A partially regenerated expensive large scale optimization method

Abstract

The present application relates to single-target large-scale expensive optimization technical field, especially to a kind of partial regeneration expensive large-scale optimization method, specific steps are as follows: first, two-stage feature selection strategy is used to reduce dimension to original large-scale optimization problem, and the important dimension extracted is used as low-dimensional subproblem, to ensure that dimension reduction processing will concentrate computing resources on important dimension, to accelerate search process and enhance algorithm convergence;Second, RBF auxiliary improved social learning particle swarm optimization algorithm is applied to generate offspring of subpopulation, and new subpopulation is formed by environmental selection and the best solution of subproblem is determined, to enhance the convergence of algorithm on subproblem, improve the convergence speed of algorithm;Meanwhile, the new subpopulation is used to update population pop;Finally, a new filling sampling strategy is provided to update optimal solution and archive Arc.In addition, the present application also provides a computer readable storage medium and computer equipment.

Description

Technical Field

[0001] This invention relates to the field of single-objective large-scale expensive optimization technology, and more particularly to an expensive large-scale optimization method with partial regeneration. Background Technology

[0002] With the development of industry and technology, optimization problems with many variables are frequently encountered in the real world, such as shop floor scheduling, casting sequence scheduling, integer linear programming, and route networks. Generally, the mathematical model for minimization problems is as follows:

[0003] min f(X)

[0004]

[0005] In the formula, Let X = (x1, x2, ..., xn) represent the search space. n ) T Let n be the number of decision variables. When n ≥ 100, it is called a Large-Scale Optimization Problem (LSOP).

[0006] Common methods for solving large-scale problems can be divided into two categories: decomposition-based large-scale optimization methods, which employ a "divide and conquer" strategy to decompose the original large-scale optimization problem into two or more subproblems that are solved independently and then combined to form the solution to the original problem; and non-decomposition-based large-scale optimization methods, which employ innovative optimization strategies to generate new solutions in order to better balance the exploration and development of large-scale optimization problems.

[0007] However, some large-scale optimization problems in practical applications are black boxes, with high computational costs and economic benefits. These problems are called large-scale expensive optimization problems, such as aerodynamic wing design, blast furnace optimization, and crashworthiness analysis in the automotive industry. Most existing algorithms for solving large-scale optimization problems cannot be directly applied to solving expensive optimization problems. On the one hand, in solving expensive optimization problems, to reduce the evaluation of the objective function and save computational costs, the common practice is to use a computationally inexpensive surrogate model to approximate the complex and expensive objective function. Commonly used surrogate models include Support Vector Machines (SVM), Multinomial Regression (PR), Gaussian Processes (GP), Artificial Neural Networks (ANN), and Radial Basis Functions (RBF) networks. However, as the problem dimension increases, more training data is needed to build an accurate surrogate model. Therefore, building an accurate surrogate model for large-scale optimization problems is extremely challenging. This is also one of the difficulties faced by non-decomposition optimization algorithms when solving expensive large-scale optimization problems. On the other hand, the performance of decomposition-based large-scale optimization algorithms largely depends on the choice of grouping strategy. However, determining the interaction relationships between decision variables requires a large amount of objective function evaluation, which is unsuitable for solving expensive optimization problems. In traditional random grouping methods, although no additional evaluation time is required, all features are considered to be equally important, and computational resources are allocated equally.

[0008] To date, there are few surrogate model-assisted evolutionary algorithms for large-scale optimization problems, and most of these algorithms can only solve expensive large-scale optimization problems of no more than 200 dimensions, also known as high-dimensional optimization problems. Existing surrogate model-assisted evolutionary algorithms for high-dimensional optimization problems can be broadly classified into two categories: optimization algorithms based on non-model search, which only use approximations provided by surrogate models for individual selection; and optimization algorithms based on model search, where the surrogate model is regarded as the optimization objective to guide the search for the optimal solution to the expensive optimization problem.

[0009] However, in large-scale and expensive optimization problems, the "curse of dimensionality" occurs due to the increase in dimensionality, which causes the search space to increase dramatically, making it very difficult to directly model and solve. Summary of the Invention

[0010] To address the shortcomings of existing technologies, this invention provides a partially regenerable, expensive, large-scale optimization method to solve the modeling and solving difficulties caused by the curse of dimensionality. Furthermore, it provides an improved social learning particle swarm optimization algorithm (SL-PSO) to generate offspring for the subpopulation, by simultaneously providing the demonstrator and the optimal solution X. gbest The system learns to update the positions of particles to balance convergence speed and population diversity.

[0011] This invention employs a surrogate model and a two-stage feature selection strategy to reduce the dimensionality of the original large-scale optimization problem, forming low-dimensional subproblems and enhancing the convergence of the algorithm. Simultaneously, it applies the RBF-assisted improved social learning particle swarm optimization algorithm SL-PSO, changing the traditional SL-PSO algorithm's module of learning from the overall average behavior to learning from the optimal solution, thus enhancing the algorithm's convergence on subproblems. Furthermore, it provides a novel filling sampling strategy to update the optimal solution and archived Arc, i.e., using the best solution of the subproblem to generate a new optimal solution in the corresponding dimension, and updating the optimal solution and archived Arc by comparing the actual evaluation results of the current optimal solution and the new optimal solution.

[0012] This invention provides an expensive, large-scale optimization method for partial regeneration, the specific steps of which are as follows:

[0013] S1. Parameter settings: The initial number of sampled solutions is N, the population size is m, and the maximum number of true fitness evaluations is FE. max ;

[0014] S2. Initial sampled solutions are generated in the decision space using Latin hypercube sampling (LHS). The true fitness of individuals in the initial sampled solutions is evaluated, and the individuals and their fitness values ​​are stored in the archive Arc. The individual with the smallest fitness value in the initial sampled solutions is defined as the optimal solution X. gbest ;

[0015] S3. Randomly initialize and generate a population pop of size m, and initialize the velocity of each individual in the population pop to v;

[0016] S4. A two-stage feature selection strategy is adopted to reduce the dimensionality of the original large-scale optimization problem. The extracted important dimensions are used as sub-problems and stored in matrix Ds. The two-stage feature selection process is completed with the assistance of a surrogate model M.

[0017] S5. The RBF-assisted improved social learning particle swarm optimization algorithm SL-PSO is applied to generate offspring of the subpopulation, and a new subpopulation is formed through environmental selection, thereby determining the optimal solution sub-X for the subproblem. best ;

[0018] S6. Replace the corresponding dimension in the population pop with the new subpopulation to form a new population pop;

[0019] S7. Utilizing the optimal solution of the sub-problem (sub-X) best Replace the optimal solution X gbest The corresponding dimensions are evaluated and the optimal solution X is updated after comparison. gbest and archive Arc;

[0020] S8. Determine if the stopping criterion is met. If so, evaluate and stop, and output the optimal solution X. gbest and its true fitness value f(X) gbest Otherwise, continue with steps S4-S8.

[0021] Preferably, in step S3, the upper and lower limits of the individual velocity in the population pop are set to 10% of the upper and lower limits of the individual position, respectively.

[0022] Preferably, step S4 specifically comprises the following steps:

[0023] Phase 1: First, randomly select sub-dimensions from the original large-scale problem and store them in the dataset Dr; second, build a proxy model based on the sub-dimensions stored in the dataset Dr; finally, randomly select a portion of samples from the archive Arc as the training set to train the proxy model M, resulting in the trained proxy model M1.

[0024] The second stage involves artificially perturbing the input data for each dimension and observing the changes in the model estimate for each dimension to extract important dimension sub-problems and store them in matrix Ds. The specific steps are as follows:

[0025] Step 1: Set all variables to their lower bound values ​​and denote them as P1;

[0026] Step 2: Set the i-th variable of vector P1 to the upper bound value and denote it as P2;

[0027] Step 3: Use the trained surrogate model M1 to estimate the fitness values ​​of P1 and P2. and

[0028] Step 4, Calculation and The difference Δ i and the difference Δ i Stored in matrix G, the difference Δ i The calculation formula is as follows;

[0029]

[0030] Step 5: Repeat steps 2-4 until all variables on vector P1 have been perturbed and then stop.

[0031] Step 6: Sort all data in matrix G in descending order and calculate the average value u;

[0032] Step 7: Extract the variables corresponding to all data greater than u in matrix G from the dataset Dr, generate subproblems based on the extracted variables, and store them in matrix Ds.

[0033] Preferably, the specific steps of step S5 are as follows:

[0034] S51. Extract the sub-population sub-pop and the velocity sub-v of the individuals in the sub-population sub-pop from the population pop and the velocity v according to the corresponding dimensions in the matrix Ds;

[0035] S52. Randomly select a part of samples from the archive Arc to train the surrogate model M to obtain a trained surrogate model M2, and use the trained surrogate model M2 to estimate the fitness value of the particles in the sub-population sub-pop;

[0036] S53. Update the sub-population sub-pop by applying the RBF-assisted improved social learning particle swarm optimization algorithm SL-PSO to generate offspring of the sub-population sub-pop, wherein the population update formula is as follows:

[0037] X i,j (t+1)=X i,j (t)+v i,j (t+1)

[0038] In the formula, i is the i-th individual in the entire group, 1≤i≤m, m represents the size of the entire population, j is the j-th dimension of the individual i, 1≤j<d, d represents the dimension of the search space, X i,j (t) is the behavior vector of the j-th dimension of the individual i in the t-th generation, v i,j (t+1) is the particle velocity update formula;

[0039] The particle velocity update formula is as follows:

[0040] v i,j (t+1)=r1(t)·v i,j (t)+r2(t)·I i,j (t)+r3(t)·C i,j (t)

[0041]

[0042] In the formula, the individual k is the demonstrator of the individual i, i<k, r1(t), r2(t) and r3(t) are random coefficients, which are randomly generated in the range of [0,1]; v i,j (t) is the velocity vector of the j-th dimension of the individual i at the t-th generation, I i,j (t) represents the behavior vector after the j-th dimension of the individual i in the t-th generation learns from the j-th dimension of the random individual k with better fitness, X k,j (t) is the behavior vector of the j-th dimension of the individual k in the t-th generation, X i,j (t) is the behavior vector of the j-th dimension of the individual i in the t-th generation; C i,j(t) represents the behavior vector of individual i in generation t after learning the j-th dimension from the optimal solution in the population, X gbest (t) represents the optimal solution X at generation t. gbest ;

[0043] S54. Use the trained surrogate model M2 again to estimate the fitness values ​​of particles in the offspring of the subpopulation.

[0044] S55. Sort all particles in the subpopulation and the offspring of the subpopulation in ascending order according to the estimated fitness value, and select the first m particles, including the m-th particle, as the new subpopulation.

[0045] S56. The particle with the best fitness value selected from the new subpopulation is defined as the optimal solution to the subproblem, sub-X. best .

[0046] Preferably, the two-stage feature selection process in step S4 is completed with the assistance of a radial basis function (RBF) model, and both the post-trained surrogate model M1 and the post-trained surrogate model M2 are radial basis function (RBF) models.

[0047] Furthermore, in the first stage, the training set used to train the agent model M is a subset of the solutions of the corresponding dimension selected from the archive Arc according to the sub-dimension in the first stage; in step S52, the samples used to train the agent model M are a subset of the solutions of the corresponding dimension selected from the archive Arc according to the sub-population dimension in step S51.

[0048] Preferably, step S7 specifically includes the following steps:

[0049] S71. Using the optimal solution to a subproblem (sub-X) best Replace the optimal solution X gbest In the corresponding dimension, a new optimal solution X is obtained. new-gbest ;

[0050] S72. Regarding the new optimal solution X new-gbest Perform a true fitness assessment and assign the new optimal solution X new-gbest Its fitness value is stored in the archive Arc;

[0051] S73. Compare the current optimal solution X gbest With the new optimal solution X new-gbest The fitness value, if the new optimal solution X new-gbest Its fitness value is better than the current optimal solution X. gbest If the fitness value is obtained, then update the current optimal solution X. gbest Otherwise, the current optimal solution X gbestNo updates will be made.

[0052] Preferably, the stopping criterion is to terminate when the maximum number of evaluations is reached; in addition, only one sub-problem is formed and optimized in each evaluation.

[0053] Preferably, before using a partially regenerated, expensive large-scale optimization method to solve a large-scale, expensive optimization problem, different training sets are loaded and preprocessed using MATLAB to ensure that the training sets can be used for large-scale, expensive optimization problems.

[0054] The present invention provides a computer-readable storage medium storing a computer program that, when executed by a processor, enables an expensive, large-scale optimization method for partial regeneration.

[0055] The present invention also provides a computer device including a memory, a processor, and a computer program stored in the memory and executable on the processor, which, when executed by the processor, enables a partially regenerable, expensive, large-scale optimization method.

[0056] Compared with the prior art, the present invention has the following beneficial effects:

[0057] 1. This invention uses a surrogate model and a two-stage feature selection strategy to reduce the dimensionality of the original large-scale optimization problem, forming a low-dimensional subproblem, thereby enhancing the convergence of the algorithm.

[0058] 2. This invention applies the RBF-assisted improved social learning particle swarm optimization algorithm SL-PSO, changing the module of the traditional SL-PSO algorithm that learns from the overall average behavior to learn from the optimal solution. The improved SL-PSO generates offspring of the subpopulation and completes the subpopulation update through environmental selection, finding the optimal solution to the subproblem, which enhances the convergence of the algorithm on the subproblem and improves the convergence speed of the algorithm.

[0059] 3. This invention provides a new filling sampling strategy for updating the optimal solution and archived Arc, that is, using the best solution of the subproblem to generate a new optimal solution in the corresponding dimension, and updating the optimal solution and archived Arc by comparing the actual evaluation results of the current optimal solution and the new optimal solution. Attached Figure Description

[0060] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are only some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0061] Figure 1 This is a flowchart illustrating an expensive, large-scale optimization method for partial regeneration in an embodiment of the present invention;

[0062] Figure 2 This is a flowchart illustrating the application of the RBF-assisted improved social learning particle swarm optimization algorithm SL-PSO to determine the optimal solution to a subproblem in an embodiment of the present invention. Detailed Implementation

[0063] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0064] In solving expensive, large-scale optimization problems, computationally inexpensive surrogate model-assisted evolutionary algorithms are typically used to approximate complex and costly objective functions, thereby reducing the evaluation of the objective function and saving computational costs. However, existing surrogate model-assisted evolutionary algorithms are only suitable for solving low-dimensional, expensive, large-scale optimization problems. As the dimensionality of expensive, large-scale optimization problems increases, the search space also expands rapidly, necessitating surrogate model-assisted evolutionary algorithms with stronger search capabilities.

[0065] Since there is currently no evolutionary algorithm that can be used to assist in solving high-dimensional, expensive, large-scale optimization problems with surrogate models, this invention reduces the dimensionality of the original large-scale optimization problem to form a low-dimensional subproblem, and then finds the optimal solution by optimizing the low-dimensional subproblem. First, this invention employs a two-stage feature selection strategy to reduce the dimensionality of the original large-scale optimization problem. The extracted important dimensions are used as low-dimensional subproblems, ensuring that computational resources are concentrated on the important dimensions while reducing the dimensionality of the original large-scale optimization problem. This accelerates the search process and enhances the algorithm's convergence. The two-stage feature selection process is assisted by a radial basis function (RBF) model. Second, this invention applies an RBF-assisted improved social learning particle swarm optimization algorithm (SL-PSO). The traditional SL-PSO algorithm, which learns from the overall average behavior, is modified to learn from the optimal solution. The improved SL-PSO generates offspring of the subpopulation, and through environmental selection, a new subpopulation is formed, determining the optimal solution for the subproblem. This enhances the algorithm's convergence on the subproblem and improves its convergence speed. Simultaneously, the new subpopulation is used to update the population `pop`. Finally, a novel filling sampling strategy is provided to update the optimal solution and the archive `Arc`. This involves replacing the corresponding dimensions of the current optimal solution with the optimal solution of the subproblem to generate a new optimal solution. The optimal solution and the archive `Arc` are updated by comparing the actual evaluation results of the current and new optimal solutions.

[0066] See Figure 1 This invention provides an expensive large-scale optimization algorithm for partial regeneration, abbreviated as ELO-PR, the specific steps of which are as follows:

[0067] S1. Parameter settings: The initial number of sampled solutions is N, the population size is m, and the maximum number of true fitness evaluations is FE. max .

[0068] In the embodiments of this application, the full English title of "An Expensive Large-scale Optimization algorithm with Partial Reproduction" is ELO-PR.

[0069] S2. Initial sampled solutions are generated in the decision space using Latin hypercube sampling (LHS). The true fitness of individuals in the initial sampled solutions is evaluated, and the individuals and their fitness values ​​are stored in the archive Arc. The individual with the smallest fitness value in the initial sampled solutions is defined as the optimal solution X. gbest .

[0070] S3. Randomly initialize and generate a population pop of size m, and initialize the velocity of each individual in the population pop to v.

[0071] In this embodiment of the application, the upper and lower limits of the individual velocity in the population pop in step S3 are set to 10% of the upper and lower limits of the individual position, respectively.

[0072] S4. A two-stage feature selection strategy is adopted to reduce the dimensionality of the original large-scale optimization problem. The extracted important dimensions are used as sub-problems and stored in matrix Ds. The two-stage feature selection process is completed with the assistance of a surrogate model M.

[0073] Feature selection is a process in machine learning used to select a relevant subset of features from a large set of available features. The underlying idea is to improve the performance, efficiency, and interpretability of machine learning models by focusing on the most informative features. Not all features in a dataset contribute equally to the prediction task; some features may be redundant, noisy, or irrelevant, which can lead to overfitting or degraded model performance. Feature selection aims to address these problems by selecting the most important and relevant features that have the strongest relationship with the target variable. Therefore, this application proposes a novel feature selection method, two-stage feature selection, to reduce the dimensionality of the original large-scale optimization problem and treat the extracted important dimensions as subproblems.

[0074] In this embodiment of the application, step S4 specifically involves the following steps:

[0075] Phase 1: First, randomly select sub-dimensions from the original large-scale problem and store them in the dataset Dr; second, build a proxy model based on the sub-dimensions stored in the dataset Dr; finally, randomly select a portion of samples from the archive Arc as the training set to train the proxy model M, resulting in the trained proxy model M1.

[0076] In the first stage of this application's embodiment, 100-dimensional data was randomly selected from the original large-scale problem and stored in the dataset Dr. Based on this 100-dimensional data, important dimensions were extracted in the second stage to generate sub-problems. In practical applications, the sub-dimensions in the first stage of this application's embodiment can be set according to actual needs.

[0077] In this embodiment of the application, in the first stage, the training set used to train the agent model M is a subset of the solutions of the corresponding dimension selected from the archive Arc according to the sub-dimension in the first stage.

[0078] The second stage involves artificially perturbing the input data for each dimension and observing the changes in the model estimate for each dimension to extract important dimension sub-problems and store them in matrix Ds. The specific steps are as follows:

[0079] Step 1: Set all variables to their lower bound values ​​and denote them as P1;

[0080] Step 2: Set the i-th variable of vector P1 to the upper bound value and denote it as P2;

[0081] Step 3: Use the trained surrogate model M1 to estimate the fitness values ​​of P1 and P2. and

[0082] Step 4, Calculation and The difference Δ i and the difference Δ i Stored in matrix G, the difference Δ i The calculation formula is as follows;

[0083]

[0084] Step 5: Repeat steps 2-4 until all variables on vector P1 have been perturbed and then stop.

[0085] Step 6: Sort all data in matrix G in descending order and calculate the average value u;

[0086] Step 7: Extract the variables corresponding to all data greater than u in matrix G from the dataset Dr, generate subproblems based on the extracted variables, and store them in matrix Ds.

[0087] Preferably, the two-stage feature selection process in step S4 is completed with the assistance of a radial basis function RBF model, wherein the trained surrogate model M1 is also a radial basis function RBF model.

[0088] It should be noted that the radial basis function RBF model is constructed by directly calling pre-written code through MATLAB software, and the code for constructing the radial basis function RBF model is prior art, so the code for constructing the radial basis function RBF model is not specifically described in the present application.

[0089] In the embodiments of the present application, the original large-scale optimization problem is dimension-reduced by adopting a surrogate model assistance and two-stage feature selection strategy, and only a small part of the dimensions are optimized in each optimization process, so as to solve the defect that the existing surrogate model-assisted algorithms are not suitable for high-dimensional large-scale optimization problems. Moreover, by taking the extracted important dimensions as sub-problems, the convergence in each optimization process can be enhanced.

[0090] S5. applying the RBF-assisted improved social learning particle swarm optimization algorithm SL-PSO to generate offspring of the sub-population, and performing environmental selection to form a new sub-population, so as to determine the optimal solution sub-X of the sub-problem best .

[0091] In the traditional SL-PSO algorithm, particles are initially sorted in ascending order according to their fitness values, and imitator particles learn the behaviors of other particles in the current group, namely demonstrator particles, which have better fitness values than the imitator particles. Except for the best particle, each particle acts as an imitator and learns from its corresponding demonstrator. It should be noted that due to the design of the algorithm, the worst particle in the population never acts as a demonstrator, and the best particle never acts as an imitator, so the position update formula of the particles is as follows:

[0092]

[0093] In the formula, i is the i-th individual in the entire group, 1≤i≤m, m represents the size of the entire population, j is the j-th dimension of the individual i, 1≤j<d, d represents the dimension of the search space, X i,j (t) is the behavior vector of the j-th dimension of the individual i in the t-th generation, ΔX i,j (t+1) is the behavior correction factor, p i (t) is the probability that the individual i learns from excellent individuals in the t-th generation, is the pre-programmed learning probability for each individual i.

[0094] In the traditional algorithm, ΔX i,j(t+1) consists of three parts: the inertial part, the part learned from random individuals, and the part learned from all individuals in the society. The calculation formula is as follows:

[0095] ΔX i,j (t+1)=r1(t)·ΔX i,j (t)+r2(t·I i,j (t)+r3(t)·Φ·C i,j (t)

[0096]

[0097] In the formula, individual k is the demonstrator of individual i, i < k, ΔX i,j (t) is the j-th dimension of the inertial behavior vector of individual i at generation t, I i,j X(t) represents the behavior vector of individual i after learning from a well-fit random individual k in generation t. k,j (t) is the j-th dimension of the behavior vector of individual k in generation t, X i,j (t) is the j-th dimension of the behavior vector of individual i in generation t; C i,j (t) represents the behavior vector of individual i after learning from all individuals in the population in generation t; Φ represents the average behavior level of the entire population in the j-th dimension; Φ is a social influence factor parameter used to control the average behavior at the group level; r1(t), r2(t), and r3(t) are random coefficients, randomly generated in the range [0,1].

[0098] However, due to the use of a surrogate model to estimate the fitness of all particles in this application, there is inherent uncertainty. The average behavior of the particles may not serve as an ideal reference point for guiding the search. Therefore, in this application, all particles in the swarm are directed towards the particle X with the optimal solution. gbest Learning. In addition, since the fitness of all particles in this application is estimated by the surrogate model, it means that the particle with the highest fitness is not necessarily the truly optimal particle. Therefore, in this application, all particles, including the particle with the best fitness value, will participate in the update process.

[0099] See Figure 2 In this embodiment of the application, step S5 specifically involves the following steps:

[0100] S51. Based on the corresponding dimension in matrix Ds, extract the subpopulation and the velocity sub-v of individuals in the subpopulation from the population pop and velocity v.

[0101] It should be noted that, in the embodiments of the present application, during the first evaluation, the population pop mentioned in steps S51 and S6 is the population pop generated by random initialization in step S3, and in the subsequent continuous evaluation and update process, the population pop mentioned in steps S51 and S6 is the population pop updated in the previous evaluation process.

[0102] S52. Randomly select a part of samples from the archive Arc to train the surrogate model M to obtain a trained surrogate model M2, and use the trained surrogate model M2 to estimate the fitness value of particles in the sub-population sub-pop.

[0103] In the embodiments of the present application, the trained surrogate model M2 is also a radial basis function RBF model.

[0104] In addition, in the embodiments of the present application, the samples used for training the surrogate model M in step S52 are a subset of solutions of corresponding dimensions selected from the archive Arc according to the dimension of the sub-population sub-pop in step S51.

[0105] S53. Update the sub-population sub-pop by applying the RBF-assisted improved social learning particle swarm optimization algorithm SL-PSO, so as to generate offspring of the sub-population sub-pop.

[0106] In the embodiments of the present application, the population update formula is as follows:

[0107] X i,j (t+1)=X i,j (t)+v i,j (t+1)

[0108] In the formula, i is the i-th individual in the entire group, 1≤i≤m, m represents the size of the entire population, j is the j-th dimension of the individual i, 1≤j<d, d represents the dimension of the search space, X i,j (t) is the behavior vector of the j-th dimension of the individual i in the t-th generation, v i,j (t+1) is the particle velocity update formula.

[0109] In the embodiments of the present application, the particle velocity update formula is as follows:

[0110] v i,j (t+1)=r1(t)·v i,j (t)+r2(t)·I i,j (t)+r3(t)·C i,j (t)

[0111]

[0112] In the formula, individual k is the demonstrator of individual i, i < k, r1(t), r2(t), and r3(t) are random coefficients, randomly generated within the range [0,1]; v i,j (t) is the velocity vector of individual i in the j-th dimension at generation t. i,j (t) represents the behavior vector of individual i in generation t after learning the j-th dimension from the j-th dimension of a well-fit random individual k. k,j (t) is the j-th dimension of the behavior vector of individual k in generation t, X i,j (t) is the j-th dimension of the behavior vector of individual i in generation t; C i,j (t) represents the behavior vector of individual i in generation t after learning the j-th dimension from the optimal solution in the population, X gbest (t) represents the optimal solution X at generation t. gbest .

[0113] In this embodiment, the improved SL-PSO modifies the traditional SL-PSO module that learns from the overall average behavior to learn from the optimal solution. By learning from both the demonstrator and the optimal solution simultaneously to update the particle's position, the convergence of the algorithm on subproblems is effectively enhanced, and the convergence speed of the algorithm is improved.

[0114] It should be noted that, since the goal of this application is to reduce dimensionality first and then optimize, the position and velocity variables in the equations should be replaced with the corresponding position and velocity of the subproblems, named sub-pop and sub-v, respectively. Similarly, X gbest Its dimensions should also be adjusted to match the dimensionality reduction of the subproblems, known as sub-X. gbest Among them, sub-X gbest It is based on the corresponding dimension in matrix Ds in X gbest Obtained by extraction.

[0115] S54. Use the trained surrogate model M2 again to estimate the fitness values ​​of particles in the offspring of the subpopulation.

[0116] S55. Sort all particles in the subpopulation and the offspring of the subpopulation in ascending order according to the estimated fitness value, and select the first m particles, including the m-th particle, as the new subpopulation.

[0117] S56. The particle with the best fitness value selected from the new subpopulation is defined as the optimal solution to the subproblem, sub-X. best .

[0118] S6. Replace the corresponding dimension in the population pop with the new subpopulation to form a new population pop.

[0119] S7. Utilizing the optimal solution of the sub-problem (sub-X) best Replace the optimal solution X gbest The corresponding dimensions are evaluated and the optimal solution X is updated after comparison. gbest And the Arc archive, the specific steps are as follows:

[0120] S71. Using the optimal solution to a subproblem (sub-X) best Replace the optimal solution X gbest In the corresponding dimension, a new optimal solution X is obtained. new-gbest .

[0121] S72. Regarding the new optimal solution X new-gbest Perform a true fitness assessment and assign the new optimal solution X new-gbest Its fitness value is stored in the archive Arc.

[0122] S73. Compare the current optimal solution X gbest With the new optimal solution X new-gbest The fitness value, if the new optimal solution X new-gbest Its fitness value is better than the current optimal solution X. gbest If the fitness value is obtained, then update the current optimal solution X. gbest Otherwise, the current optimal solution X gbest No updates will be made.

[0123] It should be noted that in this application, for the new optimal solution X... new-gbest Regardless of whether its fitness value is better than the optimal solution X gbest The fitness value, the new optimal solution X new-gbest Both their fitness values ​​are stored in the archive Arc for updating the archive Arc.

[0124] S8. Determine if the stopping criterion is met. If so, evaluate and stop, and output the optimal solution X. gbest and its true fitness value f(X) gbest Otherwise, continue with steps S4-S8.

[0125] In this embodiment of the application, the stopping criterion is to terminate when the maximum number of evaluations is reached.

[0126] Furthermore, in this embodiment, only one sub-problem is formed and optimized in each evaluation, which effectively saves optimization time.

[0127] In this embodiment of the application, before using the partially regenerated expensive large-scale optimization method to solve the large-scale expensive optimization problem, MATLAB is used to load and preprocess different training sets to ensure that the training sets can be used for large-scale expensive optimization problems.

[0128] It should be noted that the partially regenerated, expensive large-scale optimization method provided in the embodiments of this application is applicable to solving single-objective, large-scale, expensive optimization problems.

[0129] This application evaluates the performance of the proposed algorithm using 15 test functions from CEC2013 and provides empirical comparisons with some state-of-the-art large-scale expensive algorithms, including SEA-LEEA-RG, a surrogate ensemble algorithm for large-scale expensive optimization problems with random grouping; SAEA-RFS, a surrogate-assisted evolutionary algorithm for large-scale expensive problems with random feature selection; and SADE-AMSS, a surrogate-assisted differential evolution algorithm for large-scale expensive optimization problems with adaptive multi-subspace search.

[0130] Based on the separability of variables, the 15 test functions of CEC2013 can be further divided into 5 categories: fully decomposable functions F1-F3, partially decomposable functions with separable variables F4-F7, partially decomposable functions without separable variables F8-F11, overlapping functions F12-F14, and completely indivisible function F15. Furthermore, the decision variable dimension for functions F1-F12 and F15 is 1000, while the decision variable dimension for functions F13 and F14 is 905.

[0131] In the experiments of this application, the particle values ​​range from [-5, 5] in functions F1, F5, and F9; from [-32, 32] in functions F3, F6, and F10; and from [-100, 100] in the remaining functions. The initial sampled solutions had N = 500 individuals, and the initial population size was m = 100. During the formation of subproblems, the dimension sub = 100, and 3*sub particles were selected from the archive Arc to train the surrogate model. The upper and lower limits of particle velocity were set to 10% of the upper and lower limits of particle position, respectively. Furthermore, to ensure fairness in the comparison, the stopping criterion for all algorithms in this experiment was set to reaching the maximum number of evaluations, i.e., 11*d, where d represents the dimension of the search space. In addition, each algorithm was run independently 25 times.

[0132] In the experiments of this application, the significance of the results of this algorithm and other algorithms was evaluated using the Wilcoxon rank-sum test at a significance level of 0.05. The symbols "+", "≈" and "-" indicate that the performance of this algorithm is significantly better, equal, or worse than that of other algorithms, as shown in Table 1.

[0133] Table 1. Comparison of the median and median absolute deviation of this algorithm with three other algorithms on the CEC 2013 benchmark problem.

[0134]

[0135]

[0136]

[0137] As shown in Table 1, the proposed algorithm exhibits superior performance on partially separable problems F4–F11 and completely inseparable problems F15. The table clearly demonstrates that compared to the SEA-LEEA-RG algorithm, this application achieves the best performance on 11 test cases; compared to the SAEA-RFS algorithm, it achieves the best performance on 13 test cases; and compared to the SADE-AMSS algorithm, it achieves the best performance on 10 test cases.

[0138] Or, more precisely, on the completely separable problems F1–F3, the performance of this algorithm is inferior to the SEA-LEEA-RG algorithm, but superior to the SAEA-RFS and SADE-AMSS algorithms.

[0139] On partially separable problems F4–F11, except for F9, our algorithm outperforms the other three algorithms. Notably, our algorithm exhibits similar performance to SADE-AMSS on F9. Furthermore, compared to the SEA-LEEA-RG algorithm, our algorithm achieves favorable results on seven functions, showing similar performance only on F6. On all eight functions, our algorithm significantly outperforms the SAEA-RFS algorithm. On F5 and F9, our algorithm's results are roughly similar to SADE-AMSS, but in the remaining functions, our algorithm significantly outperforms SADE-AMSS.

[0140] On overlapping problems F12–F14 and completely inseparable problems F15, this algorithm achieves the best overall performance compared to the other three algorithms, showing a significant improvement. Only on F15 does it show similar performance to the SADE-AMSS algorithm.

[0141] Additionally, the present invention provides a computer-readable storage medium storing a computer program that, when executed by a processor, enables an expensive, large-scale optimization method for partial regeneration.

[0142] Furthermore, the present invention also provides a computer device including a memory, a processor, and a computer program stored in the memory and executable on the processor, which, when executed by the processor, enables a partially regenerable, expensive, large-scale optimization method.

[0143] Those skilled in the art will understand that embodiments of the present invention can be provided as methods, systems, or computer program products. Therefore, the present invention can take the form of hardware embodiments, software embodiments, or embodiments combining software and hardware aspects. Furthermore, the present invention can take the form of a computer program product implemented on one or more computer-usable storage media containing computer-usable program code, wherein the storage media includes, but is not limited to, disk storage and optical storage.

[0144] This invention is described with reference to flowchart illustrations and / or block diagrams of methods, apparatus / systems, and computer program products according to embodiments of the invention. It will be understood that each block of the flowchart illustrations and / or block diagrams, and combinations of blocks in the flowchart illustrations and / or block diagrams, can be implemented by computer program instructions. These computer program instructions can be provided to a processor of a general-purpose computer, special-purpose computer, embedded processor, or other programmable data processing apparatus to produce a machine, such that the instructions, which execute via the processor of the computer or other programmable data processing apparatus, generate instructions for implementing the flowchart illustrations and / or block diagrams. Figure 1 One or more processes and / or boxes Figure 1 A device that provides the functions specified in one or more boxes.

[0145] These computer program instructions may also be stored in a computer-readable storage medium that can direct a computer or other programmable data processing device to function in a particular manner, such that the instructions stored in the computer-readable storage medium produce an article of manufacture including instruction means, which are implemented in a process Figure 1 One or more processes and / or boxes Figure 1 The function specified in one or more boxes.

[0146] These computer program instructions may also be loaded onto a computer or other programmable data processing equipment to cause a series of operational steps to be performed on the computer or other programmable equipment to produce a computer-implemented process, thereby providing instructions that execute on the computer or other programmable equipment for implementing the process. Figure 1 One or more processes and / or boxes Figure 1 The steps of the function specified in one or more boxes.

[0147] Those skilled in the art will understand that all or part of the processes in the above embodiments can be implemented by a computer program instructing related hardware. The program can be stored in a computer-readable storage medium, and when executed, it can include the processes of the embodiments of the above methods. The storage medium can be a magnetic disk, optical disk, read-only memory (ROM), or random access memory (RAM), etc.

[0148] Finally, it should be noted that the above description is only a preferred embodiment of the present invention and is not intended to limit the present invention. Although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art can still modify the technical solutions described in the foregoing embodiments or make equivalent substitutions for some of the technical features. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the protection scope of the present invention.

Claims

1. An expensive, large-scale optimization method for partial regeneration, characterized in that, The specific steps are as follows: S1. Parameter settings: The initial number of sampled solutions is N, the population size is m, and the maximum number of true fitness assessments is [missing information]. ; S2. Initial sampled solutions are generated in the decision space using Latin hypercube sampling (LHS). The true fitness of individuals in the initial sampled solutions is evaluated, and the individuals and their fitness values ​​are stored in the archive Arc. The individual with the smallest fitness value in the initial sampled solutions is defined as the optimal solution. ; S3. Randomly initialize and generate a population pop of size m, and initialize the velocity of each individual in the population pop to v; S4. A two-stage feature selection strategy is used to reduce the dimensionality of the original large-scale optimization problem. The extracted important dimensions are treated as sub-problems and stored in matrix Ds. The two-stage feature selection process is assisted by a surrogate model M. The specific steps of step S4 are as follows: Phase 1: First, randomly select sub-dimensions from the original large-scale problem and store them in the dataset Dr; second, build a proxy model based on the sub-dimensions stored in the dataset Dr; finally, randomly select a portion of samples from the archive Arc as the training set to train the proxy model M, resulting in the trained proxy model M1. The second stage involves artificially perturbing the input data for each dimension and observing the changes in the model estimate for each dimension to extract important dimension sub-problems and store them in matrix Ds. The specific steps are as follows: Step 1: Set all variables to their lower bound values ​​and denote them as P1; Step 2: Set the i-th variable of vector P1 to the upper bound value and denote it as P2; Step 3: Use the trained surrogate model M1 to estimate the fitness values ​​of P1 and P2. ( )and ( ); Step 4, Calculation ( )and ( The difference and the difference Stored in matrix G, the difference The calculation formula is as follows; ； Step 5: Repeat steps 2-4 until all variables on vector P1 have been perturbed and then stop. Step 6: Sort all data in matrix G in descending order and calculate the average value u; Step 7: Extract the variables corresponding to all data greater than u in matrix G from the dataset Dr, generate subproblems based on the extracted variables, and store them in matrix Ds; S5. Apply the RBF-assisted improved social learning particle swarm optimization algorithm SL-PSO to generate offspring of the subpopulation, and form a new subpopulation through environmental selection, thereby determining the optimal solution to the subproblem. Specifically, step S5 involves the following steps: S51. Based on the corresponding dimension in matrix Ds, extract the subpopulation and the velocity sub-v of individuals in the subpopulation from the population pop and velocity v. S52. Randomly select a portion of samples from the archive Arc to train the surrogate model M, obtain the trained surrogate model M2, and use the trained surrogate model M2 to estimate the fitness values ​​of particles in the subpopulation. S53. Apply the RBF-assisted improved social learning particle swarm optimization algorithm SL-PSO to update the subpopulation sub-pop and generate the offspring of the subpopulation sub-pop. The population update formula is as follows: ， In the formula, i is the i-th individual in the entire population, 1≤i≤m, m represents the size of the entire population, j is the j-th dimension of individual i, 1≤ j <d, and d represents the dimension of the search space, is the behavior vector of the j-th dimension of individual i in the t-th generation, is the particle velocity update formula; The particle velocity update formula is as follows: ， ， In the formula, individual k is the demonstrator of individual i, i < k. , and The coefficients are randomized within the range [0,1]. Let the j-th dimension of individual i be the velocity vector at generation t. Let represent the behavior vector of individual i in generation t after learning the j-th dimension from the j-th dimension of a well-fit random individual k. Let j be the behavior vector of individual k in generation t. Let i be the j-th dimension of the behavior vector of individual i in generation t; Let represent the behavior vector of individual i in generation t after learning the optimal solution in the population in dimension j. The optimal solution at generation t ; S54. Use the trained surrogate model M2 again to estimate the fitness values ​​of particles in the offspring of the subpopulation. S55. Sort all particles in the subpopulation and the offspring of the subpopulation in ascending order according to the estimated fitness value, and select the first m particles, including the m-th particle, as the new subpopulation. S56. The particle with the best fitness value selected from the new subpopulation is defined as the optimal solution to the subproblem. ; S6. Replace the corresponding dimension in the population pop with the new subpopulation to form a new population pop; S7. Utilizing the optimal solution of subproblems Replace the optimal solution The corresponding dimensions are evaluated realistically, and the optimal solution is updated after comparison. and archive Arc; S8. Determine if the stopping criterion is met. If so, evaluate and stop, and output the optimal solution. and its true fitness value f ( Otherwise, continue with steps S4-S8.

2. The expensive large-scale optimization method for partial regeneration according to claim 1, characterized in that, In step S3, the upper and lower limits of the individual velocity in the population pop are set to 10% of the upper and lower limits of the individual position, respectively.

3. The expensive large-scale optimization method for partial regeneration according to claim 2, characterized in that, The two-stage feature selection process in step S4 is completed with the assistance of the radial basis function (RBF) model. Both the trained surrogate model M1 and the trained surrogate model M2 are radial basis function (RBF) models. Furthermore, in the first stage, the training set used to train the agent model M is a subset of the solutions of the corresponding dimension selected from the archive Arc according to the sub-dimension in the first stage; in step S52, the samples used to train the agent model M are a subset of the solutions of the corresponding dimension selected from the archive Arc according to the sub-population dimension in step S51.

4. The expensive large-scale optimization method for partial regeneration according to claim 3, characterized in that, The specific steps of step S7 are as follows: S71. Using the best solution to a subproblem Replace the optimal solution In the corresponding dimension, a new optimal solution is obtained. ; S72. For the new optimal solution Perform a true fitness assessment and assign the new optimal solution. Its fitness value is stored in the archive Arc; S73. Compare the current optimal solution With the new optimal solution The fitness value, if the new optimal solution Its fitness value is better than the current optimal solution. If the fitness value is obtained, then update the current optimal solution. Otherwise, the current optimal solution No updates will be made.

5. The expensive large-scale optimization method for partial regeneration according to claim 4, characterized in that, The stopping criterion is to terminate when the maximum number of evaluations is reached; furthermore, only one sub-problem will be formed and optimized in each evaluation.

6. The expensive large-scale optimization method for partial regeneration according to claim 5, characterized in that, Before using a partially regenerated, expensive large-scale optimization method to solve the large-scale expensive optimization problem, MATLAB was used to load and preprocess different training sets to ensure that the training sets could be used for the large-scale expensive optimization problem.

7. A computer-readable storage medium, characterized in that, The storage medium stores a computer program, which, when executed by a processor, enables the implementation of an expensive, large-scale optimization method for partial regeneration as described in any one of claims 1-6.

8. A computer device, characterized in that, It includes a memory, a processor, and a computer program stored in the memory and executable on the processor, which, when executed by the processor, implements the expensive, large-scale optimization method for partial regeneration as described in any one of claims 1-6.

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