Graph neural network guided constrained multi-target intelligent optimization framework

Abstract

According to the invention, a graph neural network guided constrained multi-objective intelligent optimization framework is constructed for a constrained multi-objective optimization problem. On the basis of the framework, how to learn topological structure information constructed by constraint conditions is considered, the topological structure information is embedded into a multi-objective evolution algorithm based on decomposition, and three novel evolution components, namely a topological information learning model, a weight vector judgment criterion based on population density and a weight vector updating mechanism based on clustering, are designed. Therefore, a novel constrained multi-objective evolution algorithm is obtained, and the algorithm is named as MOEA / Dlearning. The evolution framework constructed by the method has good universality, the candidate solution set is quickly evaluated by training the graph neural network, and the population development capability is enhanced by using a weight vector discrimination criterion and an updating mechanism. Based on the framework, a derived algorithm is obviously superior to a peer algorithm in multiple benchmark test problems and practical engineering problems, and a new effective tool is provided for solving the multi-objective constraint optimization problem.

Description

Technical Field

[0001] The present invention relates to a method for solving constrained multi-objective optimization problems (CMOPs) by using a graph neural network to guide an evolutionary algorithm. Background Art

[0002] Constrained multi-objective optimization problems are widely present in various practical applications, such as batch flow shop scheduling optimization problems, power system optimization problems, drone path planning problems and engineering optimization problems. However, existing optimization algorithms lack the learning of constraint characteristics, and it is difficult to quickly find the optimal solution when faced with unknown actual CMOPs.

[0003] In order to solve CMPOs, various constraint processing techniques have been proposed, and a series of constraint-based multi-objective evolutionary algorithms (CMOEAs) have been derived, which can be roughly divided into four categories:

[0004] Penalty-based CMOEAs incorporate constraints into the objective function to simplify the decision space. For example, Ma et al. proposed a displacement-based penalty mechanism to deal with constraints and designed an adaptive penalty function Ship to introduce potential solutions.

[0005] Multi-population co-evolution CMOEAs use multiple populations to search the solution space in parallel, improving the algorithm's ability to find excellent feasible solutions. For example, Kong et al. proposed a dynamic dual-population co-evolution multi-objective evolutionary algorithm DDCMEA, which enhances the ability to balance convergence and feasibility by dynamically adjusting the number of offspring in the population.

[0006] The idea of ​​multi-stage CMOEAs is to divide the process of solving CMOPs into multiple stages. For example, Ma et al. proposed a multi-stage evolutionary algorithm that enhances convergence and diversity by adding constraints one by one and step-by-step processing.

[0007] Archive-based CMOEAs maintain one or more archives to accelerate the search of the population. For example, Xia et al. proposed a dual-archive CMOEA that provides two types of individuals to help balance convergence and diversity.

[0008] The intelligent optimization method guided by graph neural network is a general method for solving constrained multi-objective optimization problems. It uses graph neural network to learn existing CMOPs and combines it with intelligent optimization algorithms to enable the algorithm to have the ability to learn problem characteristics. Summary of the invention

[0009] The purpose of this invention is to solve the actual CMOPs with unknown and irregular Pareto fronts, avoid the rapid deterioration of algorithm performance, and accelerate the optimal solution to the problem. A constrained multi-objective intelligent optimization framework guided by a graph neural network is proposed, which includes a constrained multi-objective evaluation strategy assisted by a graph neural network, a solution module integrating learning and evolution, and a weight vector discrimination criterion and update mechanism.

[0010] The technical solution of the present invention is to propose a constrained multi-objective intelligent optimization framework guided by a graph neural network, which includes the following steps:

[0011] 1. Constrained Multi-Objective Evaluation Strategy Assisted by Graph Neural Network

[0012] Using data and graph neural networks to train a quality evaluator, which can be embedded in the evolutionary algorithm, it can achieve fast and accurate evaluation of population convergence, diversity, and constraint violations. The steps can be simply summarized as follows:

[0013] First, CMOEAs are randomly selected to solve existing CMOPs to obtain sufficient training samples. Then, the information of individuals in the population is extracted to prepare for the construction of the graph structure. Next, the dominance of individuals is determined and marked. Finally, the processed data is used to build a graph neural network model. The core components involved in this framework are:

[0014] 1. Dataset construction: 10 advanced CMOEAs were randomly selected to solve 31 constrained multi-objective test problems to obtain sufficient training samples. The objective function of each test problem was set to 3, 5, 8, 10 and 15. Starting from the 4000th iteration, the results were saved every 20,000 generations, for a total of 4 times. The specific steps are as follows:

[0015] First, the data is initialized. Then, the decision variables, objective function values, and constraint violation degrees are extracted and saved from each individual. Next, the dominant or non-dominated individual is judged: if it is a dominant individual, it is marked as non-excellent (labeled as 1) and its hypervolume is calculated; if it is a non-dominated individual, it is marked as excellent (labeled as 0), and its constraint violation degree is clustered. According to the clustering results, the individuals with smaller average values ​​are marked as 0, and the remaining ones are marked as 1. All data and labels are saved. After that, data processing is performed, which includes two steps: (1) the dominant individuals are sorted in descending order according to the hypervolume value, and the individuals in the bottom 1 / 4 of the ranking are discarded; (2) the zero-filling operation is performed to find the maximum number of decision variables for all individuals with the same target number. If the number of decision variables of an individual is insufficient, a zero value is added at the end to ensure that the number of decision variables of all individuals is the same.

[0016] 2. Model construction: This paper adopts a lightweight graph neural network model GraphSAGE. The model uses node sampling and neighbor aggregation to learn node features, reducing the complexity of calculating the entire graph. In GraphSAGE, the function of applying pooling operation to aggregate neighbor features is as follows:

[0017]

[0018] In the formula, is the feature vector of node v in the ,th layer; σ represents the activation function; W l is the weight matrix of the lth layer; max is the maximum pooling operation; W pool and b are the weights and biases of the pooling layer; N(v) is the set of neighbors of node v.

[0019] Model construction mainly includes graph structure definition, graph classification model training and node classification model training. For graph structure definition, first, according to the constructed data set, the Euclidean distance between each individual and all other individuals is calculated, and these distances are sorted. Then, for each individual, the individuals that do not include themselves and are ranked in the top 4% of the distance are selected as the edges in their graph structure, with the current individual as the head node of the directed edge and the selected neighbor individuals as the tail nodes of the directed edge. In addition, the edge information is not stored in the adjacency matrix, but only the indexes of the head and tail individuals are stored, which can greatly save storage space and improve processing efficiency. Finally, based on the obtained node features and edge information, the graph structure required for training is constructed.

[0020] The graph classification model includes three graph convolution layers (SAGEConv), three graph pooling layers (pool), three fully connected layers (Line), and corresponding activation functions (ReLU), batch normalization layers, and random dropout layers (Dropout). During the forward propagation process, each convolution layer enters the ReLU activation function and then enters the pooling layer. The pooled node features are subjected to global maximum pooling and global average pooling. After the results of the two are aggregated, they are processed by the fully connected layer and random dropout layer to map the features to the final problem results.

[0021] The node classification model includes two convolutional layers. After the first convolutional layer, it enters the ReLU activation function and then the random dropout layer; after the second convolutional layer, it enters the Softmax activation function. After the first convolutional layer, the ReLU and random dropout layers are used to prevent overfitting, and after the second convolutional layer, the Softmax activation function is used to calculate the final point classification result.

[0022] 3. Model training: The specific steps are as follows: First, initialize the edges, then extract the individual decision variables, objective function values, and constraint violation degrees from the constructed data set as node features, and the current individual index is the head index of the edge. Next, according to the graph structure definition, use the Euclidean distance to calculate other individuals that are close to it (take the first 4%*N) to construct the graph required for training. Finally, perform graph classification training and node classification training to ensure that the model has the ability to identify problem types and individual quality.

[0023] 2. Solution Module for Integration of Learning and Evolution

[0024] Combining the trained graph neural network model with the Multi-objective Evolutionary Algorithm Based on Decomposition (MOEA / D), a new constrained multi-objective evolutionary algorithm based on graph neural network, MOEA / Dlearning, is proposed. The specific steps of the algorithm are as follows:

[0025] Step 1. Initialize the population and set the offspring individual set offs to an empty set.

[0026] Step 2. Randomly select two parents from the neighbor individual set to perform crossover and mutation operations to obtain the offspring individual offspring.

[0027] Step 3. Calculate the constraint violation degree CV of the offspring

[0028] Step 4. Fix CV and update ideal point Z.

[0029] Step 5. Save the old population P old , mixed offs.

[0030] Step 6. Update the population according to the scalar aggregation value and the weight vector; calculate the Euclidean distance between the individual and each weight vector, find the N individuals closest to the weight vector, and update the associated individual matrix AM.

[0031] Step 7. When the number of evaluations reaches a certain threshold, the trained graph neural network model is called to perform graph classification prediction, and the corresponding node classification model is called according to the prediction results to evaluate individual quality.

[0032] Step 8. Use the newly proposed weight vector discrimination method to determine the category of each weight vector.

[0033] Step 9. Using the different category values ​​of the weight vector, calculate the rank value of the weight vector.

[0034] Step 10. According to the number of excellent individuals associated with the weight vector, determine whether the weight vector needs to be updated.

[0035] 3. Weight Vector Discrimination Criteria and Update Mechanism

[0036] This paper designs a weight vector discrimination criterion and update mechanism based on population distribution information for CMOPs with irregular Pareto fronts to provide the optimal search direction and its neighborhood, thereby enhancing the dynamic adaptability of the algorithm. The specific steps of the weight vector discrimination criterion based on graph neural network are as follows:

[0037] Step 1. Calculate the distance between each weight vector W and its neighboring individuals to obtain the associated individuals;

[0038] Step 2. According to the number of individuals associated with each weight vector and the number of excellent individuals, weight vector discrimination is performed;

[0039] Step 3. If the weight vector has a corresponding associated individual and the number of excellent individuals is greater than 1, the weight vector is defined as a strong weight vector, and the class value Class is set to 4;

[0040] Step 4. If the weight vector has associated individuals and there is only one excellent individual, the weight vector is defined as a standard weight vector, and the class value Class is set to 3;

[0041] Step 5. If the weight vector has no associated individual, it is defined as a difference weight vector and the class value Class is set to 1.

[0042] Step 6. Calculate the weight vector rank value based on the historical Class value for weight vector update. The weight vector rank value is calculated by using the historical Class values ​​of weight vectors of three consecutive generations and using the following formula to calculate the weight vector rank value:

[0043]

[0044] In the formula, rank(λ) represents the rank value of the weight vector λ, Class t (λ) represents the category value of the t-th generation weight vector, and r represents the learning rate. The rank value can reflect the performance change trend of the weight vector and is an important indicator for evaluating its potential.

[0045] The specific steps of the weight vector update mechanism are as follows:

[0046] Step 1. Use the newly designed probability calculation formula to calculate the deletion probability p. The specific formula is as follows:

[0047] p=(-g 2 +maxFE g) / maxFE 2 +0.1,

[0048] Where p represents the probability, g represents the current number of iterations, and maxFX represents the maximum number of evaluations.

[0049] Step 2. Delete the weight vector with the smallest rank value according to probability p;

[0050] Step 3. Delete the historical Class value and its associated individuals corresponding to the weight vector;

[0051] Step 4. Update population P;

[0052] Step 5. Perform clustering operation on the remaining weight vectors;

[0053] Step 6. Use the cluster center corresponding to the largest cluster obtained by clustering as the new weight vector. BRIEF DESCRIPTION OF THE DRAWINGS

[0054] Figure 1 Schematic diagram of sampling and aggregation in the GraphSAGE model;

[0055] Figure 2 Learning model schematics;

[0056] Figure 3 Regarding the IGD and HV indicators, the statistical results of the algorithm on the test problem. DETAILED DESCRIPTION

[0057] In order to verify the effectiveness of the constrained multi-objective evolutionary algorithm MOEA / Dlearning based on graph neural network, five advanced CMOEAs were selected for comparative experiments, namely: reference point-based ANSGA-III

[24] , indicator and crowding distance-based IDBEA

[25] , decomposition-based MOEA / DD, reference vector-based RVEA, and archive-based C-TAEA. To ensure the fairness of the experiment, the parameter settings of each algorithm are directly provided by the author. Inverted Generational Distance (IGD) and Hypervolume (HV) are selected as evaluation indicators to comprehensively evaluate the convergence and diversity of the final solution. The calculation formula of IGD is as follows:

[0058]

[0059] In the formula, PF* represents the optimal Pareto frontier, y represents each solution in the true optimal Pareto frontier, x represents the solution in the closest Pareto frontier, and d(y, x) represents the Euclidean distance between y and x. The smaller the IGD, the higher the quality of the solution.

[0060] The calculation formula of HV is as follows:

[0061]

[0062] In the formula, δ represents the Lebesgue measure, which is used to measure the volume. |S| represents the number of non-dominated solution sets, v i Represents the hypervolume composed of the ideal point and the i-th solution. The larger the HV, the better the performance of the algorithm.

[0063] To verify the overall performance of MOEA / Dlearning, four C_DTLZ problems were selected as benchmark test problems, and each problem contained five different target numbers: 3, 5, 8, 10, and 15. To ensure the fairness of the algorithm, the population size of the test problem was uniformly set to 100, the maximum number of evaluations maxFE was 300000, and all algorithms were independently run 30 times on each test problem.

[0064] Figure 3 The figure shows the number of best results of IGD and HV indicators obtained by the algorithms MOEA / Dlearning, ANSGA-III, IDBEA, MOEA / DD, RVEA and C-TAEA in the benchmark problems. Among them, MOEA / Dlearning obtained 11 best IGD results for all test problems, ANSGA-III and RVEA obtained 3 each, C-TAEA obtained 2, MOEA / DD obtained 1, and IDBEA did not obtain the best IGD result. Regarding the HV indicator, MOEA / Dlearning obtained 11 best HV results, C-TAEA obtained 4, IDBEA and RVEA obtained 2 each, MOEAD-D obtained 1, and ANSGA-III did not obtain the best HV result. Compared with other algorithms, MOEA / Dlearning is more effective in solving benchmark problems, and the solutions obtained have good convergence, diversity and distribution, while other compared algorithms perform relatively poorly on many benchmark problems.

[0065] At the same time, three practical problems are selected to verify the performance of the proposed MOEA / Dlearning, namely, the automobile side collision design problem, the water resources management problem and the bulk carrier design problem. Among them, the automobile side collision design problem contains 3 objectives, 7 decision variables and 10 constraints; the water resources management problem contains 5 objectives, 3 decision variables and 7 constraints; the bulk carrier design problem contains 3 objectives, 6 decision variables and 9 constraints.

[0066] The maximum number of evaluations maxFE for the three problems is set to 200000, and each problem is run independently 30 times. Table 1 shows the mean and standard deviation of the IGD index of MOEA / Dlearning and the five comparison algorithms on three practical problems. The black data represent the optimal value. Under the Wilcasson measure, "+", "-" and "=" respectively indicate that the comparison algorithm obtains better, worse and similar results compared with MOEA / Dlearning.

[0067] Table 1 Means and standard deviations of IGD indicators for three practical problems

[0068]

[0069] As can be seen from Table 1, MOEA / Dlearning achieved two of the best IGD results in three practical problems, RVEA only achieved one, and the other four algorithms did not achieve the best IGD results. This shows that MOEA / Dlearning performs better than the other five compared algorithms on these three practical problems. The experiment not only verifies the effectiveness of MOEA / Dlearning in complex multi-objective problems, but also reflects its wide adaptability and competitiveness in practical applications. Compared with the other five advanced algorithms, MOEA / Dlearning shows higher efficiency and better solution quality when solving practical problems.

[0070] In summary, the evolutionary framework based on graph neural network and the derived MOEA / Dlearning algorithm for constrained multi-objective optimization problems in this patent have a good effect on solving CMOPs, verifying the feasibility of evolutionary algorithms for solving such problems. According to the performance of the algorithm, it can be found that the evolutionary idea based on graph neural network has significant advantages and provides an optional tool for solving unknown constrained multi-objective optimization problems.

Claims

1. A constrained multi-objective intelligent optimization framework guided by a graph neural network, characterized in that: The characteristics of unknown constrained multi-objective optimization problems can be learned, and arbitrary constrained multi-objective evolutionary algorithms can be embedded to improve the algorithm's search performance and trade-off ability. The specific steps are as follows: Step 1. Dataset construction; Step 2. Graph neural network model training; Step 3. Initialize the population P, the neighbor individual set B, and the weight vector W; Step 4. Randomly select two parent individuals from B for crossover mutation to obtain the offspring individual offspring; Step 5. Environmental selection obtains a new population P; Step 6. Use the old population to train the model; Step 7. Determine the validity of the weight vector; Step 8. Determine whether the weight vector should be updated based on historical performance; Step 9. Use cluster centers to generate weight vector W and update B; Step 10. Repeat steps 4-9 until the termination condition is met.

2. A graph neural network guided constrained multi-objective intelligent optimization framework as claimed in claim 1, characterized in that: In step 1, a training data set is constructed, and 10 advanced constrained multi-objective optimization algorithms are randomly selected to solve 31 constrained multi-objective test problems to obtain sufficient training data samples. The steps are as follows: Step 1. Initialize the decision vector x, calculate the objective function value F(x) and the constraint violation degree CV(x); Step 2. Read vector (x, F(x), CV(x)) from the population individual; Step 3. Determine whether the individual is a dominant solution; Step 4. If the individual is the dominant solution, mark the individual as 1 and calculate the hypervolume value of the individual; Step 5. If the individual is a non-dominated solution, mark the individual as 0; Step 6. Aggregate the CVs of all non-dominated solutions, and mark the one with the smallest average CV as 0, otherwise it is marked as 1; Step 7. Save the (x, F(x), CV(x)) data and labels of all individuals.

3. A graph neural network guided constrained multi-objective intelligent optimization framework as claimed in claim 1, characterized in that: In step 2, the data set constructed in claim 2 is used to train the graph neural network, and a constrained multi-objective optimization problem learning model based on the graph neural network is proposed. The specific steps are as follows: Step 1. Initialize the head index H of the edge and the tail index T of the edge; Step 2. Extract H and node data (x, F(x), CV(x)) from the dataset; Step 3. Take the first 4% of the closest individual indexes as T and sort them in ascending order; Step 4. Each H and T form an edge E; Step 5. The node data (x, F(x), CV(x)) and E form a graph; Step 6. Perform graph classification training and point classification training.

4. A graph neural network guided constrained multi-objective intelligent optimization framework as claimed in claim 3, characterized in that: In step 2, a learning model for constrained multi-objective optimization problems based on graph neural networks is proposed, and a lightweight graph neural network model GraphSAGE is selected. The model uses node sampling and neighbor aggregation to learn node features and reduce the complexity of calculating the entire graph. In GraphSAGE, a pooling operation is applied to aggregate the function of neighbor features. The specific formula is as follows: In the formula, is the feature vector of node v in layer l; σ represents the activation function; W l is the weight matrix of the lth layer; max represents the maximum pooling operation; W pool and b are the weights and biases of the pooling layer; N(v) is the set of neighbors of node v.

5. The constrained multi-objective intelligent optimization framework guided by a graph neural network as claimed in claim 1, characterized in that: In step 7, a new weight vector discrimination criterion based on graph neural network is proposed, and the steps are as follows: Step 1. Calculate the distance between each weight vector W and its neighboring individuals to obtain the associated individuals; Step 2. The weight vector is judged according to the number of individuals associated with each weight vector and the number of excellent individuals; Step 3. If the weight vector has associated individuals and the number of excellent individuals is greater than 1, the weight vector is defined as a strong weight vector, and the class value Class is set to 4; Step 4. If the weight vector has associated individuals and there is only one excellent individual, the weight vector is defined as a standard weight vector, and the class value Class is set to 3; Step 5. If the weight vector has no associated individual, it is defined as a difference weight vector, and the class value Class is set to 1; Step 6. Calculate the weight vector level value based on the historical Class value for weight vector update.

6. A graph neural network guided constrained multi-objective intelligent optimization framework as claimed in claim 5, characterized in that: In step 6, a new method for calculating the weight vector rank is proposed, which records the historical Class values ​​of the weight vectors of three consecutive generations, and calculates the rank value of the weight vector using the following formula: rank(λ)=Class t (λ)+r·Class t-1 (λ) +r 2 ·Class t-2 (λ)+...+r t-1 ·Class t-T+1 (λ), In the formula, rank(λ) represents the rank value of the weight vector λ, Class t (λ) represents the category value of the t-th generation weight vector, r represents the learning rate, and the rank value can reflect the performance change trend of the weight vector and is an important indicator for evaluating its future potential.

7. A graph neural network guided constrained multi-objective intelligent optimization framework as claimed in claim 1, characterized in that: In step 8, an updating mechanism based on the weight vector discrimination criterion is proposed, and the steps are as follows: Step 1. Calculate the deletion probability p; Step 2. Delete the weight vector with the smallest rank value according to probability p; Step 3. Delete the historical Class value and its associated individuals corresponding to the weight vector; Step 4. Update population P; Step 5. Cluster the remaining weight vectors; Step 6. Generate a new weight vector using the cluster center of the largest cluster obtained by clustering.

8. A graph neural network guided constrained multi-objective intelligent optimization framework as claimed in claim 7, characterized in that: In step 1, a new probability calculation formula is designed, and the specific formula is as follows: p=(-g 2 +maxFE·g) / maxFE 2 +0.1, Where p represents the probability, g represents the current number of iterations, and maxFX represents the maximum number of evaluations.

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