Clustering method for final elemental composition of molten steel based on support vector machine

Abstract

This invention provides a clustering method for the final elemental content of molten steel based on support vector machines, relating to the field of converter steelmaking process technology. For the clustering problem of the final elemental content of different batches of molten steel during converter steelmaking, this invention uses a quadratic surface as the cluster center to characterize each cluster. The selection of the cluster center is based on the idea of ​​kernelless quadratic least squares twin support vector machines, which can effectively characterize even complex distributions of the final elemental content of molten steel samples in the feature space. This invention can generate nonlinear quadratic clustering surfaces without introducing a kernel function. For molten steel datasets with typically large sample sizes, this saves a significant amount of time in selecting kernel functions and parameter tuning, enabling better control of steel quality.

Description

Technical Field

[0001] This invention relates to the field of converter steelmaking process technology, and in particular to a clustering method for the elemental content of molten steel at the final stage based on support vector machine. Background Technology

[0002] Steel is one of the most important industrial materials, and its composition and properties have been extensively studied. Controlling the final composition of molten steel in the converter steelmaking process is a crucial and necessary task for controlling the quality of steel products; this process is a commonly used method in steel production. Final composition elements include carbon (C), phosphorus (P), sulfur (S), and manganese (Mn), which should be accurately sensed and controlled at the end of the converter steelmaking process. During steel production, these elements enter the molten steel along with the raw materials. Some of these elements remain to improve certain properties of the steel product. For example, adding manganese helps increase the tensile strength of steel. However, some residual elements may be harmful to the steel product and negatively impact its quality. Therefore, accurate analysis and control of the final composition elements of molten steel during converter steelmaking has attracted widespread attention in the field of metallurgical research. An important task of final composition element analysis is clustering the element content in the molten steel. Clustering refers to grouping similar data points into one category and different data points into different categories. The content of various elements in molten steel may vary, thus requiring clustering to better understand their relationships and effects. Cluster analysis allows steels with similar properties to be grouped into the same category, revealing patterns and correlations between different elemental contents. This helps determine the quality and performance of steel and provides crucial guidance for metallurgical engineers and steel manufacturers. Furthermore, cluster analysis of the final elemental content of molten steel can help companies identify potential anomalies or abnormal steel—steel deviating from the norm. These abnormal steels may be caused by abnormalities in the production process or other factors, resulting in significant differences in elemental content compared to normal steel. By identifying and analyzing these abnormal steel samples, company personnel can detect potential quality problems early and take appropriate measures for adjustment and improvement. Cluster analysis of molten steel elemental content also provides steel manufacturing companies with effective quality control tools. By observing and monitoring the compositional characteristics of different clusters, it can be ensured that each batch of molten steel meets specific quality requirements and standards. This helps improve the consistency and reliability of steel products, meet customer needs, and enhance the company's competitiveness. Therefore, cluster analysis of the final elemental content of molten steel has significant practical application value, providing the steel industry with key information on quality control, product improvement, and process optimization.

[0003] Clustering is an important task in machine learning and has been widely applied in practical problems, including anomaly detection, customer segmentation, and credit risk assessment. The goal of clustering is to explore the underlying structure of data points, dividing sample points in a dataset into different groups or clusters, maximizing the similarity of data points within the same cluster and minimizing the similarity between data points in different clusters. Common clustering algorithms include partitioning clustering, hierarchical clustering, density clustering, and model-based clustering. Partitioning clustering methods divide the dataset into disjoint clusters and iteratively optimize the cluster division to minimize intra-cluster differences and maximize inter-cluster differences. Hierarchical clustering methods build a hierarchical clustering structure by progressively merging or splitting clusters. Initially, each data point is treated as a separate cluster, and then the nearest clusters are merged based on similarity metrics until all data points are merged into one large cluster, or the data points are split into multiple clusters. Density clustering methods are based on the concepts of density reachability and density connectivity, dividing high-density regions into clusters and treating low-density regions as noise or boundary points. By identifying core objects and adjacent density-reachable objects, clusters with different density levels can be formed. Model-based clustering methods assume that datasets are generated by specific probability distributions and utilize these distributions and parameters for clustering. These algorithms, based on different principles and methods, can be applied to different data types and problem scenarios. Through clustering, we can extract valuable information from large amounts of data, discover potential correlations between data points, and better solve practical problems.

[0004] The paper "Prediction of endpoint phosphorus content of molten steel in bofusing weighted k-means and gmdh neural network. Journal of Iron and Steel Research, International, 19(1):11–16, 2012" proposes a method to perceive the phosphorus content in the endpoint composition of molten steel. This method generates clusters with homogeneous data based on weighted k-means clustering and uses the entropy weight method (EWM) to calculate the weights of factors affecting the target. Based on the clustering results, it further assists in the subsequent prediction process. The paper "Twin support vector machine for clustering. IEEE Transactions on Neural Networks and Learning Systems, 26(10):2583–2588, 2015" proposes a planar clustering algorithm, namely Twin Support Vector Clustering (TWSVC). TWSVC constructs a clustering model based on the idea of ​​twin support vector machine. The implementation method is to iteratively update the cluster center plane until the convergence condition is met to obtain the final clustering result. When the data distribution exhibits a certain degree of nonlinearity, introducing a kernel function can yield a nonlinear TWSVC, which can effectively address nonlinear clustering problems. The clustering algorithms proposed in the aforementioned literature divide clusters using points and planes as cluster centers. However, considering the potential nonlinearity in the distribution of elemental composition at the endpoint of molten steel from the same batch during converter steelmaking, direct clustering in the original space is unlikely to yield good results. Therefore, a kernel function is needed to map the samples to a higher-dimensional feature space for clustering. However, the introduction of kernel functions has several drawbacks. First, there are no fixed rules for choosing a kernel function, and the clustering accuracy is highly dependent on its selection. Second, kernel functions contain parameters, and the clustering accuracy is highly dependent on these parameters, requiring significant time and effort for parameter tuning. Summary of the Invention

[0005] The technical problem to be solved by the present invention is to address the shortcomings of the prior art by providing a clustering method for the elemental content of the final components of molten steel based on support vector machine. This method clusters different batches of molten steel according to the elemental content of the final components of molten steel during the converter steelmaking process, so as to better control the quality of steel.

[0006] To solve the above-mentioned technical problems, the technical solution adopted by the present invention is as follows:

[0007] A clustering method for the final elemental content of molten steel based on support vector machine includes the following steps:

[0008] Step 1: Data preprocessing, scaling the values ​​corresponding to the element content in the steel dataset to between 0 and 1;

[0009] Step 2: Initialize sample labels, that is, give each batch of molten steel an initial label;

[0010] Step 3: Update cluster center; Select a cluster center for the clusters of molten steel with the same label, select a quadratic surface as the cluster center to characterize the same batch of molten steel, and obtain the cluster center based on the idea of ​​kernelless quadratic least squares twin support vector machine;

[0011] Step 4: Update the labels of all batches of molten steel based on the similarity between the samples and the centers of each cluster;

[0012] Step 5: Repeat steps 3 and 4 until the iteration stopping condition is met.

[0013] Furthermore, the specific method of step 1 is as follows:

[0014] For the input molten steel endpoint composition dataset X, the content of each element in the endpoint composition is scaled to between 0 and 1 according to the following rules:

[0015]

[0016] in, It is the magnitude of the original elemental content of the molten steel, x min and x max These represent the minimum and maximum values ​​of the element content in all batches of molten steel, respectively. It is the value of the element after scaling.

[0017] Furthermore, the specific method for step 2 is as follows:

[0018] Given the number of molten steel categories k, each batch of molten steel samples is given an initial label using the nearest neighbor graph NNG or random initialization, with the label range being {1,2,…,k}.

[0019] Furthermore, the specific method for step 3 is as follows:

[0020] Step 3.1: Define the cluster center quadratic surface corresponding to the cluster with label j:

[0021]

[0022] in, b j =[b1,b2,...,b n ]∈R n ,q j∈R represents the relevant parameters of the quadratic surface, and x represents a column vector composed of different element contents at the final composition of a batch of molten steel, with a dimension of n;

[0023] Step 3.2: Based on the idea of ​​kernelless quadratic least squares twin support vector machine, establish a mathematical model to solve for the cluster centers. For a cluster with label j, assume m j batches of molten steel Belonging to this cluster, while the remaining batches of molten steel Belonging to the remaining clusters, where m j Let represent the number of molten steel samples in the cluster labeled j, and m represent the total number of molten steel samples; the quadratic surface at the cluster center is determined by the following optimization problem:

[0024]

[0025]

[0026] Among them, the decision variable is the relevant parameter W of the quadratic surface. j b j q j The slack variable is ξ i c1 and c2 represent penalty coefficients, x i A column vector representing the elemental content characteristics of the i-th batch of molten steel samples;

[0027] Step 3.3: Solving the model;

[0028] Considering that the model contains matrix variables and their F-norms, we first define the following expression to simplify the model:

[0029] w j =vec(W j )=[w 11 ,w 12 ,...,w 1n ,w 22 ,w 23 ...,w 2n ,...,w nn ] T

[0030] b j =[b1,b2,...,b n ] T

[0031]

[0032]

[0033]

[0034] C = [I n(n+1) / 2 ,0 n(n+1) / 2×n ]

[0035] H = 2I n(n+1) / 2 -diag(vec(I n ))

[0036] M = C T HC

[0037] Among them, I n The first character represents an n-order identity matrix, and the second character represents a diagonal matrix. This represents the content of the nth element in the final composition of the i-th batch of molten steel samples;

[0038] Then we get:

[0039]

[0040]

[0041] The model then simplifies to the following form:

[0042]

[0043] st|s i T z j +q j |=1-ξ i i = m j +1,...m

[0044] Further definition The model can then be further simplified to the following form:

[0045]

[0046]

[0047] Where e represents a unit column vector of appropriate dimensions;

[0048] The above model is solved using the concave-convex program algorithm, which transforms the model into a series of quadratic programming subproblems that are iteratively solved to obtain a local optimum. The subproblems corresponding to the t-th iteration are as follows:

[0049]

[0050]

[0051] Where sign() represents the sign function;

[0052] definition Substituting the constraints into the objective function, we obtain the following unconstrained optimization problem:

[0053]

[0054] To each and Taking the partial derivative and setting it equal to 0, we obtain the solution to the optimization problem as follows:

[0055]

[0056] Where Q = [S j [e]

[0057] Given an initial Then, iterative solution. until Small enough to obtain the solution of the original model According to u j Derive the quadratic surface parameter W j b j q j .

[0058] Furthermore, in step 3.3, the initial Determined by the following model:

[0059]

[0060]

[0061] The above model is transformed into solving an eigenvalue problem, defined as follows: Pick For B j The eigenvector corresponding to the smallest eigenvalue is taken as...

[0062] Furthermore, the specific method for step 4 is as follows:

[0063] After obtaining the cluster centers of multiple clusters composed of all batches of molten steel samples, the molten steel samples are reassigned to the nearest cluster; the sample label update rules are as follows:

[0064]

[0065] Furthermore, the specific method of step 5 is as follows:

[0066] The molten steel samples are relabeled based on the updated cluster centers, and then the cluster centers are updated based on the relabeled samples. Steps 3 and 4 are repeated, alternating between updating the cluster centers and sample labels, until duplicate sample labels appear. The iteration stops, and the clustering results are output, dividing all batches of molten steel samples into multiple categories based on the content of the endpoint component elements.

[0067] The beneficial effects of adopting the above technical solution are as follows: The clustering method for the final element content of molten steel based on support vector machine provided by this invention, for the clustering problem of the final element content of different batches of molten steel in the converter steelmaking process, uses a quadratic surface as the cluster center to characterize each cluster. The selection of the cluster center is based on the idea of ​​kernelless quadratic least squares twin support vector machine. Even if the distribution of the final element content of molten steel samples in the feature space is relatively complex, the quadratic surface can still characterize it well. This invention can generate a nonlinear quadratic clustering surface without introducing a kernel function. For molten steel datasets with a typically large number of samples, this saves a lot of time in selecting kernel functions and parameter tuning. Attached Figure Description

[0068] Figure 1 A flowchart of the clustering method for the final element content of molten steel based on support vector machine provided in this embodiment of the invention;

[0069] Figure 2 This is a schematic diagram illustrating the geometric interpretation of the clustering model provided in an embodiment of the present invention. Detailed Implementation

[0070] The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and examples. The following examples are for illustrative purposes only and are not intended to limit the scope of the invention.

[0071] like Figure 1 As shown, the clustering method for the final elemental content of molten steel based on support vector machines includes the following steps:

[0072] Step 1: Data Preprocessing. The elemental content of the final component of molten steel varies significantly. For example, the percentage content of manganese (Mn) is usually much higher than that of sulfur (S). To treat the content of each element in the final component of molten steel fairly and to prevent the magnitude difference in elemental content from affecting the clustering results, the values ​​corresponding to the elemental content in the steel dataset are scaled between 0 and 1.

[0073] For the input molten steel endpoint composition dataset X, the content of each element in the endpoint composition is scaled to between 0 and 1 according to the following rules:

[0074]

[0075] in, It is the magnitude of the original elemental content of the molten steel, x min and x max These represent the minimum and maximum values ​​of the element content in all batches of molten steel, respectively. It is the value of the element after scaling.

[0076] Step 2: Initialize sample labels, assigning an initial label to each sample. Given the number of molten steel categories k, assign an initial label to each batch of molten steel samples using the nearest neighbor graph (NNG) or random initialization methods. The label range is {1,2,…,k}.

[0077] Step 3: Update the cluster center.

[0078] To select a cluster center for clusters of molten steel with the same label, considering the potential nonlinearity in the distribution of elemental composition at the endpoint of the same batch of molten steel, this method selects a quadratic surface as the cluster center to characterize the same batch of molten steel, and calculates the cluster center based on the idea of ​​a kernelless quadratic least squares twin support vector machine. The specific method is as follows:

[0079] Step 3.1: Define the cluster center quadratic surface corresponding to the cluster with label j:

[0080]

[0081] in, b j =[b1,b2,...,b n ]∈R n ,q j ∈R represents the relevant parameters of the quadratic surface, and x represents a column vector composed of different element contents at the end of a batch of molten steel, with a dimension of n.

[0082] Step 3.2: Based on the idea of ​​kernelless quadratic least squares twin support vector machine, establish a mathematical model to solve for the cluster centers. For a cluster with label j, assume m j batches of molten steel Belonging to this cluster, while the remaining batches of molten steel Belonging to the remaining clusters, where m j Let represent the number of molten steel samples in the cluster labeled j, and m represent the total number of molten steel samples; the quadratic surface at the cluster center is determined by the following optimization problem:

[0083]

[0084]

[0085] Among them, the decision variable is the relevant parameter W of the quadratic surface. j bj q j The slack variable is ξ i c1 and c2 represent penalty coefficients, x i A column vector representing the elemental content characteristics of the i-th batch of molten steel samples.

[0086] The model is explained as follows: Taking two dimensions as an example (i.e., assuming the number of elements in the final component of the molten steel sample is 2), the model considers both intra-class and inter-class information. The first term of the objective function indicates the distance f of the molten steel sample in the current cluster from the central plane. j (x) = 0 is close enough to achieve a high intra-class similarity, grouping molten steel samples with similar elemental contents into one class; the second term of the objective function is a penalty term, tolerating some samples violating the constraints; the last term of the objective function indicates f j (x) = 1 and f j The intervals between (x) = -1 should be as large as possible to separate different types of molten steel samples as much as possible, thereby reducing the possibility of incorrect clustering. Constraints require samples not in the current cluster to be at a specific distance from the central plane, aiming to obtain lower inter-class similarity and classify samples with significantly different elemental contents into different classes. Figure 2 This demonstrates the geometric interpretation of the proposed model.

[0087] Step 3.3: Solving the model.

[0088] Considering that the model contains matrix variables and their F-norms, we first define the following expression to simplify the model:

[0089] w j =vec(W j )=[w 11 ,w 12 ,...,w 1n ,w 22 ,w 23 ...,w 2n ,...,w nn ] T

[0090] b j =[b1,b2,...,b n ] T

[0091]

[0092]

[0093]

[0094] C = [I n(n+1) / 2 ,0 n(n+1) / 2×n ]

[0095] H = 2I n(n+1) / 2 -diag(vec(I n ))

[0096] M = C T HC

[0097] Among them, I n The first character represents an n-order identity matrix, and the second character represents a diagonal matrix. This represents the content of the nth element in the final composition of the i-th batch of molten steel samples.

[0098] Then we get:

[0099]

[0100]

[0101] The model then simplifies to the following form:

[0102]

[0103] st|s i T z j +q j |=1-ξ i i = m j +1,...m

[0104] Further definition The model can then be further simplified to the following form:

[0105]

[0106]

[0107] Where e represents a unit column vector of appropriate dimensions.

[0108] The Concave-Convex Programming (CCCP) algorithm is used to solve the above model, transforming it into a series of quadratic programming subproblems that are iteratively solved to obtain a local optimum. The subproblems corresponding to the t-th iteration are as follows:

[0109]

[0110]

[0111] Where sign() represents the sign function;

[0112] definition Substituting the constraints into the objective function, we obtain the following unconstrained optimization problem:

[0113]

[0114] To each and Taking the partial derivative and setting it equal to 0, we obtain the solution to the optimization problem as follows:

[0115]

[0116] Where Q = [S j [e]

[0117] Given an initial Then, iterative solution. until Small enough to obtain the solution of the original model According to u j Derive the quadratic surface parameter W j b j q j .

[0118] initial Determined by the following model:

[0119]

[0120]

[0121] The above model is transformed into solving an eigenvalue problem, defined as follows: Pick For B j The eigenvector corresponding to the smallest eigenvalue is taken as...

[0122] Step 4: Update sample labels based on the similarity between the sample and each cluster center. After obtaining the cluster centers of multiple clusters composed of all batches of molten steel samples, reassign the molten steel samples to the nearest cluster;

[0123] The sample label update rules are as follows:

[0124]

[0125] Step 5: Relabel the molten steel samples based on the updated cluster centers, and then update the cluster centers based on the relabeled samples. Repeat steps 3 and 4, alternating between updating cluster centers and sample labels, until duplicate sample labels appear. Stop the iteration, output the clustering results, and divide all batches of molten steel samples into multiple categories based on the content of the endpoint component elements.

[0126] This embodiment conducted experiments on a dataset of final elemental composition of molten steel provided by a steel company to verify the performance of the proposed method. The dataset contains 2003 samples, each with five features. Each sample represents a batch of molten steel. The percentage content of five elements in the molten steel was recorded as five features: sulfur (S), phosphorus (P), carbon (C), manganese (Mn), and silicon (Si).

[0127] To apply the method proposed in this embodiment to the clustering task of the final elemental composition of molten steel, the elbow method was first used to set the initial number of molten steel categories. The elbow method results showed that k=4 was the optimal initial number of molten steel categories. Since the dataset had no labels, the internal profile coefficient (SC) was used to evaluate the clustering performance. The SC value ranges from -1 to 1; a larger positive value indicates a more ideal clustering effect, with high density within each cluster and greater separation between clusters. Two planar clustering algorithms were selected as comparison models: Twin Support Vector Clustering (TWSVC) and Least Squares Twin Support Vector Clustering (LSTWSVC). For the nonlinear comparison model based on the kernel function, a Gaussian kernel function was chosen, defined as K(x1,x2)=exp{-μ||x1-x2|| 2}, where μ is the parameter in the Gaussian kernel function. All algorithms use the nearest neighbor graph method to initialize the labels of the molten steel samples. In machine learning tasks, grid search is widely used for parameter tuning, and this experiment also uses this method to select appropriate parameters for each algorithm. The parameters c in the model ∈ {2} -8 ,2 -7 ,...,2 10}, the parameter μ in the kernel function ∈ {2 -8 ,2 -7 ,...,2 7}

[0128] Table 3 shows the clustering results for all test methods, including SC and computation time.

[0129] Table 3 Clustering performance on the steel dataset

[0130]

[0131] As shown in Table 3, the proposed method achieved the highest SC value of 0.63. The results indicate that after clustering using the method proposed in this embodiment, the molten steel samples exhibit high density within their respective clusters and significant separation from samples in other clusters. It successfully groups molten steel samples with similar elemental contents into the same group and groups molten steel samples with significantly different elemental contents into different groups, resulting in good clustering results. As a clustering method that does not require a kernel function, the proposed method in this embodiment eliminates the workload of selecting an appropriate kernel function and adjusting hyperparameters. In terms of computation time, the proposed method is more efficient than the nonlinear LSTWSVC method and the nonlinear TWSVC method (running time was too long to record). In reality, the molten steel dataset has a relatively large number of samples; introducing a kernel function would result in a large kernel matrix, requiring more computation time during clustering. Furthermore, the proposed method achieves its clustering efficiency by solving a series of linear equations.

[0132] Table 4 shows the number of molten steel samples in each cluster after clustering using the method proposed in this embodiment, as well as the average value of the final component element content.

[0133] Table 4. Relevant information for each cluster after clustering the steel dataset.

[0134] cluster Sample size S(％) P(％) Si (%) C(％) Mn(%) Cluster 1 1686 0.01 0.01 0.09 0.09 0.35 Cluster 2 281 0.01 0.01 0.24 0.78 0.70 Cluster 3 36 0.01 0.01 0.88 0.13 1.34

[0135] Based on the observed results, the phosphorus and sulfur content was controlled at approximately 0.01%, which is a safe level that is harmless to steel products. Carbon, silicon, and manganese have a significant impact on the properties of steel. Carbon is the most important element determining the properties of steel. With increasing carbon content, tensile strength increases, but plasticity and impact toughness decrease. Steel with relatively high manganese content (>1.00%) has sufficient toughness, strength, and hardness, which can improve the hot working properties of the steel. The presence of silicon is crucial to the tensile strength of steel. The presence of silicon can also increase the elastic limit of steel. The average values ​​of the elemental composition at the endpoint of molten steel in each cluster indicate that cluster 1 consists of the most common steel products, as the content of alloying elements in the steel in this cluster is not significant. The steel in cluster 2 has relatively more carbon and manganese, but less silicon. This steel may have higher hardness and strength, but may be somewhat affected in terms of plasticity and impact toughness, while its elastic limit may also be lower. Furthermore, the steel in cluster 3 has more silicon and manganese, but less carbon. This type of steel may perform well in terms of plasticity, toughness, hardness, strength, and elastic limit, and has good hot working properties, but it may be slightly inferior in terms of tensile strength.

[0136] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention, and not to limit them; although the present invention has been described in detail with reference to the foregoing embodiments, those skilled in the art should understand that modifications can still be made to the technical solutions described in the foregoing embodiments, or equivalent substitutions can be made to some or all of the technical features therein; and these modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the scope defined by the claims of the present invention.

Claims

1. A clustering method for the final elemental content of molten steel based on support vector machine, characterized in that: The method includes the following steps: Step 1: Data preprocessing, scaling the values ​​corresponding to the element content in the steel dataset to between 0 and 1; Step 2: Initialize sample labels, that is, give each batch of molten steel an initial label; Step 3: Update cluster centers; select a cluster center for clusters of molten steel with the same label, choose a quadratic surface as the cluster center to characterize the same batch of molten steel, and obtain the cluster center based on the idea of ​​a kernelless quadratic least squares twin support vector machine; the specific method is as follows: Step 3.1: Define the cluster center quadratic surface corresponding to the cluster with label j: ； in, Representing the relevant parameters of a quadratic surface, A column vector representing the composition of different element contents at the final stage of a batch of molten steel, with dimension n; Step 3.2: Based on the idea of ​​kernelless quadratic least squares twin support vector machine, establish a mathematical model to solve for the cluster centers. For a cluster with label j, assume... batches of molten steel Belonging to this cluster, while the remaining batches of molten steel Belonging to the remaining clusters, among which, This represents the number of molten steel samples in the cluster labeled j. This represents the total number of molten steel samples; its cluster-centered quadratic surface is determined by the following optimization problem: ； ； Among them, the decision variables are the relevant parameters of the quadratic surface. , , slack variables are c1 and c2 represent penalty coefficients, x i A column vector representing the elemental content characteristics of the i-th batch of molten steel samples; Step 3.3: Solving the model; Considering that the model contains matrix variables and their F-norms, we first define the following expression to simplify the model: ； ； ； ； ； ； ； ； in, Represents an n-order identity matrix. Represents a diagonal matrix; This represents the content of the nth element in the final composition of the i-th batch of molten steel samples; Then we get: ； ； The model then simplifies to the following form: ； ； Further definition , , The model can then be further simplified to the following form: ； ； Where e represents a unit column vector of appropriate dimensions; The above model is solved using the concave-convex program algorithm, which transforms the model into a series of quadratic programming subproblems that are iteratively solved to obtain a local optimum. The subproblems corresponding to the t-th iteration are as follows: ； ； in, Represents a symbolic function; definition Substituting the constraints into the objective function, we obtain the following unconstrained optimization problem: ； To each and Taking the partial derivative and setting it equal to 0, we obtain the solution to the optimization problem as follows: ； in, , , , , ; Given an initial Iterative solution ,until Small enough to obtain the solution of the original model ;according to quadric surface parameters , ; Step 4: Update the labels of all batches of molten steel based on the similarity between the samples and the centers of each cluster; Step 5: Repeat steps 3 and 4 until the iteration stopping condition is met.

2. The clustering method for the final elemental content of molten steel based on support vector machine according to claim 1, characterized in that: The specific method for step 1 is as follows: For the input molten steel endpoint composition dataset X, the content of each element in the endpoint composition is scaled to between 0 and 1 according to the following rules: ； in, It refers to the original elemental content of the molten steel. and These represent the minimum and maximum values ​​of the element content in all batches of molten steel, respectively. It is the value of the element after scaling.

3. The clustering method for the final elemental content of molten steel based on support vector machine according to claim 1, characterized in that: The specific method for step 2 is as follows: Given the number of molten steel categories k, each batch of molten steel samples is given an initial label using the nearest neighbor graph NNG or random initialization, with the label range being {1,2,…,k}.

4. The clustering method for the final elemental content of molten steel based on support vector machine according to claim 1, characterized in that: In step 3.3, the initial Determined by the following model: ； ； The above model is transformed into solving an eigenvalue problem, defined as follows: ,Pick for The eigenvector corresponding to the smallest eigenvalue is taken as... .

5. The clustering method for the final elemental content of molten steel based on support vector machine according to claim 4, characterized in that: The specific method for step 4 is as follows: After obtaining the cluster centers of multiple clusters composed of all batches of molten steel samples, the molten steel samples are reassigned to the nearest cluster; the sample label update rules are as follows: 。 6. The clustering method for the final elemental content of molten steel based on support vector machine according to claim 1, characterized in that: The specific method for step 5 is as follows: The molten steel samples are relabeled based on the updated cluster centers, and then the cluster centers are updated based on the relabeled samples. Steps 3 and 4 are repeated, alternating between updating the cluster centers and sample labels, until duplicate sample labels appear. The iteration stops, and the clustering results are output, dividing all batches of molten steel samples into multiple categories based on the content of the endpoint component elements.

Images