iteration; p T represents the position of the individual with the highest fitness at the Tth iteration; a 2 is a random number belonging to (0,2π); a 3 is a random number belonging to (-1,1); a 4 is a random number belonging to (0,1); a 1 is the linearly decreasing inertia factor as shown below; Among them, q is a constant with a fixed value of 2; T max Indicates the maximum number of iterations. Based on the sine-cosine algorithm, the exploration process of the improved BWO is adjusted as follows: This improved search method improves the global search capability and local development capability of White Whale, speeds up the convergence speed, and increases the possibility of obtaining a better solution; Step 8: Algorithm flow: (1) Initialization: initial population size, dimension variables, number of iterations, upper and lower boundary limits, fitness function, etc.; if B f > 0.5, update random dimension p j And calculate the fitness value according to formula (20), if B f ≤0.5, then update the jump strength C 1 , use Levy flight, execute formula (17) to update the individual position of the beluga whale and calculate the fitness value; if B f <W f , then update the step size factor C 2 , and calculate the step length, use formula (8) to update the individual position of the beluga whale and calculate the optimal fitness value, compare the obtained fitness values, and select the position with the optimal fitness value to update the individual position of the beluga whale; Step 9: Build a simulation experiment environment: The experimental hardware is an Intel Core i5-8300H, 32GB memory, a computer with a base frequency of 2.30GHz, and the software is MATLAB R2022a; Step 10: Benchmark test function performance comparison: Parameter setting: the initial population size is 30, the number of iterations is 500, and the dimension is 50. In order to compare the performance of the MBWO algorithm, 29 benchmark test functions in the CEC2017 test function set were selected for testing; Step 11: Six comparison algorithms are selected: GWO, GA, WOA, HHO, ACO, and BWO. To ensure the rationality of the algorithm, 30 experiments are conducted for each benchmark test function and the average value is taken; Step 12: It can be seen from the experimental data that in high-dimensional unimodal functions, multimodal functions, mixed functions, and composite functions, the MBWO algorithm converges faster, has a smaller optimal value, and takes less time to find the optimal solution than the other five algorithms; Step 13: Compare the performance of the improved algorithms; Step 14: In order to prevent the algorithm from falling into the local optimal value and improve the convergence speed, a hybrid scheme based on nonlinear balance factor, dimensional learning strategy and sine-cosine optimization algorithm is proposed to optimize the algorithm, and the performance is compared with the six original algorithms on the benchmark test function.

2. According to the multi-strategy fusion Beluga optimization algorithm of claim 1, Features: In the step 1, the initialization process: swarm intelligence optimization algorithms are often generated randomly. The population initialization of the MBWO algorithm is to randomly generate some point spaces in the search space of the problem. These points are individuals. Assuming that the population size is M and the problem dimension is N, the initialization formula is as follows: x ij =lb j +r 1 ×(ub j -lb j ); i=1,2,…,M; j=1,2,…,N,x ij represents the j-th dimension variable of individual i, lb j andub j Respectively represent the lower and upper bounds of the j-th dimension variable, r 1 is a random number between [0, 1].

3. The White Whale optimization algorithm of multi-strategy fusion according to claim 1, Features: In step 5, the nonlinear balance factor enables the improved BWO algorithm to show better local development capabilities after an effective global exploration phase. The improved balance factor is more inclined to enter the development phase, thereby significantly improving the algorithm's solution capability.

4. The White Whale optimization algorithm of multi-strategy fusion according to claim 1, Features: In step 6, the dimension learning strategy, the search agent updates its position by learning the optimal local values ​​of different neighborhoods and the information of randomly selected search members in the group. This enhances the local search capability of the White Whale optimization algorithm in the later stage and speeds up the search speed.

5. The White Whale optimization algorithm of multi-strategy fusion according to claim 1, Features: In step seven, the sine-cosine optimization algorithm, by adaptively adjusting its amplitude, effectively balances global exploration and local development capabilities during the search process. After exploring the area near a solution, the agent can run around another solution in the next cycle, making full use of the search space defined by the sine and cosine functions. SCA requires the search agent to make larger changes in the early stages of optimization and gradually reduce the amplitude of changes in the later stages of optimization. The geometric calculation characteristics of the sine-cosine algorithm provide more accurate path planning for the position update of the White Whale Optimization Algorithm (BWO), significantly enhance the global search capability of the algorithm, and accelerate the overall search process.

6. The White Whale optimization algorithm of multi-strategy fusion according to claim 1, Features: In the step eleven, it can be seen from the data that the MBWO algorithm has better optimization ability, that is, faster convergence speed, on the single-peak function F1, simple multi-peak functions F4-F7, F9, F10, mixed functions F11-F17, F19, and composite functions F20, F22-F30. The optimal value of the algorithm on functions F3, F8, F18, and F21 is slightly lower than that of the comparison algorithm. It can be seen that the nonlinear balance factor, dimensional learning strategy, sine-cosine optimization algorithm and other methods adopted by the algorithm can improve the optimization ability of the algorithm.

7. The White Whale optimization algorithm of multi-strategy fusion according to claim 1, Features: In step twelve, after comparing the performance of the six original algorithms in the benchmark function test, the MBWO algorithm showed significant optimization ability and faster convergence speed on 17 test functions. It can be seen that the improved strategy has the advantage of compensating for the original algorithm's easy falling into local optimality, and the global search ability is enhanced; the stability is enhanced.

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