Calculation method and system for solving large-scale multi-objective optimization problem

Abstract

The invention relates to the technical field of computer algorithms, in particular to a calculation method and system for solving a large-scale multi-objective optimization problem. According to the technical scheme, the method comprises the following steps that an initialized population meeting the problem solving requirement is generated according to input parameters, variables to be optimized of each individual in the population are anchor variables and anchor point variables instead of variable data needed for solving the problem, and the number of the variables is greatly smaller than the number of the variable data needed for solving the problem; when the problem of large-scale multi-objective optimization is solved, variable complex correlation can be better coped, optimization efficiency is improved, optimization precision and difficulty are flexibly controlled, visual and effective result display is provided, and the method has remarkable advantages in multiple aspects.

Description

Technical Field

[0001] The present invention relates to the field of computer algorithm technology, and in particular to a computing method and system for solving large-scale multi-objective optimization problems. Background Art

[0002] Multi-objective optimization problems are a type of optimization problem that is common in scientific research and engineering applications. This type of problem requires taking into account multiple optimization objectives at the same time, and requires the algorithm to be able to provide a set of optional solutions that can make different trade-offs among multiple optimization objectives. For example, in the design of automotive parts, we need to consider not only the structural strength of the parts, but also multiple objectives such as the process difficulty and manufacturing cost. We need to make decisions based on the preference for different objectives based on specific needs. When the specific needs are not yet determined, we require the algorithm to provide a set of compromise solutions that can cover potential needs, so as to help make quick decisions after the specific needs are determined.

[0003] The multi-objective optimization system process mainly includes three components: evolutionary operator, problem evaluation, and environmental selection. The evolutionary operator usually uses a genetic algorithm or a differential evolution algorithm to generate a new offspring population based on the current population. The problem evaluation is to calculate the value obtained by each individual in the population on each target through the constructed objective function. The environmental selection is to evaluate the value obtained by the individual on the target and delete some solutions with poor quality. After the environmental selection, the remaining population will generate a new offspring population through the evolutionary operator again and enter a new cycle until the number of cycles reaches the preset requirement. After the cycle ends, the remaining population is the solution that meets our requirements.

[0004] Generally speaking, when the number of variables to be optimized is less than or equal to 100, we call it a general multi-objective optimization problem, and when the number of variables to be optimized exceeds 100, we call it a large-scale multi-objective optimization problem. Compared with general multi-objective optimization problems, the optimization difficulty of large-scale multi-objective optimization problems is significantly increased. Since large-scale multi-objective optimization problems need to optimize a very large number of variables at the same time, their search space is almost infinite. It will become extremely difficult to find the correct evolutionary direction in such a vast search space. The consequence of not being able to find the correct evolutionary direction is that the evolution stagnates and cannot further converge to the direction with a better target value.

[0005] Currently, there is a solution to this problem based on decision variable analysis. This solution is to observe the correlation between the variables to be optimized and their target values ​​one by one, divide the variables to be optimized into convergence-related variables and diversity-related variables, and then optimize them separately. After the variables are classified, their evolutionary direction can be roughly determined. Therefore, this method can clarify the optimization direction for evolution and reduce the difficulty of convergence.

[0006] The main problems with this method are:

[0007] 1. In many problems, there is no obvious linear correlation between the variables to be optimized and the target values ​​obtained. Therefore, when faced with problems with complex correlations between some variables, it is difficult to correctly classify the variables, resulting in a serious decline in performance.

[0008] 2. The method of observing the variables to be optimized one by one requires a very large number of problem evaluations to obtain relatively satisfactory results, and the number of evaluations required increases with the number of variables to be optimized. This means that when solving problems with a very large number of variables to be optimized (more than 1,000), the optimization efficiency of this method will drop sharply.

[0009] Therefore, the present application proposes a computational method and system for solving large-scale multi-objective optimization problems. Summary of the invention

[0010] The purpose of the present invention is to propose a computational method and system for solving large-scale multi-objective optimization problems in order to address the problem in the background technology that when faced with problems with complex correlations between some variables, it is difficult to correctly classify the variables, resulting in a serious degradation in performance.

[0011] On the one hand, the present application provides a computational method for solving large-scale multi-objective optimization problems, comprising the following steps:

[0012] Generate an initialized population that meets the requirements of the problem according to the input parameters, where the variables to be optimized for each individual in the population are anchor variables and anchor point variables rather than variable data required to solve the problem, and the number of variables is much less than the number of variable data required to solve the problem;

[0013] Adopting the evolutionary strategy based on genetic operators, a new offspring population is generated based on the anchor variables and anchor point variables of the current population. The offspring population also only contains the anchor variables and anchor point variables, and does not contain the variable data for solving the problem.

[0014] According to the anchor variable and anchor point variable of each individual in the population, the complete variable data of each individual in the population is calculated based on the selected interpolation method, and then evaluated by the problem function to obtain the data of each individual on each target;

[0015] According to the non-dominated sorting method, the convergence of each individual in the population is evaluated according to the data of each individual on each target, and some individuals with poor convergence are deleted;

[0016] According to the data of the remaining population on each target after non-dominated selection, a normalized space is established, and the diversity of each individual in the population is evaluated according to the Euclidean distance between the reference vector and each individual in the population, and some individuals with poor diversity are deleted;

[0017] The variable data and target value data of the population obtained after the optimization is completed are output, and the optimization effect is displayed in an intuitive way.

[0018] On the other hand, the present application provides a computing system for solving large-scale multi-objective optimization problems, comprising:

[0019] The initialization module generates an initialization population that meets the requirements of the problem according to the input parameters, where the variables to be optimized for each individual in the population are anchor variables and anchor point variables rather than the variable data required to solve the problem, and the number of variables is much less than the number of variable data required to solve the problem;

[0020] The offspring generation module uses an evolutionary strategy based on genetic operators to generate a new offspring population based on the anchor variables and anchor point variables of the current population. The offspring population also only contains anchor variables and anchor point variables, and does not contain variable data for solving the problem;

[0021] The interpolation module calculates the complete variable data of each individual in the population based on the selected interpolation method according to the anchor variable and anchor point variable of each individual in the population, and then evaluates it through the problem function to obtain the data of each individual on each target;

[0022] The non-dominated selection module evaluates the convergence of each individual in the population according to the data of each individual on each target, and deletes some individuals with poor convergence;

[0023] The environmental selection module establishes a normalized space based on the data of the remaining population on each target after non-dominated selection, evaluates the diversity of each individual in the population based on the Euclidean distance between the reference vector and each individual in the population, and deletes some individuals with poor diversity;

[0024] The result output module outputs the variable data and target value data of the population obtained after the optimization is completed, and displays the optimization effect in an intuitive way.

[0025] Optionally, the variables to be optimized in the initialization population generated by the initialization module are anchor variables and anchor point variables, rather than variable data required to solve the problem, wherein the number of anchor variables and anchor point variables can be defined by the user, and 3-15 are recommended, which is much less than the variable data required to solve the problem.

[0026] Optionally, a new offspring population is generated based on the anchor variables and anchor point variables of the current population using genetic operators. The offspring population only contains the anchor variables and anchor point variables and does not contain variable data for solving the problem. This greatly reduces the number of variables that need to be optimized compared to traditional methods.

[0027] Optionally, the interpolation module will generate complete variable data required to solve the problem based on the anchor variables and anchor point variables of each individual in the current population using an interpolation method, and calculate the values ​​obtained by each individual on each target through the problem function.

[0028] Optionally, the anchor variables are all floating point numbers in the range of 0-1, the first digit of the anchor variable is 1, the last digit is the number of variables in the problem, and the rest are real numbers greater than 1 and less than the number of variables in the problem, and are arranged in ascending order.

[0029] Optionally, the interpolation methods used in the interpolation module include linear interpolation, nearest neighbor interpolation, and spline curve interpolation, which are defined according to user needs; the number of anchor variables and anchor point variables used is defined according to user needs to control optimization accuracy and optimization difficulty.

[0030] Optionally, let the anchor variables of the individuals be v = v1, v2, …, v s , the anchor variable is p=p1,p2,…,p s , then the calculation formula for the complete variable data is:

[0031]

[0032] Among them, x k Represents the kth data in the complete variable data, and the value of k is between p i and p i+1 between.

[0033] Optionally, let the anchor variables of the individuals be v = v1, v2, …, v s , the anchor variable is p=p1,p2,…,p s , then the calculation formula for the complete variable data is:

[0034]

[0035] Among them, x k Represents the kth data in the complete variable data, and the value of k is between p i and p i+1 between

[0036] Optionally, let the anchor variables of the individuals be v = v1, v2, …, v s , the anchor variable is p=p1,p2,…,p s , then the calculation formula for the complete variable data is:

[0037] x k =v i +b i (kp i )+c i (kp i ) 2 +d i (kp i ) 3

[0038] Among them, b i , c i , d i The value of can be obtained through the spline curve interpolation formula, x k Represents the kth data in the complete variable data, and the value of k is between p i and p i+1 between.

[0039] Compared with the prior art, the present invention has at least one of the following beneficial technical effects:

[0040] Anchor variables and anchor point variables are used to indirectly deal with problems, without relying on direct analysis of the correlation between the variables to be optimized and the target values ​​for classification. This avoids the problem of inaccurate classification due to complex correlations, thereby ensuring that the entire optimization process can be carried out stably and effectively, and improving adaptability to various complex situations.

[0041] When initializing the population and subsequently generating the offspring population, the process revolves around anchor variables and anchor point variables, which are far fewer in number than the variable data actually required to solve the problem. For example, the number of anchor variables and anchor point variables is usually only 3-15. Compared with traditional methods, this greatly reduces the scale of variables that need to be processed during operations such as evolution, reduces the amount of calculation and optimization difficulty, avoids the problem of evolutionary stagnation caused by too many dependent variables, and significantly improves optimization efficiency.

[0042] The interpolation method is used to obtain complete variable data based on anchor variables and anchor point variables. There is no need to evaluate the actual variables one by one in large quantities like the traditional method, which effectively controls the number of problem evaluations. This allows the optimization efficiency to be maintained relatively high when dealing with large-scale multi-objective optimization problems.

[0043] When solving large-scale multi-objective optimization problems, the present invention can better cope with complex correlations of variables, improve optimization efficiency, flexibly control optimization accuracy and difficulty, and provide intuitive and effective result display, and has significant advantages in many aspects. BRIEF DESCRIPTION OF THE DRAWINGS

[0044] Figure 1 is a flowchart of a computational method for solving large-scale multi-objective optimization problems;

[0045] Figure 2 This is the final optimization target value data display chart when the number of targets is 2;

[0046] Figure 3 This is the final optimization target value data display chart when the number of targets is 3;

[0047] Figure 4 The figure below shows the final data display of the variables to be optimized when the number of variables to be optimized is 1000. DETAILED DESCRIPTION

[0048] The technical solution of the present invention is further described below in conjunction with the accompanying drawings and specific embodiments.

[0049] Example 1

[0050] like Figure 1 As shown, the present invention provides a computational method for solving large-scale multi-objective optimization problems, comprising the following steps:

[0051] Generate an initialized population that meets the requirements of the problem according to the input parameters, where the variables to be optimized for each individual in the population are anchor variables and anchor point variables rather than variable data required to solve the problem, and the number of variables is much less than the number of variable data required to solve the problem;

[0052] Adopting the evolutionary strategy based on genetic operators, a new offspring population is generated based on the anchor variables and anchor point variables of the current population. The offspring population also only contains the anchor variables and anchor point variables, and does not contain the variable data for solving the problem.

[0053] According to the anchor variable and anchor point variable of each individual in the population, the complete variable data of each individual in the population is calculated based on the selected interpolation method, and then evaluated by the problem function to obtain the data of each individual on each target;

[0054] According to the non-dominated sorting method, the convergence of each individual in the population is evaluated according to the data of each individual on each target, and some individuals with poor convergence are deleted;

[0055] According to the data of the remaining population on each target after non-dominated selection, a normalized space is established, and the diversity of each individual in the population is evaluated according to the Euclidean distance between the reference vector and each individual in the population, and some individuals with poor diversity are deleted;

[0056] The variable data and target value data of the population obtained after the optimization is completed are output, and the optimization effect is displayed in an intuitive way.

[0057] Example 2

[0058] The present invention discloses a computing system for solving large-scale multi-objective optimization problems, which is composed of an initialization module, a child generation module, an interpolation module, an environment selection module and a result output module.

[0059] The input requirements of this system are: population size, number of variables to be optimized, number of targets, number of anchor points, problem function, number of iterations, and interpolation function code. After the input is completed, the system will first enter the initialization module. In the initialization module, the system will perform the following operations in sequence:

[0060] 1. According to the number of anchor points and population of the input problem function, randomly generate anchor variables of the population, the number of which is the same as the number of input anchor points, the upper limit of the anchor variable is uniformly set to 1, and the lower limit is uniformly set to zero;

[0061] 2. According to the number of anchor points and population of the input problem function, randomly generate anchor point variables of the population. The number is the same as the number of input anchor points. The first digit of the anchor point variable is 1, and the last digit is the number of variables to be optimized. The remaining anchor point variables are real numbers greater than 1 and less than the number of variables to be optimized, and are arranged in ascending order.

[0062] After the initialization is completed, the next generation module is entered. In the next generation module, the system will perform the following operations in sequence:

[0063] 1. Randomly select two individuals;

[0064] 2. Generate offspring of the anchor variables of the two individuals using a genetic algorithm;

[0065] 3. Generate offspring of the anchor variables of the two individuals using a genetic algorithm;

[0066] 4. Rearrange the anchor variables in ascending order;

[0067] 5. Set the first position of the anchor variable to 1 and the last position to the number of variables to be optimized;

[0068] 6. Repeat the above steps until all individuals are selected;

[0069] 6. Merge the generated offspring population with the population retained in the previous iteration to form a new population.

[0070] After entering the interpolation module, the system will perform the following operations in sequence:

[0071] 1. Select the interpolation formula according to the input interpolation function code. If the input of the interpolation function code is 1 or no input, linear interpolation is used. If the input of the interpolation function code is 2, nearest neighbor interpolation is used. If the input of the interpolation function code is 3, spline curve interpolation is used.

[0072] 2. Use the selected interpolation function to generate the complete variables to be optimized based on the values ​​of each individual's anchor variable and anchor point variable. If linear interpolation is used, then according to the formula

[0073]

[0074] Calculate each variable to be optimized, where v i , v i+1 are the i-th and i+1-th anchor variables of the individual, respectively, p i , p i+1 are the i-th and i+1-th anchor variables of the individual, respectively, x k Represents the kth data in the complete variable data, and the value of k is between p i and p i+1 If the nearest neighbor interpolation is used, then according to the formula

[0075]

[0076] Calculate each variable to be optimized, where v i , v i+1 are the i-th and i+1-th anchor variables of the individual, respectively, p i , p i+1 are the i-th and i+1-th anchor variables of the individual, respectively, x k Represents the kth data in the complete variable data, and the value of k is between p i and p i+1 If spline interpolation is used, first establish a cubic polynomial equation

[0077] x k =v i +b i (kp i )+c i (kp i )2 +d i (kp i ) 3

[0078] For each pair of anchor variable and anchor point variable, we have

[0079] P i (p i )=v i

[0080] as well as

[0081] P i (p i+1 )=v i+1

[0082] The constraints of the spline curve require P i (p i+1 ) is similar to P in slope and curvature i+1 (p i+1 ) remain consistent, that is

[0083] P' i (p i+1 )=P' i+1 (p i+1 )

[0084] as well as

[0085] P” i (p i+1 )=P” i+1 (p i+1 )

[0086] Considering that the curvature of the two endpoints of the curve should be 0, substituting the above formula into the cubic polynomial equation, we can get

[0087] P i (p) = a i +b i (pp i )+b i (pp i ) 2 +b i (pp i ) 3 =v i

[0088] Considering the existence condition p = p i , can be obtained

[0089] a i =v i

[0090] Then assume that h i =pi+1 -p i , due to P i (p i )=v i , can be obtained

[0091]

[0092] Substitute into the formula P' i (p i+1 )=P' i+1 (p i+1 ), we can get

[0093]

[0094] as well as

[0095] P' i+1 (p i+1 )=b i+1 +2c i (p i+1 -p i+1 )+3d i (p i+1 -p i+1 ) 2 =b i+1

[0096] So we can get

[0097]

[0098] Substituting it into the formula P' i (p i+1 )=P' i+1 (p i+1 ), we can get

[0099] 2c i +6h i d i =2c i+1

[0100] Assume m i =P" i (x i )=2c i , can be obtained

[0101]

[0102] Substitute into the formula P i (p) = a i +b i (pp i )+b i (pp i ) 2 +bi (pp i ) 3 =v i , can be obtained

[0103]

[0104] Substitute into the formula Available

[0105]

[0106] Then we can construct the linear equation

[0107]

[0108] This equation has a unique solution when every m i After the calculation is completed, the coefficients in the spline curve interpolation formula can be obtained.

[0109] 3. Substitute the obtained variables to be optimized into the problem function to obtain the values ​​obtained by each individual on each target;

[0110] After entering the non-dominated selection module, the system will perform the following operations in sequence:

[0111] 1. Use the non-dominated sorting method to sort each individual in the population according to its target value;

[0112] 2. Count the number of individuals in each non-dominated frontier from low to high until the total number of individuals counted is greater than the input population size

[0113] 3. Delete the individuals that are not included in the statistics from the population;

[0114] After entering the environment selection module, the system will perform the following operations in sequence:

[0115] 1. Recalculate the maximum point of the target space based on the target value of each individual in the population;

[0116] 2. Establish a normalized space based on the maximum and minimum points of the target space;

[0117] 3. Normalize the target values ​​of all individuals in the population;

[0118] 4. Establish reference vector in normalized space

[0119] 5. Calculate the Euclidean distance between each individual in the population and each reference vector;

[0120] 6. Aggregate each individual with its closest reference vector;

[0121] 7. Find the reference vector with the most aggregated individuals and delete the solution farthest from the reference vector.

[0122] 8. Repeat step 7 until the number of remaining individuals is equal to the input population size

[0123] 9. Output a new generation of population.

[0124] After that, the system will determine whether the current number of iterations has reached the maximum number of iterations. If not, the system will return to the offspring generation module and use the current population as the parent of the offspring for a new cycle until the current number of iterations reaches the maximum number of iterations. After reaching the maximum number of iterations, the system enters the result output module and performs the following operations in sequence:

[0125] 1. Draw the target value data of the output population;

[0126] 2. Save the variable data of the final output population in the dec.mat file;

[0127] 3. Save the target value data of the final output population in the obj.mat file;

[0128] 4. Exit the system program.

[0129] In this embodiment, the module is based on the non-dominated sorting technology and the environment selection technology of NSGA-III, which can effectively take into account multiple optimization objectives and solve the problem; the generated image displays the values ​​of the individuals obtained on each objective in an intuitive way, providing assistance for users to make decisions.

[0130] Anchor variables and anchor point variables are used to indirectly deal with problems, without relying on direct analysis of the correlation between the variables to be optimized and the target values ​​for classification. This avoids the problem of inaccurate classification due to complex correlations, thereby ensuring that the entire optimization process can be carried out stably and effectively, and improving adaptability to various complex situations.

[0131] When initializing the population and subsequently generating the offspring population, the process revolves around anchor variables and anchor point variables, which are far fewer in number than the variable data actually required to solve the problem. For example, the number of anchor variables and anchor point variables is usually only 3-15. Compared with traditional methods, this greatly reduces the scale of variables that need to be processed during operations such as evolution, reduces the amount of calculation and optimization difficulty, avoids problems such as evolutionary stagnation caused by too many dependent variables, and significantly improves optimization efficiency.

[0132] When faced with problems with a large number of variables (more than 1,000), existing methods use the method of observing the variables to be optimized one by one, which requires a large number of problem evaluation times, and the number of times increases sharply with the increase of variables, and the optimization efficiency decreases sharply. However, this solution uses interpolation to obtain complete variable data based on anchor variables and anchor point variables. It does not need to evaluate the actual variables one by one in large quantities like traditional methods, and effectively controls the number of problem evaluations, so that when dealing with large-scale multi-objective optimization problems, it can still maintain a relatively high optimization efficiency.

[0133] The interpolation methods used in the solution (such as linear interpolation, nearest neighbor interpolation, spline interpolation, etc.) and the number of anchor variables and anchor point variables can be defined according to user needs. Users can flexibly select and adjust these parameters based on the characteristics of the specific problem being solved, the expected optimization accuracy, and the acceptable optimization difficulty, so that the optimization process can not only meet the expected accuracy requirements, but also be carried out efficiently within the acceptable difficulty range, enhancing the applicability of the solution to different application scenarios.

[0134] Results presentation and practicality.

[0135] It has a result output module that can output the variable data and target value data of the population obtained after the optimization is completed, and display the optimization effect in an intuitive way. This is convenient for users to directly view and understand the optimization results and quickly make corresponding decisions based on the results. Whether it is analyzing the optimization results in scientific research scenarios or implementing specific plans based on the optimization results in engineering applications, it is more practical and operational.

[0136] When solving large-scale multi-objective optimization problems, the present invention can better cope with complex correlations of variables, improve optimization efficiency, flexibly control optimization accuracy and difficulty, and provide intuitive and effective result display, and has significant advantages in many aspects.

[0137] Calculation method and model code

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[0162] The above specific embodiments are only several optional embodiments of the present invention. Based on the technical solutions of the present invention and the relevant inspirations of the above embodiments, those skilled in the art can make various alternative improvements and combinations to the above specific embodiments.

Claims

1. A computational method for solving large-scale multi-objective optimization problems, characterized in that: The following steps are involved: Generate an initialized population that meets the requirements of the problem according to the input parameters, where the variables to be optimized for each individual in the population are anchor variables and anchor point variables rather than variable data required to solve the problem, and the number of variables is much less than the number of variable data required to solve the problem; Adopting the evolutionary strategy based on genetic operators, a new offspring population is generated based on the anchor variables and anchor point variables of the current population. The offspring population also only contains the anchor variables and anchor point variables, and does not contain the variable data for solving the problem. According to the anchor variable and anchor point variable of each individual in the population, the complete variable data of each individual in the population is calculated based on the selected interpolation method, and then evaluated by the problem function to obtain the data of each individual on each target; According to the non-dominated sorting method, the convergence of each individual in the population is evaluated according to the data of each individual on each target, and some individuals with poor convergence are deleted; According to the data of the remaining population on each target after non-dominated selection, a normalized space is established, and the diversity of each individual in the population is evaluated according to the Euclidean distance between the reference vector and each individual in the population, and some individuals with poor diversity are deleted; The variable data and target value data of the population obtained after the optimization is completed are output, and the optimization effect is displayed in an intuitive way.

2. A computing system for solving large-scale multi-objective optimization problems, characterized in that: include: The initialization module generates an initialization population that meets the requirements of the problem according to the input parameters, where the variables to be optimized for each individual in the population are anchor variables and anchor point variables rather than the variable data required to solve the problem, and the number of variables is much less than the number of variable data required to solve the problem; The offspring generation module uses an evolutionary strategy based on genetic operators to generate a new offspring population based on the anchor variables and anchor point variables of the current population. The offspring population also only contains anchor variables and anchor point variables, and does not contain variable data for solving the problem; The interpolation module calculates the complete variable data of each individual in the population based on the selected interpolation method according to the anchor variable and anchor point variable of each individual in the population, and then evaluates it through the problem function to obtain the data of each individual on each target; The non-dominated selection module evaluates the convergence of each individual in the population according to the data of each individual on each target, and deletes some individuals with poor convergence; The environmental selection module establishes a normalized space based on the data of the remaining population on each target after non-dominated selection, evaluates the diversity of each individual in the population based on the Euclidean distance between the reference vector and each individual in the population, and deletes some individuals with poor diversity; The result output module outputs the variable data and target value data of the population obtained after the optimization is completed, and displays the optimization effect in an intuitive way.

3. A computing system for solving large-scale multi-objective optimization problems according to claim 2, characterized in that: The variables to be optimized in the initialization population generated by the initialization module are anchor variables and anchor point variables, wherein the number of anchor variables and anchor point variables can be defined by the user, and 3-15 are recommended.

4. A computing system for solving large-scale multi-objective optimization problems according to claim 2, characterized in that: Genetic operators are used to generate new offspring populations based on the anchor variables and anchor point variables of the current population. The offspring populations only contain the anchor variables and anchor point variables.

5. A computing system for solving large-scale multi-objective optimization problems according to claim 2, characterized in that: The interpolation module will generate complete variable data required to solve the problem based on the anchor variables and anchor point variables of each individual in the current population using the interpolation method, and calculate the values ​​obtained by each individual on each target through the problem function.

6. A computing system for solving large-scale multi-objective optimization problems according to claim 2, characterized in that: The anchor variables are all floating-point numbers ranging from 0 to 1. The first digit of the anchor variable is 1, the last digit is the number of variables in the problem, and the rest are real numbers greater than 1 and less than the number of variables in the problem, and are arranged in ascending order.

7. A computing system for solving large-scale multi-objective optimization problems according to claim 2, characterized in that: The interpolation methods used in the interpolation module include linear interpolation, nearest neighbor interpolation, and spline curve interpolation, which are defined according to user needs; the number of anchor variables and anchor point variables used is defined according to user needs to control optimization accuracy and optimization difficulty.

8. The linear interpolation method used in the interpolation module according to claim 7, characterized in that: Assume that the individual anchor variable is v = v1, v2, ..., v s , the anchor variable is p=p1,p2,…,p s , then the calculation formula for the complete variable data is: Among them, x k Represents the kth data in the complete variable data, and the value of k is between p i and p i+1 between.

9. The nearest neighbor interpolation method used in the interpolation module according to claim 7, characterized in that: Assume that the individual anchor variable is v = v1, v2, ..., v s , the anchor variable is p=p1,p2,…,p s , then the calculation formula for the complete variable data is: Among them, x k Represents the kth data in the complete variable data, and the value of k is between p i and p i+1 between.

10. The spline curve interpolation method used in the interpolation module according to claim 2, characterized in that: Assume that the individual anchor variable is v = v1, v2, ..., v s , the anchor variable is p=p1,p2,…,p s , then the calculation formula for the complete variable data is: x k =v i +b i( k-p i) +c i( k-p i) 2 +d i( k-p i) 3 Among them, b i , c i , d i The value of can be obtained through the spline curve interpolation formula, x k Represents the kth data in the complete variable data, and the value of k is between p i and p i+1 between.

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