Evolution constraint optimization method based on urgency perception constraint processing technology

Abstract

The invention discloses an evolutionary constraint optimization method based on urgency perception constraint processing technology, which can effectively process complex and diverse constraint conditions in an optimization problem and improve the solving efficiency and the solution quality, and comprises the following steps: S1, initializing a population and constraining urgency weight; s2, calculating a fitness function and an original constraint default degree of the population; s3, obtaining a filial generation population according to variation, crossover and selection operations, and updating the urgency weight of a population constraint condition; s4, updating population individuals according to the used constraint processing technology; s5, repeating the steps S3-S4 until a termination condition is met; the method has good universality, is relatively simple to implement, has good calculation efficiency and expandability, and is easier to popularize in practical application.

Description

Technical Field

[0001] The present invention belongs to the field of intelligent computing, and in particular relates to an evolutionary constraint optimization method based on urgency-aware constraint processing technology. Background Art

[0002] Many practical optimization problems involve complex constraints, which make the feasible region smaller and the search process more difficult. Constraints define a non-empty feasible domain, which contains many restrictive conditions or constraints. The solution must strictly follow these constraints to solve a specific optimization problem. Generally speaking, these constraints can be divided into two categories: equality constraints and inequality constraints. The former are presented in the form of mathematical equations, and the latter are given in the form of mathematical inequalities. Both types of constraints require the decision variables to meet the corresponding conditions. The goal of solving constrained optimization problems is to find the decision variable values ​​that make the objective function reach the optimal (minimum or maximum) under the premise of satisfying all constraints. When solving constrained optimization problems, candidate solutions that meet all constraints are feasible individuals, and candidate solutions that do not meet any constraints are infeasible individuals.

[0003] Dealing with infeasible individuals is a major challenge for constrained optimization problems. To address this problem, commonly used constraint processing techniques include penalty functions, feasibility rules, constraint methods, random sorting methods, and multi-objective methods. These methods use the information of infeasible individuals in different ways to guide the search process to converge to the optimal solution. No constraint processing technique is superior to other techniques for all problems. Therefore, for a specific constrained optimization problem, a lot of trial and error experiments are required to select the most appropriate constraint processing technique and adjust the relevant parameters. This shows that solving constrained optimization problems requires in-depth research on different constraint processing methods and selecting the best method based on the characteristics of the specific problem. This is the key to improving the efficiency of solving constrained optimization problems.

[0004] Evolutionary constrained optimization based on urgency-aware constraint processing attempts to guide the algorithm to prioritize the most urgent constraints during the search process by evaluating the urgency of the constraints. The core idea of ​​this method is that different constraints may have different impacts on the optimization goal, and the constraints that have a greater impact on the goal should be prioritized. In this way, a wider range of feasible areas can be explored while ensuring that the most critical constraints are met, thereby improving the efficiency and solution quality of the algorithm. This constraint processing method based on urgency perception brings new ideas and possibilities to solving complex constrained optimization problems. It can more flexibly and effectively utilize infeasible solution information and improve solution efficiency. This is a promising method worthy of further exploration. Summary of the invention

[0005] The purpose of the present invention is to provide an evolutionary constrained optimization method based on urgency-aware constraint processing technology to solve the problem of easily falling into local optimality in constrained optimization problems, and to obtain the optimal solution more quickly during the constraint optimization process.

[0006] In order to solve the above technical problems, the present invention adopts the following technical solutions and implementation steps:

[0007] The evolutionary constrained optimization method based on urgency-aware constraint processing technology is an emerging constraint optimization method. It guides the search process by dynamically evaluating the urgency of the constraints, thereby alleviating the common local optimal dilemma of constrained optimization problems. It includes the following steps:

[0008] Step S1, initializing the population and constraint urgency weights;

[0009] Step S2, calculating the fitness function and the original constraint violation degree of the population;

[0010] Step S3, obtaining a progeny population according to mutation, crossover and selection operations, and updating the urgency weight of the population constraint condition;

[0011] Step S4, updating the population individuals according to the constraint processing technology used;

[0012] Step S5, repeat steps S3-S4 until the termination condition is met.

[0013] Furthermore, the step S1 of initializing the population and the constraint urgency weights includes the following steps:

[0014] Step S11, define the maximum number of iterations T and the population size of the population as NP;

[0015] Step S12: construct an initialization population P = {x 1 ,x 2 ,...,x NP}, when using the nature-inspired algorithm to solve the constrained optimization problem, the constrained optimization problem has multiple constraints. The number of constraints for the constrained optimization problem is set to m, and the constraint urgency weight is initialized

[0016] Furthermore, the step S2 of calculating the fitness function and the original constraint violation degree of the population includes the following steps:

[0017] Step S21: Substitute the population P into the constrained optimization problem to obtain the value of each individual x i The fitness f(x) of (i=1,2,...,NP) i );

[0018] Step S22: The population P calculates each individual x according to various constraints in the constrained optimization problem. i The constraint violation degree g of the jth constraint condition j (x i )(j=1,2,...,m).

[0019] Furthermore, the step S3 obtains the offspring population according to the mutation, crossover and selection operations, and updates the urgency weight of the population constraint condition, including the following steps:

[0020] Step S31, execute the loop and initialize the current number of iterations t=1;

[0021] Step S32: Perform mutation, crossover and selection operations on the population P to generate a descendant population Q = {v 1 ,v 2 ,...,v NP};

[0022] Step S33: After substituting the population Q into the constrained optimization problem, we can get the value of each individual v i (i=1,2,...,NP) fitness f(v i ) and each individual v i The constraint violation degree g of the jth (j=1,2,...,m) constraint condition j (v i );

[0023] Step S34: traverse each individual v in the population Q i , according to each individual v i For each constraint violation g j (v i ), find the individual v in the population Q that does not satisfy the jth constraint condition i The number of a j , as shown in formula (1);

[0024] In formula (2), p j (v i ) means that when individual v i When the jth constraint is satisfied, the value is 0. i When the jth constraint is not satisfied, the value is 1;

[0025] Step S35: For the jth (j=1, 2, ..., m) constraint, the above formula calculates the number of species with a j Individuals do not satisfy this constraint. jDividing by the total number of individuals in the entire population that do not meet all constraints, we get the proportion of violation of the jth constraint, which can reflect the relative importance of the jth constraint in the entire optimization problem; the present invention can set the urgency weight of the constraint according to this proportion, and individuals that violate important constraints will be subject to a larger fitness penalty. The urgency weight of each constraint is calculated. j , the specific formula is shown in (3);

[0026] Step S36, the total value of constraint violation of each individual indicates the distance between the current individual and the feasible domain. The smaller the total value of constraint violation, the closer the current individual is to the feasible domain and the optimal solution. In order to make the search direction of the population search algorithm explore more areas with larger constraint violation, the constraint violation of each individual can be adjusted. The present invention updates the population Q = {v 1 ,v 2 ,...,v NP} individual v i Each constraint violation degree Conv(v i ), the specific formula is shown in (4);

[0027] Step S37, update population p = {x 1 ,x 2 ,...,x NP} individual x i Each constraint violation Conv(x i ), the specific formula is shown in (5);

[0028] Furthermore, the step S4 of updating the population individuals according to the constraint processing technology used includes the following steps:

[0029] Step S41: According to the selected constraint processing technology, all individuals in population P and population Q are compared. If (Conv(x i ),f(x i ))<(Conv(v i ),f(v i ))(i.e. individual v in Q i Compared with the individual x in the original population P i Better), then replace the original individual x of P i , otherwise keep individual x from population P i ; Common constraint processing techniques include penalty function, feasibility rule, ε constraint method, random sorting method and multi-objective method;

[0030] The penalty function method introduces a penalty term σ to penalize the violation of the constraint, which converts the constrained optimization method into an unconstrained term, (Conv(x i ),f(x i ))<(Conv(v i ),f(v i )) can be expressed by formula (6);

[0031] The feasibility rule-based method is a popular constraint processing technique. The feasibility rule adopts three feasibility criteria: (1) for two feasible solutions, the individual with smaller fitness is given priority; (2) for a feasible individual and an infeasible individual, the feasible individual is given priority; (3) for two infeasible individuals, the individual with smaller constraint violation is given priority. According to the three feasibility criteria, (Conv(x i ),f(x i ))<(Conv(v i ),f(v i ))As shown in formula (7);

[0032] The ε constraint method uses the ε parameter to control the tolerance level of the relaxation of the constraint, indicating that the individual is considered feasible at this level; the ε level can be used to explore the search space well in the early stage; in the later stage, in order to obtain feasible individuals, the level will be set to 0 at a certain point in the evolution; the ε constraint method (Conv(x i ),f(x i ))<(Conv(v i ),f(v i )) can be expressed by formula (8);

[0033] In formula (9), ε(0) = ε(x θ ), ε(x θ ) is the θth individual with the best constraint violation in the first generation, cp is the parameter for controlling the level of decline, T c To control the generation parameters, t is the current iteration number;

[0034] Random sorting (SR) is a random-based method to balance fitness values ​​and constraint violations. The main difference between SR and other constraint handling techniques is that the search bias in SR is guided by a probability parameter pf; in this method, the overall sorting follows a probabilistic procedure: two individuals are compared according to their objective function with probability pf, otherwise they are compared using the overall constraint violation. After the population is sorted by SR, the highest-ranked part is selected for recombination and its features are shared with the next generation. (Conv(x i ),f(x i ))<(Conv(v i ),f(v i )) can be expressed by formula (10);

[0035] In the multi-objective constraint processing method, the constraint violation function is regarded as the objective function, and the constraint optimization problem is transformed into a multi-objective optimization problem; i ),f(x i ))<(Conv(v i ),f(v i )) can be expressed by formula (11):

[0036] Further, the step S5 repeats the steps S3-S4 until the termination condition is met, including the following steps:

[0037] Step S51, judging whether the number of iterations t satisfies the termination condition T; if so, obtaining the optimal value in the final population as the final optimization result; otherwise, returning to step S3;

[0038] Beneficial effects of the present invention:

[0039] An evolutionary constrained optimization method based on urgency-aware constraint processing technology is proposed. The present invention is combined with constraint processing technology and added to the heuristic algorithm to better balance the objective function and constraint violation, enhance the diversity and convergence of the algorithm, and effectively identify the most urgent constraints in the current iteration and give priority to satisfying these constraints. Compared with the method of treating all constraints equally, this can push the solution to move inside the feasible domain faster and improve the convergence speed. At the same time, it can also better guide the search towards the optimal solution, thereby improving the accuracy of the final convergence.

[0040] The present invention can pay attention to the most urgent constraints in a targeted manner. This can avoid falling into local optimality and improve the applicability of the algorithm for nonlinear problems. In addition, this method can better cope with changes in constraints in the problem and enhance the robustness of the algorithm by dynamically adjusting the degree of attention to different constraints. As a result, the algorithm can still maintain good performance when facing complex and changeable constraints. Finally, by giving priority to satisfying the most urgent constraints, a solution closer to the center of the feasible domain can be generated, thereby improving the quality of the obtained solution. The present invention combines many advantages, such as improving convergence performance, enhancing the ability to handle complex constraints, improving algorithm robustness, and improving the quality of solutions. The present invention is a very valuable constraint processing method. BRIEF DESCRIPTION OF THE DRAWINGS

[0041] In order to more clearly illustrate the technical solution of the present invention, the following briefly introduces the drawings required for implementation. These drawings constitute a part of the specification. The drawings of the present invention are used together with the solution of the present invention to explain the technical solution of the present invention, but do not constitute a limitation on the technical solution of the present invention.

[0042] Figure 1 It is a schematic diagram of the overall process of the present invention.

[0043] Figure 2 This is the algorithm for calculating constraint urgency weights of the present invention. Specific implementation methods

[0044] The following is combined with Figure 1 , an evolutionary constrained optimization method based on urgency-aware constraint processing technology, specifically comprising the following steps:

[0045] Step S1, initializing the population and constraint urgency weights;

[0046] Step S2, calculating the fitness function and the original constraint violation degree of the population;

[0047] Step S3, obtaining the offspring population according to the crossover, mutation and selection operators, and updating the urgency weight of the population constraint condition;

[0048] Step S4, updating the population individuals according to the constraint processing technology used;

[0049] Step S5, repeat steps S3-S4 until the termination condition is met.

[0050] Among them, step S1 initializing the population and the constraint urgency weight specifically includes:

[0051] Step S11, define the maximum number of iterations T and the population size of the population as NP;

[0052] Step S12: construct an initialization population P = {x1 ,x 2 ,…,x NP}, when using the nature-inspired algorithm to solve the constrained optimization problem, the constrained optimization problem has multiple constraints. The number of constraints for the constrained optimization problem is set to m, and the constraint urgency weight is initialized

[0053] Among them, step S2 calculates the fitness function of the population and the original constraint violation degree specifically includes:

[0054] Step S21: Substitute the population P into the constrained optimization problem to obtain each individual x i The fitness f(x) of (i=1,2,…,NP) i );

[0055] Step S22: The population P can calculate each individual x according to various constraints in the constrained optimization problem. i The constraint violation degree g of the jth constraint condition j (x i )(j=1,2,…,m).

[0056] Among them, step S3 obtains the offspring population according to the mutation, crossover and selection operations, and updates the urgency weight of the population constraint condition specifically including:

[0057] Step S31, execute the loop and initialize the current number of iterations t=1;

[0058] Step S32: Perform mutation, crossover and selection operations on the population P to generate a descendant population Q = {v 1 ,v 2 ,...,v NP};

[0059] Step S33: After substituting the population Q into the constrained optimization problem, we can get the value of each individual v i (i=1,2,…,NP) fitness f(v i ) and each individual v i The constraint violation degree g of the jth (j=1,2,…,m) constraint condition j (v i );

[0060] Step S34: traverse each individual v in the population Q i , according to each individual v i For each constraint violation g j (v i ), find the number a of individuals in population Q that do not satisfy the jth constraint condition j , as shown in formula (1);

[0061] In formula (2), p j (v i ) means that when individual v i When the jth constraint is satisfied, the value is 0. i When the jth constraint is not satisfied, the value is 1;

[0062] Step S35: For the jth (j=1, 2, ..., m) constraint condition, the above formula calculates the number of species with a j Individuals do not satisfy this constraint. j Divided by the total number of individuals in the entire population that do not meet all constraints, the violation ratio of the jth constraint is obtained. This ratio can reflect the relative importance of the jth constraint in the entire optimization problem. The present invention can set the urgency weight of the constraint according to this ratio. Individuals that violate important constraints will be subject to a larger fitness penalty. The urgency weight of each constraint is calculated. j , the specific formula is shown in (3);

[0063] Step S36, the total value of constraint violation of each individual indicates the distance between the current individual and the feasible domain. The smaller the total value of constraint violation, the closer the current individual is to the feasible domain and the optimal solution. In order to make the search direction of the population search algorithm explore more areas with larger constraint violation, the present invention can be achieved by adjusting the constraint violation of each individual. The present invention updates the population Q = {v 1 ,v 2 ,…,v NP} individual v i Each constraint violation degree Conv(v i ), the specific formula is shown in (4);

[0064] Step S37, update population p = {x 1 ,x 2 ,…,x NP} individual x i Each constraint violation Conv(x i ), the specific formula is shown in (5);

[0065] Wherein, step S4 of updating the population individuals according to the constraint processing technology used specifically includes:

[0066] Step S41: According to the selected constraint processing technology, all individuals in population P and population Q are compared. If (Conv(xi ),f(x i ))<(Conv(v i ),f(v i ))(i.e. individual v in Q i Compared with the individual x in the original population P i Better), then replace the original individual x of P i , otherwise keep individual x from population P i ; Optional constraint processing techniques include penalty function, feasibility rule, ε constraint method, random sorting method and multi-objective method;

[0067] The penalty function method introduces a penalty term σ to penalize the violation of the constraint, which converts the constrained optimization method into an unconstrained term, (Conv(x i ),f(x i ))<(Conv(v i ),f(v i )) can be expressed by formula (6);

[0068] The feasibility rule-based method is a popular constraint processing technique. The feasibility rule adopts three feasibility criteria: (1) for two feasible solutions, the individual with smaller fitness is given priority; (2) for a feasible individual and an infeasible individual, the feasible individual is given priority; (3) for two infeasible individuals, the individual with smaller constraint violation is given priority. According to the three feasibility criteria, we can define (Conv(x i ),f(x i ))<(Conv(v i ),f(v i ))As shown in formula (7);

[0069] The ε constraint method uses the ε parameter to control the tolerance level of the relaxation of the constraint, indicating that the individual is considered feasible at this level; the ε level can be used to explore the search space well in the early stage; in the later stage, in order to obtain feasible individuals, the level will be set to 0 at a certain point in the evolution; the ε constraint method (Conv(x i ),f(x i ))<(Conv(v i ),f(v i )) can be expressed by formula (8);

[0070] In formula (9), ε(0) = ε(x θ ), ε(x θ) is the θth individual with the best constraint violation in the first generation, cp is the parameter for controlling the level of decline, T c To control the generation parameters, t is the current iteration number;

[0071] Random sorting (SR) is a random-based method to balance fitness values ​​and constraint violations. The main difference between SR and other constraint handling techniques is that the search bias in SR is guided by a probability parameter pf; in this method, the overall sorting follows a probabilistic procedure, and two individuals are compared according to their objective function with probability pf, otherwise they are compared using the overall constraint violation. The population is sorted by SR and the highest-ranked part is selected for recombination, thereby sharing its features with the next generation. (Conv(x i ),f(x i ))<(Conv(v i ),f(v i )) can be expressed by formula (10);

[0072] In the multi-objective constraint processing method, the constraint violation function is regarded as the objective function, and the constraint optimization problem is transformed into a multi-objective optimization problem; i ),f(x i ))<(Conv(v i ),f(v i )) can be expressed by formula (11);

[0073] Wherein, S5 repeats steps S3-S4 until the termination condition is met, which specifically includes:

[0074] Step S51, determine whether the number of iterations t satisfies the termination condition T. If so, obtain the optimal value in the final population as the final optimization result; otherwise, return to step S3.

[0075] It can be seen from the above technical solutions that the present invention has the following advantages: the evolutionary constrained optimization method based on urgency-aware constraint processing technology can significantly improve the convergence speed and accuracy of the algorithm by dynamically identifying and preferentially satisfying the most urgent constraints at present when constraining optimization problems. Compared with the traditional method of treating all constraints equally, it can push the solution into the feasible domain faster and better guide the search towards the optimal solution. At the same time, this method is also very applicable to processing complex nonlinear constraints, and can focus on the most critical constraints in a targeted manner to avoid falling into suboptimal areas. In addition, it can also improve the robustness of the algorithm by dynamically adjusting the attention paid to different constraints, so that it can still maintain stable performance when facing complex and variable constraints. Finally, preferentially satisfying the most urgent constraints can generate high-quality solutions that are closer to the center of the feasible domain, thereby significantly improving the quality of the final solution. In summary, the evolutionary constrained optimization method based on urgency-aware constraint processing technology is a very valuable constraint optimization technology, which is worthy of further research and promotion in practical applications.

Claims

1. An evolutionary constrained optimization method based on urgency-aware constraint processing technology, characterized in that: The following steps are involved: Step S1, initializing the population and constraint urgency weights; Step S2, calculating the fitness function and the original constraint violation degree of the population; Step S3, obtaining a progeny population according to mutation, crossover and selection operations, and updating the urgency weight of the population constraint condition; Step S4, updating the population individuals according to the constraint processing technology used; Step S5, repeat steps S3-S4 until the termination condition is met.

2. The evolutionary constrained optimization method based on urgency-aware constraint processing technology according to claim 1 is characterized in that: The step S1 of initializing the population and the constraint urgency weights includes: Step S11, define the maximum number of iterations T and the population size of the population as NP; Step S12: construct an initialization population P = {x1, x2, ..., x NP }, when using the nature-inspired algorithm to solve the constrained optimization problem, the constrained optimization problem has multiple constraints. The number of constraints for the constrained optimization problem is set to m, and the constraint urgency weight is initialized 3. The evolutionary constrained optimization method based on urgency-aware constraint processing technology according to claim 1 is characterized in that: The step S2 of calculating the fitness function of the population and the original constraint violation degree includes: Step S21: Substitute the population P into the constrained optimization problem to obtain each individual x i The fitness f(x) of (i=1,2,...,NP) i ); Step S22: The population P can calculate each individual x according to various constraints in the constrained optimization problem. i The constraint violation degree g of the jth (j=1,2,...,m) constraint condition j (x i ).

4. The evolutionary constrained optimization method based on urgency-aware constraint processing technology according to claim 1 is characterized in that: The step S3 obtains the offspring population according to the mutation, crossover and selection operations, and updates the urgency weight of the population constraint condition, including: Step S31, execute the loop and initialize the current number of iterations t=1; Step S32: Perform mutation, crossover and selection operations on the population P to generate a descendant population Q = {v1, v2, ..., v NP }; Step S33: After substituting the population Q into the constrained optimization problem, we can get the value of each individual v i The fitness f(v i ) and each individual v i The constraint violation degree g of the jth (j=1,2,...,m) constraint condition j (v i ); Step S34: traverse each individual v in the population Q i , according to each individual v i The constraint violation degree g j (v i ), find the individual v in population Q that does not satisfy the jth constraint condition i The number of a j , as shown in formula (1); In formula (2), p j (v i ) means that when individual v i When the jth constraint is satisfied, the value is 0. i When the jth constraint is not satisfied, the value is 1; Step S35: For the jth (j=1, 2, ..., m) constraint, the above formula calculates the number of species with a j Individuals do not satisfy this constraint. j Dividing by the total number of individuals in the entire population that do not meet all constraints, we get the proportion of violation of the jth constraint, which can reflect the relative importance of the jth constraint in the entire optimization problem; the present invention can set the urgency weight of the constraint according to this proportion, and individuals that violate important constraints will be subject to a larger fitness penalty. The urgency weight of each constraint is calculated. j , the specific formula is shown in (3); Step S36, the total value of constraint violation of each individual indicates the distance between the current individual and the feasible domain. The smaller the total value of constraint violation, the closer the current individual is to the feasible domain and the optimal solution. In order to make the search direction of the population search algorithm explore more areas with larger constraint violation, the present invention can be achieved by adjusting the constraint violation of each individual. The present invention updates the population Q = {v1, v2, ..., v NP } individual v i Each constraint violation degree Conv(v i ), the specific formula is shown in (4); Step S37, update population p = {x1, x2, ..., x NP } individual x i Each constraint violation Conv(x i ), the specific formula is shown in (5); 5. The evolutionary constrained optimization method based on urgency-aware constraint processing technology according to claim 1, characterized in that: The step S4 updates the population individuals according to the constraint processing technology used, including: Step S41: According to the selected constraint processing technology, all individuals in population P and population Q are compared. If individual v in Q i Compared with the individual x in the original population P i Better Then replace the original individual x of P i , otherwise keep individual x from population P i ; Optional constraint processing techniques include penalty function, feasibility rule, ε constraint method, random sorting method and multi-objective method; The penalty function method introduces a penalty term σ to penalize the violation of constraints, which converts the constrained optimization method into an unconstrained term. It can be expressed by formula (6); The feasibility rule-based method is a popular constraint handling technique. The feasibility rule adopts three feasibility criteria: (1) for two feasible solutions, the individual with smaller fitness is given priority; (2) for a feasible individual and an infeasible individual, the feasible individual is given priority; (3) for two infeasible individuals, the individual with smaller constraint violation is given priority. According to the three feasibility criteria, we can define As shown in formula (7); The ε constraint method uses the ε parameter to control the tolerance level of the relaxation of the constraint, indicating that individuals at this level are considered feasible; in the early stage, the ε level can be used to explore the search space well; in the later stage, in order to obtain feasible individuals, the level will be set to 0 at a certain point in the evolution; the ε constraint method It can be expressed by formula (8); In formula (9), ε(0) = ε(x θ ), ε(x θ ) is the θth individual with the best constraint violation in the first generation, cp is the parameter for controlling the level of decline, T c To control the generation parameters, t is the current iteration number; Random sorting (SR) is a random-based method to balance fitness values ​​and constraint violations. The main difference between SR and other constraint handling techniques is that the search bias in SR is guided by a probability parameter pf; in this method, the overall sorting follows a probabilistic procedure: two individuals are compared according to their objective function with probability pf, otherwise they are compared using the overall constraint violation. After the population is sorted by SR, the highest-ranked part is selected for recombination, thereby sharing its characteristics with the next generation. It can be expressed by formula (10); In the multi-objective constraint processing method, the constraint violation function is regarded as the objective function, and the constraint optimization problem is transformed into a multi-objective optimization problem. It can be expressed by formula (11); 6. The evolutionary constrained optimization method based on urgency-aware constraint processing technology according to claim 1, characterized in that: The step S5 repeats steps S3-S4 until the termination condition is met, including: Step S51, determine whether the number of iterations t satisfies the termination condition T. If so, obtain the optimal value in the final population as the final optimization result; otherwise, return to step S3.

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