Multi-objective particle swarm optimization method based on angle competition and auxiliary evolution mechanism

Abstract

The invention discloses a multi-objective particle swarm optimization method based on angle competition and an auxiliary evolution mechanism. The method comprises the following steps: 1, constructing a multi / super multi-objective particle swarm optimization framework; 2, a population initialization module randomly initializes N particles to obtain a population P; 3a, determining a leader set by a competition updating module by taking the population P as a benchmark; then generating a competitive offspring population based on the population P and the leader set; 3b, an auxiliary updating module selects an assistance pool for executing an auxiliary updating mechanism according to the population P and the previous assistance pool, and then executes an auxiliary updating strategy to generate an auxiliary offspring population; 4, combining the population P, the competitive offspring population and the auxiliary offspring population by an environment selection module to form a new population Q, performing environment selection, and selecting N elite particles from the new population Q to enter next iteration; and 5, repeating the steps 3-4, and when the maximum number of iterations is reached, ending to obtain an optimal particle swarm. The method has the effect that the capability of solving the super-multi-objective optimization problem by the particle swarm optimization can be improved.

Description

Technical Field

[0001] The present invention relates to the technical field of multi-objective particle swarm optimization, and in particular to a multi-objective particle swarm optimization method based on angle competition and auxiliary evolution mechanism. Background Art

[0002] The neural network architecture search problem is a real-world optimization problem that combines multi-objective and super-multi-objective optimization. Its optimization objective system typically encompasses multiple objectives, such as classification accuracy, computational complexity, and parameter count. Due to conflicts between these different objectives, it is often difficult to simultaneously achieve a global optimal state. As the objective dimension expands toward super-multi-objective goals, traditional multi-objective evolutionary algorithms face difficulties in solving them. Particle swarm optimization, with its rapid convergence rate and efficient computational power, has demonstrated strong competitiveness in addressing multi-objective optimization problems. When faced with multi-objective optimization problems, the performance of most current particle swarm algorithms is highly dependent on the global or individual optimal particle. Furthermore, most particle swarm algorithms have a relatively simple search and update strategy. When faced with super-multi-objective problems, the strategies of most multi-objective particle swarm algorithms are difficult to effectively extend to the super-multi-objective domain.

[0003] Disadvantages of existing technologies: The existing particle swarm algorithm is relatively simple in the selection of search strategies and update strategies, and has limitations in dealing with complex multi-objective optimization problems. Summary of the Invention

[0004] The present invention provides a multi-objective particle swarm optimization method based on angle competition and auxiliary evolution mechanism, which can improve the ability of particle swarm algorithm to solve super multi-objective optimization problems.

[0005] To achieve the above objectives, the present invention provides a multi-objective particle swarm optimization method based on angle competition and assisted evolution mechanism, the key of which is to include the following steps:

[0006] Step 1: Construct a multi-objective / super-multi-objective particle swarm optimization framework CCAMOPSO, which is provided with a population initialization module. The population initialization module is respectively connected to a competition update module, an auxiliary update module and an environment selection module. The competition update module and the auxiliary update module are respectively connected to the environment selection module.

[0007] Step 2: The population initialization module randomly initializes N particles to construct a population P and sets the number of iterations to zero; the population initialization module passes the population P to the competition update module, the auxiliary update module and the environment selection module;

[0008] Step 3a: The competitive update module uses the N particles in the population P as a benchmark and adopts the leader particle selection mechanism based on angle information to determine the leader set; then, based on the population P and the selected leader set, a dual-angle-guided competitive particle update mechanism is used to generate the competitive offspring population Offspring1 and pass it to the environment selection module;

[0009] Step 3b: The auxiliary update module first selects an auxiliary pool to perform the auxiliary update mechanism based on the current population P and the previous generation auxiliary pool. It then performs further screening on the selected auxiliary pool to ensure the diversity and convergence of high-quality parents. Subsequently, it executes the auxiliary update strategy to generate the auxiliary offspring population Offspring2 and passes it to the environment selection module.

[0010] Step 4: The environment selection module merges the current population P, the competitive offspring population Offspring1, and the auxiliary offspring population Offspring2 to form a new population Q, and performs environment selection to select N elite particles from the new population Q as the population P to enter the next iteration operation;

[0011] Step 5: Repeat steps 3-4. When the maximum number of iterations is reached, end the iteration operation and then use the N elite particles obtained in the last iteration as the optimal particle swarm.

[0012] Through the above design, the present invention proposes a new multi- / super-multi-objective particle swarm optimization framework, named CCAMOPSO. Under the CCAMOPSO framework system, particles are still guided by angle information to drive the population to implement competitive update operations. However, considering that the single competition mechanism itself has insurmountable defects, the present invention innovatively introduces an assisted evolution mechanism. In the assisted evolution process, in order to further optimize the quality of particles, CCAMOPSO proposes a non-competitive cooperative particle assisted update mechanism, which can effectively help particles break through the dilemma of being easily trapped in local optimality when solving super-multi-objective optimization problems. At the same time, in order to improve the convergence of the assistance pool, the present invention uses I ε+ The indicator pre-selects the auxiliary pool to ensure the convergence of the solution set. On this basis, a fusion I ε+ The assisted pool reselection strategy of the index and angle information is designed to achieve a balance between the convergence and diversity of the assisted particles. In addition, in order to enhance the selection pressure in the environmental selection process, the algorithm adopts I ε+ The dominance relationship replaces the traditional Pareto dominance relationship to improve the particle selection effect, thereby improving the performance of the entire algorithm and enhancing the ability of the particle swarm algorithm to solve multi-objective optimization problems.

[0013] As an advantage: the competition update module is provided with a particle swarm selection module and a dual-angle competition update module connected in sequence;

[0014] The particle swarm selection module is used to determine a leader set from the population P using a leader particle selection mechanism based on angle information;

[0015] The dual-angle competition update module is used to generate a competitive offspring population Offspring1 based on the population P and the leader set using a dual-angle-guided competitive particle update mechanism.

[0016] The competitive update module employs a competitive mechanism to eliminate the impact of global and individual optima on swarm particles. Specifically, during the competition process, the particle swarm selection module uses an angle-based leader particle selection mechanism to accurately select a set of leaders, which it then uses to guide the particle update process. Furthermore, to effectively mitigate the potential negative impact of global and individual optimal particles on algorithm performance, the dual-angle competitive update module employs a dual-angle-guided competitive particle update mechanism for particle updates. This mechanism significantly improves the algorithm's adaptability and stability in complex optimization scenarios by dynamically balancing its exploration and development capabilities.

[0017] Preferably, the particle swarm selection module determines a leader set from the population P through a leader particle selection mechanism based on angle information, specifically in the following steps:

[0018] A1: The particle swarm selection module divides all particles in the population P into layers based on the non-dominated sorting algorithm, and extracts the non-dominated frontier solution set U of each layer. c , U c = non-dominated sorting (P);

[0019] A2: Take the first-layer non-dominated frontier solution set U1 as the candidate set Leader_T of the leadership set;

[0020] When the population size length of the candidate set Leader_T is less than or equal to the set leader set length nums (where nums = N / 10, N is the population size), all particles in the candidate set Leader_T are added to the leader set Leader;

[0021] When the population size length of the candidate set Leader_T is greater than the set leader set length nums, the boundary value protection mechanism of the angle constraint is used to select boundary value particles from the candidate set Leader_T and save them in the leader set Leader;

[0022] A3: When the population size length of the leader set Leader is greater than the set leader set length nums, some candidate particles are randomly deleted from the leader set Leader so that the population size length of the leader set Leader is equal to the set leader set length nums;

[0023] When the population size of the leader set Leader is less than the set leader set length nums, the angle information between the leader set and the candidate set is used as the judgment criterion. The particle with the largest angle information in the candidate set (i.e., the solution with the best diversity) is removed from the candidate set and added to the leader set. This angle-information-guided leader set selection method ensures that the leader set maintains a certain size while maintaining good diversity and convergence, thereby guiding population evolution. This continues until the population size of the leader set Leader equals the set leader set length nums.

[0024] When the population size length of the leader set Leader is equal to the set leader set length nums, the particle swarm selection module outputs the leader set Leader to the dual-angle competition update module.

[0025] As an example, the particle swarm selection module selects boundary value particles from the candidate set Leader_T and saves them into the leader set Leader through the boundary value protection mechanism of angle constraint. The specific steps are as follows:

[0026] B1: The particle swarm selection module performs normalization processing on each particle in the candidate set Leader_T. The normalization expression is:

[0027]

[0028] Among them, f i ′(x a ) represents particle x a The normalized value of the i-th target in ; m represents the total number of targets; f i (x a ) represents particle x a The value of the i-th target in ; represents the ideal point, i.e., the minimum value of the i-th objective of all particles in the normalized population; represents the lowest point, i.e., the maximum value of the i-th target of all particles in the normalized population; particle x a The normalized target vector is F′(x a )=(f1′(x a ),f2′(x a ),...,f m ′(x a ));

[0029] After normalization, the angle information between any two particles in the candidate set Leader_T is calculated. The calculation expression is as follows:

[0030]

[0031] in, Represents particle x a and x b The vector angle between them; arccos represents the inverse cosine function;

[0032] B2: The particle swarm selection module calculates the first fitness value Sum of each particle in the candidate set Leader_T based on the convergence evaluation index. The calculation expression is:

[0033]

[0034] Among them, Sum(x) represents the first fitness value of particle x. The smaller the Sum value, the better the convergence of the particle.

[0035] B3: Use the boundary value identification strategy to identify the boundary value particles of the candidate set Leader_T and add the boundary value particles to the leader set Leader; the boundary value particles are the particles corresponding to the maximum and minimum values ​​of the normalized values ​​of each target in the candidate set Leader_T, and the total number of boundary value particles is 2m;

[0036] B4: For each boundary value particle x in the leader set Leader, introduce angle information as the similarity measurement basis and find the neighbor particle neighbor(x) that is closest to the boundary value particle x in the candidate set Leader. That is, select the particle with the closest angle distance to the boundary value particle as the neighbor particle neighbor(x);

[0037] B5: The particle swarm selection module compares the first fitness value Sum of the boundary value particle x and the neighbor particle neighbor(x). When the first fitness value of the neighbor particle neighbor(x) is greater than or equal to the first fitness value of the boundary value particle x, it indicates that the boundary value particle x has a relatively better fitness performance in the current solution space. Therefore, according to the boundary value protection mechanism, the boundary value particle x is retained in the leader set Leader; when the first fitness value of the neighbor particle neighbor(x) is less than the first fitness value of the boundary value particle x, that is, the convergence of the neighbor particle neighbor(x) is better than that of the boundary value particle x, according to the boundary value protection mechanism, the boundary value particle x is removed from the leader set Leader, and the neighbor particle neighbor(x) is included in the leader set Leader to achieve optimal adjustment of the leader set;

[0038] B6: Repeat steps B4-B5 until the first fitness value comparison between all boundary value particles and their neighbor particles is completed.

[0039] The particle swarm selection module introduces minimum angle information to conduct multiple rounds of screening and comparison of non-boundary solution particles, and ultimately selects individuals that excel in global search capability, local development capability, and population diversity maintenance, and incorporates them into the final leadership set, thereby ensuring that the leadership set can fully play its guiding role and promote the evolution of the entire population.

[0040] Preferably, the dual-angle competition update module generates a competitive offspring population Offspring1 through a dual-angle-guided competitive particle update mechanism, and the specific steps are as follows:

[0041] C1: The dual-angle competition update module generates a random variable randnum, randnum∈[0,1];

[0042] When randnum≤0.5, C2a is updated with the minimum angle;

[0043] When randnum>0.5, C2b is updated with the maximum angle;

[0044] C2a, minimum angle update;

[0045] Randomly select from the leader set Elite particles are used to calculate the angle information between the elite particles and the particles to be updated in the population P according to formula (2). The elite particle with the smallest angle value with the particle to be updated is regarded as the winning particle. The speed and position of the particle to be updated are updated based on the winning particle. The update expression is as follows:

[0046] v y,k (t+1)=r1v y,k (t)+r2.x w,k (t)-x y,k (t) / (4)

[0047] x y,k (t+1)=x y,k (t)+v y,k (t+1) (5)

[0048] Among them, v y,k represents the velocity vector of the particle y to be updated; r1 and r2 are random variables, randomly generated by uniform distribution in [0,1]; x w,k represents the position vector of the winning particle w; x y,k represents the position vector of the particle y to be updated; k represents the scaling factor, k = 0.05; t represents the time;

[0049] C2b, maximum angle update;

[0050] Randomly select from the leader set Elite particles, calculate the angle information between the elite particles and the particles to be updated in the population P according to formula (2), and take the elite particle with the largest angle value with the particle to be updated as the winning particle. Based on the winning particle, the speed and position of the particle to be updated are updated according to formulas (4) and (5);

[0051] C3: Repeat steps C1-C2 to complete the update of all particles in population P and generate the competitive offspring population Offspring1.

[0052] The dual-angle competitive update module constructs a dynamic evolutionary strategy by comprehensively utilizing the minimum and maximum angle information between the leader particle and the particle to be updated. Specifically, the minimum angle information is used to guide the population toward the global optimal region, ensuring the algorithm's exploration capability, while the maximum angle information is used to maintain population diversity and prevent premature convergence. This dual-angle information dynamically adjusts the evolutionary direction of the particles, allowing the particles to moderately explore new search spaces while maintaining a certain degree of similarity with the leader particle, thereby achieving an optimal balance between maintaining diversity and improving convergence.

[0053] Preferably, the auxiliary update module is provided with an assisting pool pre-selection module, an assisting pool re-selection module and an assisting pool update module connected in sequence;

[0054] The assisting pool pre-selection module is used to select the ε+ Indicator, calculate the second fitness value F of each particle in the population, and make a preliminary selection of the assisting pool based on the second fitness value F;

[0055] The assisting pool reselection module is used to select the ε+ Indicator and angle information are used to conduct secondary selection of the initially selected assistance pool;

[0056] The assisting pool updating module is used to update the secondary selected assisting pool through the assisting pool updating strategy to generate an auxiliary offspring population Offspring2.

[0057] As a preference: the auxiliary pool pre-selection module adopts an I-based ε+ The indicator's assistance pool pre-selection strategy performs preliminary selection of the assistance pool. The specific steps are as follows:

[0058] D1: The assisting pool pre-selection module obtains a population P and a previous generation assisting pool population Pool with a population size length of N, then merges the population P with the previous generation assisting pool population Pool to obtain an assisting pool population Matingpool with a merged population size length of 2N, and normalizes each particle in the assisting pool population Matingpool;

[0059] D2: Calculate the second fitness value F of each particle in the assisting pool population Matingpool. The calculation expression is:

[0060]

[0061] Among them, I ε+ (x a ,x b ) indicates I ε+ Indicator, used to describe a solution x a Dominate another solution x in the goal space b The minimum distance required; m is the number of targets; ε is the minimum distance that gives a Pareto set approximation in each dimension, so that the other approximation is weakly dominated; i is the i-th target; P is the population P; e is the base of the natural logarithm function; f i ′(x a ) represents particle x a The normalized value of the i-th target in f i ′(x b ) represents particle x b The normalized value of the i-th target in F(x a ) represents particle x a The second fitness value of , the larger the second fitness value, the better the convergence of the particle; k represents the scaling factor, k = 0.05;

[0062] D3: When the population size of the assisting pool population Matingpool is greater than N, find the particle with the smallest second fitness value from the assisting pool population Matingpool and delete it; then update the second fitness values ​​of the remaining particles in the assisting pool population Matingpool according to formula (8), and the calculation expression is:

[0063]

[0064] Among them, F * (x) represents the second fitness value of particle x after the update; F(x) represents the second fitness value of particle x before the update; x * The particle with the smallest second fitness value;

[0065] D4: Repeat step D3. When the population size length of the assisting pool population Matingpool is equal to N, the assisting pool pre-selection module outputs the assisting pool population Matingpool with a population size length of N to the assisting pool re-selection module.

[0066] In the updating mechanism of the assistance pool, CCAMOPSO first adopts I ε+ Indicators are selected, but I ε+As an index related to convergence, the selected particles usually have advantages in terms of convergence. However, some particles in the population show good diversity while maintaining good convergence. In the cooperative update process, both parties involved in the cooperation should be of high quality to ensure the effectiveness of the update operation and the global search ability. Therefore, a secondary selection needs to be carried out on the initially selected assistance pool.

[0067] Preferably, the assistance pool re-selection module adopts a dual-index-guided assistance pool re-selection strategy for the secondary selection of the assistance pool. The specific steps are as follows:

[0068] E1: The assistance pool re-selection module obtains the assistance pool population Matingpool, normalizes each particle in the assistance pool population Matingpool, and then calculates the second fitness value F of each particle in the assistance pool population Matingpool according to formula (6) and formula (7);

[0069] E2: Sort all the particles in the assistance pool population Matingpool according to the second fitness value F to obtain the ranking rank of the second fitness value of each particle; then calculate the angular information between any two particles through formula (2) to obtain the minimum angle Angle of each particle;

[0070] E3: Randomly select two particles x and y from the assistance pool population Matingpool for comparison;

[0071] If the second fitness value F(x) > F(y) and the minimum angle value Angle(x) > Angle(y), that is, the convergence and diversity of particle x are both better than those of particle y, then add particle x to the candidate set;

[0072] If the second fitness value F(x) < F(y) and the minimum angle value Angle(x) < Angle(y), then add particle y to the candidate set;

[0073] Otherwise, randomly select one particle from particles x and y to add to the candidate set;

[0074] E4: Calculate the acceptance probability of the candidate set particles according to formula (9), and the calculation expression is:

[0075] p = rank(i) / N (9)

[0076] Where, p represents the acceptance probability of the candidate set particles; rank(i) represents the ranking of the second fitness value of the candidate set particles; N represents the population size length of the assistance pool population Matingpool; rand(1) is a random probability, rand(1) ∈ [0,1];

[0077] When p>rand(1), the candidate set particle is added to the secondary assisting pool population R_Matingpool; otherwise, a particle is randomly selected from the assisting pool population Matingpool and added to the secondary assisting pool population R_Matingpool;

[0078] E5: Repeat steps E3-E4. When the population size length of the secondary assisting pool population R_Matingpool is equal to the population size preset value nums, output the secondary assisting pool population R_Matingpool to the assisting pool update module.

[0079] The assisting pool reselection module will ε+ The combination of indicators and angle information comprehensively considers the convergence and diversity of the algorithm, further improving the quality of the selected particles.

[0080] Preferably, the assistance pool updating module adopts an assistance pool updating strategy based on cooperative learning to update the secondary selected assistance pool, and the specific steps are as follows:

[0081] F1: The assisting pool update module randomly selects two particles x from the secondary assisting pool population R_Matingpool a ,x b , and update the two particles in a mutual learning manner according to formula (10) and formula (11), and then update the updated particle x′ a ,x′ b Add to the set P'; the particle update calculation expression is as follows:

[0082] x a,j ′=a j *x a,j +(1-a j )*x b,j (10)

[0083] x b,j ′=a j *x b,j +(1-a j )*x a,j (11)

[0084] Among them, x a,j ′ represents the updated particle x a The jth decision variable of x a,j Represents particle x before update a The jth decision variable of x b,j ′ represents the updated particle x b The jth decision variable of x b,j Represents particle x before update b The jth decision variable ofj is a random variable, a j is randomly generated as 0 or 1; j represents the decision variable, j∈[1,n];

[0085] F2: Repeat step F1. When the population size length of set P′ is equal to the population size preset value nums, perform polynomial mutation on set P′ to generate an auxiliary offspring population Offspring2.

[0086] In order to correct the shortcomings of the evolutionary strategy of the competitive particle swarm algorithm in solving multi-objective problems, such as fast convergence speed and easy falling into local optimality, the assistance pool update module adopts a new update strategy to update the selected assistance pool so as to generate candidate solutions more efficiently.

[0087] Because the particles in the assisting pool already have good convergence and diversity, they no longer use a competition-based update method. Instead, they adopt a cooperative learning method. This cooperative learning method changes the traditional one-way learning process of competing particles, enables information transfer between two particles, and helps particles in the population escape local optima.

[0088] As a preference: the environment selection module adopts an I ε+ The environmental selection mechanism of the dominance relationship selects N elite particles from the new population Q. The specific steps are as follows:

[0089] G1: The environment selection module performs non-dominated sorting on all particles in the new population Q based on the non-dominated sorting algorithm, and then normalizes each particle in the sorted population R;

[0090] G2: The environment selection module calculates the diversity index d(x, C) of each particle in the population R according to formula (12). The calculation expression is as follows:

[0091]

[0092] Where ρ represents the curvature of the curve / surface C; f i (x) represents the vector of the i-th target in particle x on the curve / surface C;

[0093] G3: Calculate the convergence index I of population R according to formula (6) ε+ , find two populations R with the lowest convergence index I ε+ Particles x and y;

[0094] When I ε+ When (x,y)<0, delete particle x from population R;

[0095] When I ε+When (x, y) > 0, particle y is deleted from population R;

[0096] When I ε+ When (x, y) = 0, the diversity indices d(x, C) of particles x and y are compared;

[0097] When d(x, C) > d(y, C), particle x is deleted from population R;

[0098] When d(x, C) < d(y, C), particle y is deleted from population R;

[0099] When d(x, C) = d(y, C), either particle x or particle y is randomly deleted from population R;

[0100] G4: Repeat step G3. When the population size of population R is equal to N, the environmental selection module outputs N elite particles in population R.

[0101] The non - dominated sorting algorithm uses I ε+ indices to judge the dominance relationship between two particles in the objective space. Based on I ε+ The dominance relationship of the indices is defined as follows:

[0102] Definition: Given two particle vectors x1 and x2 and their corresponding fitnesses I(x a ) and I(x b ), when and only when any one of the following formulas (13) or (14) is satisfied, then the vector x a is dominated by the vector x b (denoted as x a <x b );

[0103] I(x a , x b ) < I(x b , x a ) (13)

[0104]

[0105] I ε+ quantifies the minimum distance required for one solution to dominate another in the objective space and determines their dominance relationship based on this. During the environmental selection process, for two randomly selected individuals x and y, if x dominates y, then y will be removed from the population; conversely, if y dominates x, then x will be removed from the population. If there is no direct dominance relationship between x and y, that is, they are non - dominated (or non - inferior) to each other, then these two individuals will be regarded as equivalent, and the diversity index will be further used to distinguish them in order to make decisions in the subsequent selection process.

[0106] The diversity index uses the Euclidean distance between an individual and a curve / surface C to distinguish the diversity of two individuals. The characteristics of the curve / surface C are:

[0107]

[0108] The environment selection module combines I ε+ The index and distance d(x,C) information are used to eliminate the worst solution and finally select elite particles with higher quality, further improving the level of multi-objective particle swarm optimization.

[0109] The beneficial effects of the present invention are:

[0110] 1. The present invention adopts a particle update method that combines angle competition update and auxiliary update, which effectively solves the problem that particle breakthroughs are prone to falling into local optimality when solving multi-objective optimization problems.

[0111] 2. In order to improve the convergence of the assist pool, the present invention uses I ε+ The indicator pre-selects the auxiliary pool to ensure the convergence of the solution set. On this basis, a fusion I ε+ The assistance pool reselection strategy of indicator and angle information achieves a balance between the convergence and diversity of the assisted particles.

[0112] 3. In order to enhance the selection pressure in the environmental selection process, the present invention adopts I ε+ The dominance relationship replaces the traditional Pareto dominance relationship to improve the particle selection effect, thereby improving the performance of the entire algorithm and enhancing the ability of the particle swarm algorithm to solve multi-objective optimization problems. BRIEF DESCRIPTION OF THE DRAWINGS

[0113] Figure 1 Schematic diagram of the multi-objective / super-multi-objective particle swarm optimization framework CCAMOPSO in the embodiment;

[0114] Figure 2 Schematic diagram of the competitive particle update mechanism with dual-angle guidance in the embodiment;

[0115] Figure 3 2 is a graph showing the comparison of HV values ​​in the UF test in the embodiment. DETAILED DESCRIPTION

[0116] The present invention will be further described in detail below with reference to the accompanying drawings and specific examples. The following examples or drawings are used to illustrate the present invention, but are not intended to limit the scope of the present invention.

[0117] A multi-objective particle swarm optimization method based on angle competition and assisted evolution mechanism, characterized by comprising the following steps:

[0118] Step 1: Construct a multi-objective / super-multi-objective particle swarm optimization framework CCAMOPSO, such as Figure 1 As shown, the multi-objective / super-multi-objective particle swarm optimization framework CCAMOPSO is provided with a population initialization module, which is respectively connected to a competition update module, an auxiliary update module and an environment selection module, and the competition update module and the auxiliary update module are respectively connected to the environment selection module;

[0119] Step 2: The population initialization module randomly initializes N particles to construct a population P and sets the number of iterations to zero; the population initialization module passes the population P to the competition update module, the auxiliary update module and the environment selection module;

[0120] Step 3a: The competitive update module uses the N particles in the population P as a benchmark and adopts the leader particle selection mechanism based on angle information to determine the leader set; then, based on the population P and the selected leader set, a dual-angle-guided competitive particle update mechanism is used to generate the competitive offspring population Offspring1 and pass it to the environment selection module;

[0121] Step 3b: The auxiliary update module first selects an auxiliary pool to perform the auxiliary update mechanism based on the current population P and the previous generation auxiliary pool, and then further screens the selected auxiliary pool; then, it executes the auxiliary update strategy to generate the auxiliary offspring population Offspring2 and passes it to the environment selection module;

[0122] Step 4: The environment selection module merges the current population P, the competitive offspring population Offspring1, and the auxiliary offspring population Offspring2 to form a new population Q, and performs environment selection to select N elite particles from the new population Q as the population P to enter the next iteration operation;

[0123] Step 5: Repeat steps 3-4. When the maximum number of iterations is reached, end the iteration operation and then use the N elite particles obtained in the last iteration as the optimal particle swarm.

[0124] The competition update module is provided with a particle swarm selection module and a dual-angle competition update module connected in sequence;

[0125] The particle swarm selection module is used to determine a leader set from the population P using a leader particle selection mechanism based on angle information;

[0126] The dual-angle competition update module is used to generate a competitive offspring population Offspring1 based on the population P and the leader set using a dual-angle-guided competitive particle update mechanism.

[0127] The particle swarm selection module determines the leader set from the population P through a leader particle selection mechanism based on angle information. The specific steps are as follows:

[0128] A1: The particle swarm selection module divides all particles in the population P into layers based on the non-dominated sorting algorithm, and extracts the non-dominated frontier solution set U of each layer. c , U c = non-dominated sorting (P);

[0129] A2: Take the first-layer non-dominated frontier solution set U1 as the candidate set Leader_T of the leadership set;

[0130] When the population size length of the candidate set Leader_T is less than or equal to the set leader set length nums, all particles in the candidate set Leader_T are added to the leader set Leader;

[0131] When the population size length of the candidate set Leader_T is greater than the set leader set length nums, the boundary value protection mechanism of the angle constraint is used to select boundary value particles from the candidate set Leader_T and save them in the leader set Leader;

[0132] A3: When the population size length of the leader set Leader is greater than the set leader set length nums, some candidate particles are randomly deleted from the leader set Leader so that the population size length of the leader set Leader is equal to the set leader set length nums;

[0133] When the population size length of the leader set Leader is less than the set leader set length nums, the angle information between the leader set and the candidate set is selected as the judgment criterion, and the particle with the largest angle information in the candidate set is removed from the candidate set and added to the leader set; until the population size length of the leader set Leader is equal to the set leader set length nums;

[0134] When the population size length of the leader set Leader is equal to the set leader set length nums, the particle swarm selection module outputs the leader set Lead to the dual-angle competition update module.

[0135] The particle swarm selection module selects boundary value particles from the candidate set Leader_T and saves them to the leader set Leader through the boundary value protection mechanism of angle constraint. The specific steps are as follows:

[0136] B1: The particle swarm selection module performs normalization processing on each particle in the candidate set Leader_T. The normalization expression is:

[0137]

[0138] Among them, f i ′(x a ) represents particle x a The normalized value of the i-th target in ; m represents the total number of targets; f i (x a ) represents particle x a The value of the i-th target in; represents the ideal point, i.e., the minimum value of the i-th objective of all particles in the normalized population; represents the lowest point, i.e., the maximum value of the i-th target of all particles in the normalized population; particle x a The normalized target vector is F′(x a )=(f1′(x a ),f2′(x a ),...,f m ′(x a ));

[0139] After normalization, the angle information between any two particles in the candidate set Leader_T is calculated. The calculation expression is as follows:

[0140]

[0141] in, Represents particle x a and x b The vector angle between them; arccos represents the inverse cosine function;

[0142] B2: The particle swarm selection module calculates the first fitness value Sum of each particle in the candidate set Leader_T based on the convergence evaluation index. The calculation expression is:

[0143]

[0144] Among them, Sum(x) represents the first fitness value of particle x. The smaller the Sum value, the better the convergence of the particle.

[0145] B3: Use the boundary value identification strategy to identify the boundary value particles of the candidate set Leader_T and add the boundary value particles to the leader set Leader; the boundary value particles are the particles corresponding to the maximum and minimum values ​​of the normalized values ​​of each target in the candidate set Leader_T, and the total number of boundary value particles is 2m;

[0146] B4: For each boundary value particle x in the leader set Leader, introduce angle information as the similarity measurement basis and find the neighbor particle neighbor(x) that is closest to the boundary value particle x in the candidate set Leader_T;

[0147] B5: The particle swarm selection module compares the first fitness value Sum of the boundary value particle x with that of the neighbor particle neighbor(x). When the first fitness value of the neighbor particle neighbor(x) is greater than or equal to the first fitness value of the boundary value particle x, the boundary value particle x is retained in the leader set Leader; when the first fitness value of the neighbor particle neighbor(x) is less than the first fitness value of the boundary value particle x, the boundary value particle x is removed from the leader set Leader, and the neighbor particle neighbor(x) is included in the leader set Leader;

[0148] B6: Repeat steps B4-B5 until the first fitness value comparison between all boundary value particles and their neighbor particles is completed.

[0149] The dual-angle competition update module generates the competitive offspring population Offspring1 through a dual-angle-guided competitive particle update mechanism. The specific steps are as follows:

[0150] C1: The dual-angle competition update module generates a random variable randnum, randnum∈[0,1];

[0151] When randnum≤0.5, C2a is updated with the minimum angle;

[0152] When randnum>0.5, C2b is updated with the maximum angle;

[0153] C2a, minimum angle update;

[0154] Randomly select from the leader set Elite particles are used to calculate the angle information between the elite particles and the particles to be updated in the population P according to formula (2), and the elite particle with the smallest angle value with the particles to be updated is regarded as the winning particle, as shown in Figure 2 As shown, based on the winning particle, the speed and position of the particle to be updated are updated. The update expression is as follows:

[0155]

[0156] Among them, v y,k represents the velocity vector of the particle y to be updated; r1 and r2 are random variables, randomly generated by uniform distribution in [0,1]; x w,k represents the position vector of the winning particle w; x y,k represents the position vector of the particle y to be updated; k represents the scaling factor, k = 0.05; t represents the time;

[0157] C2b, maximum angle update;

[0158] Randomly select from the leader set Elite particles, calculate the angle information between the elite particles and the particles to be updated in the population P according to formula (2), and take the elite particle with the largest angle value with the particle to be updated as the winning particle. Based on the winning particle, the speed and position of the particle to be updated are updated according to formulas (4) and (5);

[0159] C3: Repeat steps C1-C2 to complete the update of all particles in population P and generate the competitive offspring population Offspring1.

[0160] The auxiliary update module is provided with an assisting pool pre-selection module, an assisting pool re-selection module and an assisting pool update module connected in sequence;

[0161] The assisting pool pre-selection module is used to select the ε+ Indicator, calculate the second fitness value F of each particle in the population, and make a preliminary selection of the assisting pool based on the second fitness value F;

[0162] The assisting pool reselection module is used to select the ε+ Indicator and angle information are used to conduct secondary selection of the initially selected assistance pool;

[0163] The assisting pool updating module is used to update the secondary selected assisting pool through the assisting pool updating strategy to generate an auxiliary offspring population Offspring2.

[0164] The auxiliary pool pre-selection module adopts the ε+ The indicator's assistance pool pre-selection strategy performs preliminary selection of the assistance pool. The specific steps are as follows:

[0165] D1: The assisting pool pre-selection module obtains a population P and a previous generation assisting pool population Pool with a population size length of N, then merges the population P with the previous generation assisting pool population Pool to obtain an assisting pool population Matingpool with a merged population size length of 2N, and normalizes each particle in the assisting pool population Matingpool;

[0166] D2: Calculate the second fitness value F of each particle in the assisting pool population Matingpool. The calculation expression is:

[0167] I ε+ (x a ,x b )=min v (f i ′(x a )-ε)≤f i ′(x b ), 1≤i≤m (6)

[0168]

[0169] Among them, I ε+ (x a ,x b ) indicates I ε+ Indicator, used to describe a solution x a Dominate another solution x in the goal space b The minimum distance required; m is the number of targets; v is the minimum distance that gives a Pareto set approximation in each dimension, so that the other approximation is weakly dominated; i is the i-th target; P is the population P; e is the base of the natural logarithm function; f i ′(x a ) represents particle x a The normalized value of the i-th target in f i ′(x b ) represents particle x b The normalized value of the i-th target in F(x a ) represents particle x a The second fitness value of , the larger the second fitness value, the better the convergence of the particle; k represents the scaling factor, k = 0.05;

[0170] D3: When the population size of the assisting pool population Matingpool is greater than N, find the particle with the smallest second fitness value from the assisting pool population Matingpool and delete it; then update the second fitness values ​​of the remaining particles in the assisting pool population Matingpool according to formula (8), and the calculation expression is:

[0171]

[0172] Among them, F * (x) represents the second fitness value of particle x after the update; F(x) represents the second fitness value of particle x before the update; x * The particle with the smallest second fitness value;

[0173] D4: Repeat step D3. When the population size length of the assisting pool population Matingpool is equal to N, the assisting pool pre-selection module outputs the assisting pool population Matingpool with a population size length of N to the assisting pool re-selection module.

[0174] The assisting pool reselection module adopts a dual-indicator-guided assisting pool reselection strategy to perform secondary selection of the assisting pool. The specific steps are as follows:

[0175] E1: The assistance pool re-selection module obtains the assistance pool population Matingpool, normalizes each particle in the assistance pool population Matingpool, and then calculates the second fitness value F of each particle in the assistance pool population Matingpool according to Formula (6) and Formula (7).

[0176] E2: Sort all the particles in the assistance pool population Matingpool according to the second fitness value F to obtain the ranking rank of the second fitness value of each particle; then calculate the angular information between any two particles through Formula (2) to obtain the minimum angle Angle of each particle.

[0177] E3: Randomly select two particles x and y from the assistance pool population Matingpool for comparison.

[0178] If the second fitness value F(x) > F(y) and the minimum angle value Angle(x) > Angle(y), then add particle x to the candidate set.

[0179] If the second fitness value F(x) < F(y) and the minimum angle value Angle(x) < Angle(y), then add particle y to the candidate set.

[0180] Otherwise, randomly select one particle from particles x and y to add to the candidate set.

[0181] E4: Calculate the acceptance probability of the candidate set particles according to Formula (9), and the calculation expression is:

[0182] p = rank(i) / N (9)

[0183] Where, p represents the acceptance probability of the candidate set particles; rank(i) represents the ranking of the second fitness value of the candidate set particles; N represents the population size length of the assistance pool population Matingpool; rand(1) is the random probability, and rand(1) ∈ [0, 1].

[0184] When p > rand(1), add the candidate set particles to the secondary assistance pool population R_Matingpool; otherwise, randomly select one particle from the assistance pool population Matingpool and add it to the secondary assistance pool population R_Matingpool.

[0185] E5: Repeat steps E3 - E4. When the population size length of the secondary assistance pool population R_Matingpool is equal to the preset population size value nums, output the secondary assistance pool population R_Matingpool to the assistance pool update module.

[0186] The assistance pool update module uses an assistance pool update strategy based on cooperative learning to update the secondary selected assistance pool. The specific steps are as follows:

[0187] F1: The assisting pool update module randomly selects two particles x from the secondary assisting pool population R_Matingpool a ,x b , and update the two particles in a mutual learning manner according to formula (10) and formula (11), and then update the updated particle x′ a ,x′ b Add to the set P'; the particle update calculation expression is as follows:

[0188] x a,j ′=a j *x a,j +(1-a j )*x b,j (10)

[0189] x b,j ′=a j *x b,j +(1-a j )*x a,j (11)

[0190] Among them, x a,j ′ represents the updated particle x a The jth decision variable of x a,j Represents particle x before update a The jth decision variable of x b,j ′ represents the updated particle x b The jth decision variable of x b,j Represents particle x before update b The jth decision variable of j is a random variable, a j is randomly generated as 0 or 1; j represents the decision variable, j∈[1,n];

[0191] F2: Repeat step F1. When the population size length of set P′ is equal to the population size preset value nums, perform polynomial mutation on set P′ to generate an auxiliary offspring population Offspring2.

[0192] The environment selection module adopts the ε+ The environmental selection mechanism of the dominance relationship selects N elite particles from the new population Q. The specific steps are as follows:

[0193] G1: The environment selection module performs non-dominated sorting on all particles in the new population Q based on the non-dominated sorting algorithm, and then normalizes each particle in the sorted population R;

[0194] G2: The environment selection module calculates the diversity index d(x, C) of each particle in population R according to formula (12), and the calculation expression is as follows:

[0195]

[0196] where ρ represents the curvature of the curve / surface C; f i (x) represents the vector of the i-th objective in particle x on the curve / surface C;

[0197] G3: Calculate the convergence index I of population R according to formula (6) ε+ , and find two particles x and y with the lowest convergence index I ε+ in population R;

[0198] When I ε+ (x, y) < 0, delete particle x from population R;

[0199] When I ε+ (x, y) > 0, delete particle y from population R;

[0200] When I ε+ (x, y) = 0, compare the diversity indices d(x, C) of particles x and y;

[0201] When d(x, C) > d(y, C), delete particle x from population R;

[0202] When d(x, C) < d(y, C), delete particle y from population R;

[0203] When d(x, C) = d(y, C), randomly delete particle x or particle y from population R;

[0204] G4: Repeat step G3. When the population size length of population R is equal to N, the environment selection module outputs N elite particles in population R.

[0205] Next, through specific experiments, the CCAMOPSO method in this invention is compared with five state-of-the-art multi-objective evolutionary algorithms on the 3, 5, 8, 10-objective test problems of DTLZ1-7, WFG1-9 and MaF1-6, and the HV values of each algorithm are calculated.

[0206] The five comparison algorithms are the multi-objective evolutionary algorithm MaOEA-IGD based on the IGD index, the evolutionary multi-objective algorithm PREA that explores the characteristics of indicators and uses the promising region, the multi-objective genetic algorithm TSNSGAII based on non-dominated sorting, the multi-objective optimization algorithm MOEA / D-UR based on the decomposition strategy, and the multi-objective optimization evolutionary algorithm HEA based on the hyper-domination degree.

[0207] 1. The comparison results on DTLZ, WFG and MaF problems are as follows:

[0208] (1) The comparison results for the three target test problems are shown in Table 1. The average ranking data show that the CCAMOPSO algorithm has the best performance among the compared algorithms and ranks first. Furthermore, the results of the Wilcoxon rank sum test confirm that CCAMOPSO has significant advantages over the NSGA-II-SDR, MaOEA-IGD, and PREA algorithms. Specifically, CCAMOPSO ranked first on 13 out of a total of 22 test instances, and ranked second on seven test functions: DTLZ5, DTLZ7, WFG1, WFG3, MaF1, MaF3, and MaF6.

[0209] CCAMOPSO demonstrated excellent performance on the challenging DTLZ3 problem, thanks to its ability to effectively leverage the convergence properties of elite individuals, accelerating the population's approach to the Pareto frontier. In contrast, hpaEA performed particularly well on DTLZ1 and DTLZ2 (especially when the number of targets m = 3), but lagged behind on other benchmarks. NSGA-II-SDR performed similarly to MOEA / D-UR, while MaOEA-IGD performed relatively poorly among these algorithms.

[0210] (2) The comparison results for the five target test problems are shown in Table 2. It is worth noting that CCAMOPSO achieved the best average ranking among the compared algorithms and significantly outperformed the other algorithms. Specifically, CCAMOPSO successfully won the first place on 12 out of 22 test instances and ranked second on the three test functions WFG3, WFG9, and MaF3. In addition, when dealing with multimodal problems such as DTLZ3 and MaF3, CCAMOPSO's performance did not weaken as the number of targets increased.

[0211] As for other algorithms, hpaEA continued to perform well on the DTLZ1 and DTLZ2 problems (when the number of objectives m = 5), while PREA's average ranking was close behind CCAMOPSO, achieving three first-place finishes and five second-place finishes across 22 test sets. NSGA-II-SDR, TSNSGAII, MOEA / D-UR, and SSCEA performed similarly on the multi-objective optimization problem with five objectives, but hpaEA's performance declined compared to its counterparts with fewer objectives.

[0212] (3) The comparison results for the eight target test problems are shown in Table 3. As the number of targets gradually increases, it can be clearly seen from the average ranking data that CCAMOPSO still significantly occupies the top position. Specifically, CCAMOPSO achieved the highest ranking in 14 of the 22 test instances and ranked second on the two test functions WFG2 and WFG3. In addition, when dealing with multi-modal problems such as MaF3, CCAMOPSO still shows the best performance.

[0213] hpaEA's average ranking was 7.091, and its performance dropped significantly when handling multimodal problems such as DTLZ3 and MaF3. This suggests that hpaEA may have convergence issues as the number of objectives increases, leading to a decline in overall performance. On the other hand, TS-NSGAII, MOEA / D-UR, and HEA showed relatively similar performance levels on the 8-objective multi-objective optimization problem (MaOP).

[0214] (4) The comparison results for the ten target test problems are shown in Table 4. As the number of targets continues to increase, it can be clearly observed from the average ranking data that CCAMOPSO still shows the best performance among all the tested algorithms. Comprehensive evaluation shows that CCAMOPSO achieved the top two results in 13 out of a total of 22 test problems. In particular, CCAMOPSO performed well in solving the MaF problem, achieving the best results on MaF3, MaF4, and MaF5, and achieving an excellent result of HV value equal to 1 on the MaF3 problem.

[0215] PREA's average ranking has declined. Further analysis reveals that PREA's performance on multimodal problems such as DTLZ3 and MaF3 degrades significantly, indicating that PREA suffers from convergence problems as the number of objectives increases. NSGA-IISDR, MOEA / D-UR, and TS-NSGA-II perform relatively similarly on 10-objective multi-objective optimization problems. In contrast, hpaEA's performance degrades significantly with increasing the number of objectives.

[0216] Table 1 Three-objective HV results of the involved algorithms on DTLZ, MaF and WFG

[0217]

[0218]

[0219] Table 2 5-target HV results of the involved algorithms on DTLZ, MaF and WFG

[0220]

[0221]

[0222] Table 3 8-target HV results of the involved algorithms on DTLZ, MaF and WFG

[0223]

[0224]

[0225] Table 4 10-target HV results of the involved algorithms on DTLZ, MaF and WFG

[0226]

[0227]

[0228] Overall, the following results are presented: 1) For irregular problems (such as WFG1, WFG2, and DTLZ7), the CCAMOPSO algorithm demonstrates good performance, as shown in the table. In contrast, decomposition-based algorithms (such as MOEA / D-UR) perform relatively poorly due to their reliance on reference vectors, which may not be applicable to irregular PFs. Furthermore, the MaOEA-IGD algorithm's strategy for maintaining diversity is insufficient, resulting in low population coverage. 2) For degenerate problems (such as WFG3 and MaF6), CCAMOPSO's performance is slightly inferior, indicating that the algorithm lacks a clear advantage in solving such problems. 3) For concave PF problems (including WFGs 4–9, DTLZ2–4, MaF2, and MaF5), the CCAMOPSO algorithm demonstrates competitive results, demonstrating its advantage in solving these problems. 4) For convex problems (such as MaF3), most existing algorithms face challenges in balancing population convergence and diversity. CCAMOPSO ranked second on the 3-objective and 5-objective MaF3 problems, and first on the 8-objective and 10-objective problems, fully demonstrating CCAMOPSO's significant advantage in handling convex problems. 5) Regarding linear problems (such as DTLZ1), the data in the table shows that CCAMOPSO's performance is slightly inferior to algorithms such as hapaEA, PREA, and HEA, indicating that its performance on linear problems needs to be improved.

[0229] In summary, the CCAMOPSO algorithm shows different performance characteristics in different types of problems, especially excellent performance in irregular problems and convex problems, but further optimization is needed when dealing with degenerate problems and linear problems.

[0230] (2) The comparison results on ZDT, UF and IMOP problems are as follows:

[0231] Previous experiments have validated the algorithm's effectiveness in handling multi- and super-multi-objective problems using test suites such as DTLZ, WFG, and MaF. To further explore the proposed algorithm's generalization capabilities in complex optimization scenarios, this experiment focuses on two- and three-objective optimization problems with complex characteristics such as non-uniform distribution and degradation, as found in the ZDT, UF, and IMOP test suites. This aims to demonstrate that the algorithm can maintain superior performance on low-dimensional multi-objective problems while solving super-multi-objective optimization problems.

[0232] Table 5 shows the mean and standard deviation of the hypervolume (HV) values ​​of the proposed algorithm (CCAMOPSO) and nine state-of-the-art multi-objective optimization algorithms on the test sets ZDT, UF, and IMOP. The optimal HV value for each test case is highlighted in bold. CCAMOPSO achieved optimal performance on 11 of the 22 test problems. Overall ranking results show that CCAMOPSO performed best, followed closely by hpaEA, PREA, and SSCEA; MaOEA / IGD and HEA performed slightly worse.

[0233] To further evaluate CCAMOPSO's ability to handle problems with irregular Pareto fronts (PFs), we compared CCAMOPSO with the ten aforementioned algorithms on the IMOP test set. The results showed that CCAMOPSO achieved the best results on five of the eight test cases in the IMOP test set, and ranked second on two others. Its performance was only slightly inferior on the IMOP5 problem.

[0234] To further analyze the experimental results, the experimental results of all the comparison algorithms on the UF1 to UF9 test cases are presented in the form of box plots. Figure 3 These box plots intuitively show the distribution of the performance of each algorithm, from which it can be clearly seen that the comprehensive performance of the CCAMOPSO algorithm is excellent.

[0235] Table 5 HV results of the algorithms on ZDT, UF and IMOP

[0236]

[0237]

[0238] The foregoing description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Those skilled in the art will readily appreciate that various modifications and variations of the present invention are possible. Any modifications, equivalent substitutions, or improvements made within the spirit and principles of the present invention are intended to be within the scope of protection of the present invention.

Claims

1. A multi-objective particle swarm optimization method based on angle competition and assisted evolution mechanism, characterized in that: The following steps are involved: Step 1: Construct a multi-objective / super-multi-objective particle swarm optimization framework CCAMOPSO, which is provided with a population initialization module. The population initialization module is respectively connected to a competition update module, an auxiliary update module and an environment selection module. The competition update module and the auxiliary update module are respectively connected to the environment selection module. Step 2: The population initialization module randomly initializes N particles to construct a population P and sets the number of iterations to zero; the population initialization module passes the population P to the competition update module, the auxiliary update module and the environment selection module; Step 3a: The competitive update module uses the N particles in the population P as a benchmark and adopts the leader particle selection mechanism based on angle information to determine the leader set; then, based on the population P and the selected leader set, a dual-angle-guided competitive particle update mechanism is used to generate the competitive offspring population Offspring1 and pass it to the environment selection module; Step 3b: The auxiliary update module first selects an auxiliary pool to perform the auxiliary update mechanism based on the current population P and the previous generation auxiliary pool, and then further screens the selected auxiliary pool; then, it executes the auxiliary update strategy to generate the auxiliary offspring population Offspring2 and passes it to the environment selection module; Step 4: The environment selection module merges the current population P, the competitive offspring population Offspring1, and the auxiliary offspring population Offspring2 to form a new population Q, and performs environment selection to select N elite particles from the new population Q as the population P to enter the next iteration operation; Step 5: Repeat steps 3-4. When the maximum number of iterations is reached, end the iteration operation and then use the N elite particles obtained in the last iteration as the optimal particle swarm.

2. The multi-objective particle swarm optimization method based on angle competition and assisted evolution mechanism according to claim 1 is characterized in that: The competition update module is provided with a particle swarm selection module and a dual-angle competition update module connected in sequence; The particle swarm selection module is used to determine a leader set from the population P using a leader particle selection mechanism based on angle information; The dual-angle competition update module is used to generate a competitive offspring population Offspring1 based on the population P and the leader set using a dual-angle-guided competitive particle update mechanism.

3. The multi-objective particle swarm optimization method based on angle competition and assisted evolution mechanism according to claim 2 is characterized in that: The particle swarm selection module determines the leader set from the population P through a leader particle selection mechanism based on angle information. The specific steps are as follows: A1: The particle swarm selection module divides all particles in the population P into layers based on the non-dominated sorting algorithm, and extracts the non-dominated frontier solution set U of each layer. c , U c = non-dominated sorting (P); A2: Take the first-layer non-dominated frontier solution set U1 as the candidate set Leader_T of the leader set; When the population size length of the candidate set Leader_T is less than or equal to the set leader set length nums, all particles in the candidate set Leader_T are added to the leader set Leader; When the population size length of the candidate set Leader_T is greater than the set leader set length nums, the boundary value protection mechanism of the angle constraint is used to select boundary value particles from the candidate set Leader_T and save them in the leader set Leader; A3: When the population size length of the leader set Leader is greater than the set leader set length nums, some candidate particles are randomly deleted from the leader set Leader so that the population size length of the leader set Leader is equal to the set leader set length nums; When the population size length of the leader set Leader is less than the set leader set length nums, the angle information between the leader set Leader and the candidate set Leader_T is selected as the judgment criterion, and the particle with the largest angle information in the candidate set Leader_T is removed from the candidate set Leader_T and added to the leader set Leader; until the population size length of the leader set Leader is equal to the set leader set length nums; When the population size length of the leader set Leader is equal to the set leader set length nums, the particle swarm selection module outputs the leader set Leader to the dual-angle competition update module.

4. The multi-objective particle swarm optimization method based on angle competition and assisted evolution mechanism according to claim 3 is characterized in that: The particle swarm selection module selects boundary value particles from the candidate set Leader_T and saves them to the leader set Leader through the boundary value protection mechanism of angle constraint. The specific steps are as follows: B1: The particle swarm selection module performs normalization processing on each particle in the candidate set Leader_T. The normalization expression is: Among them, f′ i (x a ) represents particle x a The normalized value of the i-th target in ; m represents the total number of targets; f i (x a ) represents particle x a The value of the i-th target in ; represents the ideal point, i.e., the minimum value of the i-th objective of all particles in the normalized population; represents the lowest point, i.e., the maximum value of the i-th target of all particles in the normalized population; particle x a The normalized target vector is F′(x a )=(f1′(x a ),f2′(x a ),...,f′ m (x a )); After normalization, the angle information between any two particles in the candidate set Leader_T is calculated. The calculation expression is as follows: in, Represents particle x a and x b The vector angle between them; arccos represents the inverse cosine function; B2: The particle swarm selection module calculates the first fitness value Sum of each particle in the candidate set Leader_T based on the convergence evaluation index. The calculation expression is: Among them, Sum(x) represents the first fitness value of particle x. The smaller the Sum value, the better the convergence of the particle. B3: Use the boundary value identification strategy to identify the boundary value particles of the candidate set Leader_T and add the boundary value particles to the leader set Leader; the boundary value particles are the particles corresponding to the maximum and minimum values ​​of the normalized values ​​of each target in the candidate set Leader_T, and the total number of boundary value particles is 2m; B4: For each boundary value particle x in the leader set Leader, introduce angle information as the similarity measurement basis and find the neighbor particle neighbor(x) that is closest to the boundary value particle x in the candidate set Leader_T; B5: The particle swarm selection module compares the first fitness value Sum of the boundary value particle x with that of the neighbor particle neighbor(x). When the first fitness value of the neighbor particle neighbor(x) is greater than or equal to the first fitness value of the boundary value particle x, the boundary value particle x is retained in the leader set Leader; when the first fitness value of the neighbor particle neighbor(x) is less than the first fitness value of the boundary value particle x, the boundary value particle x is removed from the leader set Leader, and the neighbor particle neighbor(x) is included in the leader set Leader; B6: Repeat steps B4-B5 until the first fitness value comparison between all boundary value particles and their neighbor particles is completed.

5. The multi-objective particle swarm optimization method based on angle competition and assisted evolution mechanism according to claim 2 is characterized in that: The dual-angle competition update module generates the competitive offspring population Offspring1 through a dual-angle-guided competitive particle update mechanism. The specific steps are as follows: C1: The dual-angle competition update module generates a random variable randnum, randnum∈[0,1]; When randnum≤0.5, C2a is updated with the minimum angle; When randnum>0.5, C2b is updated with the maximum angle; C2a, minimum angle update; Randomly select from the leader set Elite particles are generated, and the angle information between the elite particles and the particles to be updated in the population P is calculated. The elite particle with the smallest angle value with the particle to be updated is regarded as the winning particle. The speed and position of the particle to be updated are updated based on the winning particle. The update expression is as follows: v y,k (t+1)=r1v y,k (t)+r2.x w,k (t)-x y,k (t) / (4) x y,k (t+1)=x y,k (t)+v y,k (t+1) (5) Among them, v y,k represents the velocity vector of the particle y to be updated; r1 and r2 are random variables, randomly generated by uniform distribution in [0,1]; x w,k represents the position vector of the winning particle w; x y,k represents the position vector of the particle y to be updated; k represents the scaling factor, k = 0.05; t represents the time; C2b, maximum angle update; Randomly select from the leader set Elite particles are generated, and the angle information between the elite particles and the particles to be updated in the population P is calculated. The elite particle with the largest angle value with the particle to be updated is regarded as the winning particle. Based on the winning particle, the speed and position of the particle to be updated are updated using formulas (4) and (5); C3: Repeat steps C1-C2 to complete the update of all particles in population P and generate the competitive offspring population Offspring1.

6. The multi-objective particle swarm optimization method based on angle competition and assisted evolution mechanism according to claim 1 is characterized in that: The auxiliary update module is provided with an assisting pool pre-selection module, an assisting pool re-selection module and an assisting pool update module connected in sequence; The assisting pool pre-selection module is used to select the ε+ Indicator, calculate the second fitness value F of each particle in the population, and make a preliminary selection of the assisting pool based on the second fitness value F; The assisting pool reselection module is used to select the ε+ Indicator and angle information are used to conduct secondary selection of the initially selected assistance pool; The assisting pool updating module is used to update the secondary selected assisting pool through the assisting pool updating strategy to generate an auxiliary offspring population Offspring2.

7. The multi-objective particle swarm optimization method based on angle competition and assisted evolution mechanism according to claim 6 is characterized in that: The auxiliary pool pre-selection module adopts the ε+ The indicator's assistance pool pre-selection strategy performs preliminary selection of the assistance pool. The specific steps are as follows: D1: The assisting pool pre-selection module obtains a population P and a previous generation assisting pool population Pool with a population size length of N, then merges the population P with the previous generation assisting pool population Pool to obtain an assisting pool population Matingpool with a merged population size length of 2N, and normalizes each particle in the assisting pool population Matingpool; D2: Calculate the second fitness value F of each particle in the assisting pool population Matingpool. The calculation expression is: I ε+ (x a ,x b )=min ε (f i ′(x a )-ε)≤f i ′(x b ),1≤i≤m (6) Among them, I ε+ (x a ,x b ) indicates I ε+ Indicator used to describe a solution x a Dominate another solution x in the goal space b The minimum distance required; m is the number of targets; ε is the minimum distance that gives a Pareto set approximation in each dimension, so that the other approximation is weakly dominated; i is the i-th target; P is the population P; e is the base of the natural logarithm function; f′ i (x a ) represents particle x a The normalized value of the i-th target in ; f′ i (x b ) represents particle x b The normalized value of the i-th target in F(x a ) represents particle x a The second fitness value of , the larger the second fitness value, the better the convergence of the particle; k represents the scaling factor, k = 0.05; D3: When the population size of the assisting pool population Matingpool is greater than N, find the particle with the smallest second fitness value from the assisting pool population Matingpool and delete it; then update the second fitness values ​​of the remaining particles in the assisting pool population Matingpool according to formula (8), and the calculation expression is: Among them, F * (x) represents the second fitness value of particle x after the update; F(x) represents the second fitness value of particle x before the update; x * The particle with the smallest second fitness value; D4: Repeat step D3. When the population size length of the assisting pool population Matingpool is equal to N, the assisting pool pre-selection module outputs the assisting pool population Matingpool with a population size length of N to the assisting pool re-selection module.

8. The multi-objective particle swarm optimization method based on angle competition and assisted evolution mechanism according to claim 6 is characterized in that: The assisting pool reselection module adopts a dual-indicator-guided assisting pool reselection strategy to perform secondary selection of the assisting pool. The specific steps are as follows: E1: The assisting pool reselection module obtains the assisting pool population Matingpool, normalizes each particle in the assisting pool population Matingpool, and then calculates the second fitness value F of each particle in the assisting pool population Matingpool; E2: Sort all particles in the assisting pool according to the second fitness value F to obtain the second fitness value ranking of each particle; then calculate the angle information between any two particles using formula (2) to obtain the minimum angle Angle of each particle; E3: Randomly select two particles x and y from the assisting pool population Matingpool for comparison; If the second fitness value F(x)>F(y) and the minimum angle value Angle(x)>Angle(y), then add particle x to the candidate set; If the second fitness value F(x) < F(y) and the minimum angle value Angle(x) < Angle(y), then add particle y to the candidate set; Otherwise, randomly select one particle from particles x and y and add it to the candidate set; E4: Calculate the acceptance probability of the candidate set particles according to formula (9), and the calculation expression is: p = rank(i) / N (9) Where, p represents the acceptance probability of the candidate set particles; rank(i) represents the ranking of the second fitness value of the candidate set particles; N represents the population size length of the assistance pool population Matingpool; rand(1) is a random probability, and rand(1) ∈ [0, 1]; When p > rand(1), add the candidate set particles to the secondary assistance pool population R_Matingpool; otherwise, randomly select one particle from the assistance pool population Matingpool and add it to the secondary assistance pool population R_Matingpool; E5: Repeat steps E3 - E4. When the population size length of the secondary assistance pool population R_Matingpool is equal to the preset population size value nums, output the secondary assistance pool population R_Matingpool to the assistance pool update module.

9. The multi-objective particle swarm optimization method based on angle competition and assisted evolution mechanism according to claim 6, characterized in that: The assistance pool update module updates the second - selected assistance pool using an assistance pool update strategy based on cooperative learning. The specific steps are as follows: F1: The assisting pool update module randomly selects two particles x from the secondary assisting pool population R_Matingpool a ,x b , and update the two particles in a mutual learning manner according to formula (10) and formula (11), and then update the updated particle x′ a ,x′ b Add to the set P'; the particle update calculation expression is as follows: x a,j ′=a j *x a,j +(1-a j )*x b,j (10) x b,j ′=a j *x b,j +(1-a j )*x a,j (11) Among them, x a,j ′ represents the updated particle x a The jth decision variable of x a,j Represents particle x before update a The j-th decision variable of x b,j ′ represents the updated particle x b The jth decision variable of x b,j Represents particle x before update b The jth decision variable of j is a random variable, a j is randomly generated as 0 or 1; j represents the decision variable, j∈[1,n]; F2: Repeat step F1. When the population size length of the set P′ is equal to the preset population size value nums, perform polynomial mutation on the set P′ to generate the auxiliary offspring population Offspring2.

10. The multi-objective particle swarm optimization method based on angle competition and assisted evolution mechanism according to claim 1, characterized in that: The environment selection module adopts the ε+ The environmental selection mechanism of the dominance relationship selects N elite particles from the new population Q. The specific steps are as follows: G1: The environmental selection module performs non - dominated sorting on all particles in the new population Q based on the non - dominated sorting algorithm, and then normalizes each particle in the sorted population R; G2: The environmental selection module calculates the diversity index d(x, C) of each particle in the population R according to formula (12), and the calculation expression is as follows: Where ρ represents the curvature of the curve / surface C; f i (x) represents the vector of the i-th target in particle x on the curve / surface C; G3: Calculate the convergence index I of the population R ε+ , find two populations R with the lowest convergence index I ε+ Particles x and y; When I ε+ When (x,y)<0, delete particle x from population R; When I ε+ When (x,y)>0, delete particle y from population R; When I ε+ When (x,y)=0, the diversity index d(x,C) of particles x and y is compared; When d(x, C) > d(y, C), delete particle x from the population R; When d(x, C) < d(y, C), delete particle y from the population R; When d(x, C) = d(y, C), randomly delete particle x or particle y from the population R; G4: Repeat step G3. When the population size length of the population R is equal to N, the environmental selection module outputs N elite particles in the population R.

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