Cut plane selection method and system in mixed integer programming

Abstract

The invention discloses a cut plane selection method and system in mixed integer programming, and the method comprises the steps: carrying out the coding to obtain the global structure information of a whole branch and bound tree, inputting the global structure information into a multi-layer graph neural network, and enabling the graph neural network to carry out the local and global connection of a node neighborhood, dynamically calculating the feature representation of each variable node, constraint node and cut plane node, and outputting a feature vector representing the global state of the whole branch and bound tree; and carrying out joint modeling and inference on the candidate cutting plane set based on the feature vector, determining an optimal cutting plane selection strategy by adopting a reinforcement learning mode, and carrying out cutting plane selection based on the optimal cutting plane selection strategy. According to the method, the accumulation and cooperative influence of the cut plane among different nodes in the whole solving process can be accurately captured, the overall collaboration of cut plane allocation is remarkably improved, and the computing resources and time overhead required by high-dimensional information coding and model reasoning are reduced.

Description

Technical Field

[0001] This invention relates to the field of computer artificial intelligence technology, and in particular to a method and system for selecting cutting planes in mixed integer programming. Background Technology

[0002] Mixed-integer programming (MIP) is a class of mathematical optimization models widely used in numerous industrial and engineering fields, such as production scheduling, supply chain management, and network optimization. Current mainstream MIP solvers (such as Gurobi and SCIP) generally employ branch-and-bound or branch-and-cut (B&C) algorithms. To accelerate the solution process, these algorithms automatically generate and add cutting planes during the solution process to tighten the relaxation space, reduce branch nodes, and decrease computation time.

[0003] Existing technologies mainly employ the following two types of cutting plane selection strategies:

[0004] (1) Selection of cutting planes based on artificial design heuristics:

[0005] Mixed-integer programming solvers, such as SCIP, employ manually defined scoring functions to filter a large number of candidate cut planes generated in each round, ultimately selecting a subset of cut planes to update the model. While this method is simple, it struggles to adapt to different problem structures and often fails to fully utilize global information, resulting in limited cut selection efficiency. This is particularly problematic in large-scale instances, where slow solution speeds and a large number of nodes may occur.

[0006] (2) Local cutting plane selection based on machine learning:

[0007] Machine learning-based local cut plane selection strategies typically model individual branch nodes, using neural networks to score and filter cut planes at nodes, or employing reinforcement learning frameworks to optimize cut selection actions. However, these strategies only utilize local information from the current node, failing to consider the entire branch-bound tree structure and cut history. This leads to a lack of global coordination in cut plane selection between nodes, resulting in technical bottlenecks such as cut plane redundancy, insufficient information utilization, and inadequate generalization ability for large-scale problems.

[0008] In summary, the existing cutting plane selection methods have the following main technical problems:

[0009] (1) Based on local node information, it is impossible to coordinate the allocation of cutting planes globally, which limits the optimization efficiency in solving complex large-scale MIP problems.

[0010] (2) Local selection strategies cannot track the historical role of the cutting plane in the entire solution process, making it difficult to achieve collaborative optimization between cutting planes, resulting in some cutting decisions being repeated, invalid, or even weakening the solution performance.

[0011] (3) When the problem size increases, the solution speed of traditional heuristic rules and local learning methods decreases significantly, while the number of nodes and the consumption of computing resources increase significantly, making it difficult to adapt to the needs of high efficiency and intelligent optimization in actual industrial scenarios.

[0012] In view of this, the present invention is hereby proposed. Summary of the Invention

[0013] The purpose of this invention is to provide a cutting plane selection method and system in mixed integer programming, so as to reduce the computational resources and time overhead required for high-dimensional information encoding and model inference, and solve the problems existing in the prior art.

[0014] The objective of this invention is achieved through the following technical solution:

[0015] A method for choosing a cutting plane in mixed-integer programming includes:

[0016] The global structural information of the entire branch and bounding tree is obtained by encoding and input into a multi-layer graph neural network. The graph neural network uses the local and global connections of the node neighborhood to dynamically calculate the feature representation of each variable node, constraint node, and cut plane node, and outputs a feature vector representing the global state of the entire branch and bounding tree.

[0017] Based on the feature vectors, a set of multiple candidate cut plane nodes is input into a neural network with sequential modeling capabilities for joint modeling and inference, so as to simultaneously characterize the interaction, sequential dependence and global collaborative optimization effect between cut planes. Subsequently, a reinforcement learning framework is introduced on top of joint modeling and inference to learn the optimal cut plane selection strategy and implement the cut plane selection accordingly.

[0018] The encoding process for obtaining the global structural information of the entire branch-bound tree includes:

[0019] The entire branch and bound tree is encoded as a bipartite graph containing variable nodes, original constraint nodes, historical cut plane nodes, and candidate cut plane nodes. The variable nodes represent the decision variables of the mixed integer programming problem, the original constraint nodes describe the constraints of the original mathematical model, the historical cut plane nodes record all added cut planes and their attributes representing the generation time and usage effect, and the candidate cut plane nodes represent the set of cut planes to be decided. The nodes are connected by directed edges that encode various structural relationships and features.

[0020] The feature representation of the variable node includes: upper and lower bounds of the variable, coefficients of the objective function, and type; the feature representation of the original constraint node includes: left-hand side of the constraint, right-hand side of the constraint, coefficient sparsity, and type; the feature representation of the historical cutting plane node includes: historical addition time, cutting plane effect, and parallelism with the objective function.

[0021] The process of encoding to obtain the global structural information of the entire branch-bound tree includes:

[0022] The search tree is parsed to extract path and branch information, and the set of cut planes is parsed to distinguish between historical cuts and candidate cuts;

[0023] Variable nodes, original constraint nodes, historical cut plane nodes and candidate cut plane nodes are constructed sequentially, and connections between variables and various constraint nodes are established. The coefficients on the edges are written, and branch information is added to the connections of the historical cut planes.

[0024] Global structural information is obtained by encoding the edge features of the branch constraints.

[0025] The joint modeling and inference are implemented through the input transformer module using pooling operations, and the process of determining the optimal cutting plane selection strategy using reinforcement learning includes:

[0026] Using the overall features output by the graph neural network and the Transformer module as the current state, and taking the binary selection and sequential sorting of candidate cut planes as actions, an immediate reward function and a termination reward function are used as reward functions. The immediate reward is improved based on the optimal solution gap, and the termination reward is set based on the solution time difference with the mixed integer programming solver. A reinforcement learning algorithm is used to jointly optimize the parameters of the policy network and the value network, and interacts with the actual solver to obtain the optimal cut plane selection strategy.

[0027] The process of selecting the cutting plane based on the optimal cutting plane selection strategy includes:

[0028] Upon entering a branch node, the system automatically constructs a global graph structure of the current node and its historical states, and dynamically calculates the feature representations of all variable nodes, constraint nodes, added cut plane nodes, and candidate cut plane nodes.

[0029] Based on the calculated feature representation, the optimal cutting plane selection strategy is used to automatically decide the cutting plane to be selected and the order of addition at each node.

[0030] A cutting plane selection system in mixed integer programming includes:

[0031] The feature vector acquisition unit is used to encode and obtain the global structural information of the entire branch and bound tree, and input it into the multi-layer graph neural network. The graph neural network uses the local and global connections of the node neighborhood to dynamically calculate the feature representation of each variable node, constraint node, and cut plane node, and outputs a feature vector representing the global state of the entire branch and bound tree.

[0032] The optimal cut plane selection strategy determination unit is used to jointly model and infer the candidate cut plane set composed of multiple candidate cut plane nodes based on the feature vector, so as to obtain the interaction, order dependency and global cut plane cooperative optimization effect between cut planes; and to determine the optimal cut plane selection strategy by using reinforcement learning based on the results of the joint modeling and inference.

[0033] The cutting plane selection unit is used to select the cutting plane based on the optimal cutting plane selection strategy.

[0034] The process of encoding the global structural information of the entire branch and bound tree in the feature vector acquisition unit includes:

[0035] The entire branch and bound tree is encoded as a bipartite graph containing variable nodes, original constraint nodes, historical cut plane nodes, and candidate cut plane nodes. The variable nodes represent the decision variables of the mixed integer programming problem, the original constraint nodes describe the constraints of the original mathematical model, the historical cut plane nodes record all added cut planes and their attributes representing the generation time and usage effect, and the candidate cut plane nodes represent the set of cut planes to be decided. The nodes are connected by directed edges that encode various structural relationships and features.

[0036] The processing steps performed by the optimal cutting plane selection strategy determination unit include:

[0037] Using the overall features output by the graph neural network and the Transformer module as the current state, and taking the binary selection and sequential sorting of candidate cut planes as actions, an immediate reward function and a termination reward function are used as reward functions. The immediate reward is based on the gap improvement setting, and the termination reward is based on the solution time difference with the baseline solver. A reinforcement learning algorithm is used to jointly optimize the parameters of the policy network and the value network, and interacts with the actual MIP solver to obtain the optimal cut plane selection strategy.

[0038] The process of selecting a cutting plane based on the optimal cutting plane selection strategy in the cutting plane selection unit includes:

[0039] Upon entering a branch node, the system automatically constructs a global graph structure of the current node and its historical states, and dynamically calculates the feature representations of all variable nodes, constraint nodes, added cut plane nodes, and candidate cut plane nodes.

[0040] Based on the feature representation obtained by dynamic calculation, the optimal cutting plane selection strategy is used to automatically decide the cutting plane to be selected and the order of addition at each node.

[0041] Compared with existing technologies, the cutting plane selection method and system in mixed integer programming provided by this invention can accurately capture the cumulative and synergistic effects of cutting planes among different nodes throughout the solution process, significantly improve the overall synergy of cutting plane allocation, and effectively avoid common local selection problems such as repeated cutting decisions and low cutting utilization. Furthermore, it can dynamically adjust the selection and arrangement of cutting planes, adaptively make optimal decisions for different problem instances and different solution stages, and significantly reduce the computational resources and time overhead required for high-dimensional information encoding and model inference. Attached Figure Description

[0042] To more clearly illustrate the technical solutions of the embodiments of the present invention, the drawings used in the following description of the embodiments will be briefly introduced. 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.

[0043] Figure 1 This is a schematic flowchart illustrating the representation of a search tree as a bipartite graph, provided as an embodiment of the present invention.

[0044] Figure 2 This is a schematic diagram of the process for selecting the cutting plane provided in an embodiment of the present invention. Detailed Implementation

[0045] The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the specific content of the present invention. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments, which do not constitute a limitation of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the protection scope of the present invention.

[0046] First, the following explanations are provided for the terms that may be used in this article:

[0047] The term "and / or" means that either or both can be achieved simultaneously. For example, X and / or Y means that it includes both "X" or "Y" as well as the three cases of "X and Y".

[0048] The terms "comprising," "including," "containing," "having," or other similar semantic descriptions should be interpreted as non-exclusive inclusion. For example, including a technical feature element (such as raw material, component, ingredient, carrier, dosage form, material, size, part, component, mechanism, device, step, process, method, reaction conditions, processing conditions, parameter, algorithm, signal, data, product or article of manufacture, etc.) should be interpreted as including not only the expressly listed technical feature element, but also other technical feature elements that are not expressly listed and are well-known in the art.

[0049] The term "composed of" excludes any technical features not expressly listed. When used in a claim, it closes the claim to exclude all technical features other than those expressly listed, except for associated conventional impurities. If the term appears only in a clause of a claim, it limits the claim to the elements expressly listed in that clause; elements recited in other clauses are not excluded from the overall claim.

[0050] Unless otherwise explicitly specified or limited, the terms "installation," "connection," "linking," and "fixing," etc., should be interpreted broadly. For example, they can refer to fixed connections, detachable connections, or integral connections; they can refer to mechanical connections or electrical connections; they can refer to direct connections or indirect connections through an intermediate medium; and they can refer to the internal connection between two components. Those skilled in the art can understand the specific meaning of the above terms in this document according to the specific circumstances.

[0051] When concentration, temperature, pressure, size, or other parameters are expressed as numerical ranges, such ranges should be understood to specifically disclose all ranges formed by any pairing of upper limits, lower limits, or preferred values ​​within that range, regardless of whether the range is explicitly stated; for example, if the numerical range "2 to 8" is stated, then that range should be interpreted to include ranges such as "2 to 7", "2 to 6", "5 to 7", "3 to 4 and 6 to 7", "3 to 5 and 7", "2 and 5 to 7", etc. Unless otherwise stated, the numerical ranges described herein include both their endpoints and all integers and fractions within that range.

[0052] The terms “center,” “longitudinal,” “lateral,” “length,” “width,” “thickness,” “upper,” “lower,” “front,” “back,” “left,” “right,” “vertical,” “horizontal,” “top,” “bottom,” “inner,” “outer,” “clockwise,” and “counterclockwise” indicate the orientation or positional relationship based on the orientation or positional relationship shown in the accompanying drawings. They are used only for the convenience and simplification of description and do not imply that the device or component referred to must have a specific orientation, or be constructed and operated in a specific orientation. Therefore, they should not be construed as limitations on this document.

[0053] This invention provides a method and apparatus for global cut plane selection in mixed integer programming based on graph neural networks and reinforcement learning. It realizes global allocation and dynamic selection of cut planes in large-scale mixed integer programming (MIP) problems. It can realize the transformation of cut plane selection from local heuristic to global adaptive through global branch and bound tree structure modeling, high-dimensional feature extraction of graph neural networks and reinforcement learning strategy optimization.

[0054] Specifically, the core technical means in the specific implementation process of the embodiments of the present invention mainly include:

[0055] (1) Cutting plane selection method based on global information

[0056] This invention employs a global cut plane selection method that differs from traditional local cut plane selection. By uniformly modeling the entire branch-and-cut tree, it can comprehensively utilize the global structural information between branch nodes and the historical records of cut planes to achieve optimized allocation of cut planes across all nodes, thereby significantly improving the overall solution efficiency of mixed integer programming problems.

[0057] (2) Graph structure representation of branch-bound tree-cut plane bidirectional coding

[0058] This invention abstracts the relationship between the branch-bound tree and the cut plane into a bipartite graph structure, uniformly encodes the original constraints, cut plane constraints, and branch constraints of each node, and combines node features, cut plane features, constraint features, and their temporal and relational relationships to effectively capture global optimization information and achieve efficient state representation under complex MIP instances.

[0059] (3) Adaptive cutting plane selection mechanism based on graph neural network and reinforcement learning

[0060] This invention introduces a Graph Neural Network (GNN) to extract features from the aforementioned graph structure and designs a cutting plane selection algorithm based on reinforcement learning to achieve joint optimization of cutting plane combination selection and order ranking. This processing mechanism can not only dynamically adapt to different problem structures, but also achieve optimal long-term performance in cutting plane combination decisions.

[0061] The above-described technical implementation methods support efficient updating and training mechanisms for large-scale MIP instances. Specifically, the cutting plane selection method adopted in this embodiment supports rapid dynamic structure updates of branch and bound trees, efficiently and accurately embedding new cutting plane constraints and accelerating global solution computation. This effectively reduces computational overhead and training sample requirements for large-scale problems, improving model generalization ability and industrial deployment adaptability. Furthermore, the above-described technical implementation methods can significantly improve MIP solution efficiency and resource utilization.

[0062] For ease of understanding, the specific implementation methods of the embodiments of the present invention will be described in detail below.

[0063] In embodiments of the present invention, such as Figure 1 and Figure 2 As shown, the corresponding processing steps may include:

[0064] (1) First, each node of the branch and bound tree of the MIP problem is described in a structured manner to encode the global structure information of the entire branch and bound tree;

[0065] In this embodiment of the invention, the branch and bound search tree and cut plane information are uniformly modeled as a bipartite graph, where one side is the variable node and the other side is the constraint node;

[0066] The variable nodes cover all variables in the problem, and their node characteristics are the coefficients of the variable in the objective function.

[0067] The constraint nodes are further divided into three types: original constraint nodes, historical cut plane nodes, and candidate cut plane nodes; the common features of the three types of constraint nodes are the comparison sign (less than or equal to, equal to, or greater than or equal to) and the right-hand side value; among them, historical cut plane nodes additionally record the timestamp (the round or time of addition) and the cut plane effect, which are used to characterize their generation time and usage effect;

[0068] The bipartite graph establishes connections only between variable nodes and constraint nodes. For each variable connected to a constraint, the coefficient of that variable within that constraint is recorded as an edge feature. For each variable connected to a historical cut plane, in addition to recording the coefficient, branch information of the search tree is appended, namely the comparison sign and right-hand side value of the branch constraint on that variable. For each variable connected to a candidate cut plane, the coefficient of that variable within that candidate cut is recorded. All the above node and edge features are uniformly normalized and indexed for consistency before output to facilitate subsequent graph representation learning and cut plane selection.

[0069] like Figure 1 As shown, the corresponding processing flow may include the following steps:

[0070] (11) Perform structural parsing on the search tree to extract path and branch information;

[0071] (12) Analyze the set of cutting planes to distinguish between historical cuts and candidate cuts;

[0072] (13) Construct nodes for variables and constraints, construct nodes for cut planes, construct edges for variables and constraints, and construct edges for variables and cut planes;

[0073] Specifically, variable nodes (features of objective function coefficients), original constraint nodes (features of comparison signs and right-hand side values), historical cut plane nodes (features of comparison signs, right-hand side values, timestamps, and cut plane effects), and candidate cut plane nodes (features of comparison signs and right-hand side values) are constructed sequentially. Based on this, connections between variables and the above-mentioned constraint nodes are established, coefficients on the edges are written according to the aforementioned rules, and branch information is added to the connections involving historical cut planes.

[0074] (14) Obtain global structural information by encoding the edge features of the branch constraints, that is, complete the feature normalization and output of the graph.

[0075] (2) Input the global structure information into the multi-layer graph neural network, and extract features by the multi-layer graph neural network (such as the convolutional neural network). That is, the graph neural network uses the local and global connections of the node neighborhood to dynamically calculate the feature representation of each variable node, constraint node, and cut plane node, and outputs the feature vector representing the global state of the entire branch and bound tree.

[0076] Specifically, after completing the structured description of each node of the branch and bound tree for the MIP problem, i.e., after modeling and obtaining the corresponding global structure, such as... Figure 2As shown, the global structure is input into a multi-layer graph neural network (GNN) to achieve high-dimensional feature embedding of various nodes and aggregation of global information. The graph neural network utilizes the local and global connections of the node neighborhood to dynamically calculate the feature representation of each variable node, constraint node, and cut plane node, and outputs a feature vector representing the global state of the entire branch and bound tree.

[0077] For example, the corresponding variable node inputs include the upper and lower bounds of the variable, the coefficients of the objective function, and the type; the original constraint node inputs are the left and right ends of the constraint (i.e., the left-hand and right-hand constraint terms), the sparsity of the coefficients, and the type; the historical cutting plane node encodes its historical addition time, cutting plane effect, parallelism with the objective function, and other multi-dimensional attributes.

[0078] After each round of graph convolution, the information of each node flows and merges in the graph, ultimately obtaining a high-dimensional feature vector that represents the global state of the entire branch-bound tree.

[0079] (3) Based on the feature vectors, the set of candidate cut planes consisting of multiple candidate cut plane nodes is input into a neural network with sequential modeling capabilities for joint modeling and inference, such as... Figure 2 As shown, this involves performing the corresponding cutting plane feature aggregation, as well as selecting and sorting the cutting planes;

[0080] Specifically, in this embodiment of the invention, the aggregated features can be fed into the Transformer module through pooling (such as mean pooling) to jointly model and infer the candidate cut plane set. The Transformer structure can capture the interactions, order dependencies, and global synergistic effects between cut planes, providing richer information input for subsequent decision networks.

[0081] (4) Based on the results of the joint modeling and inference, the optimal cutting plane selection strategy is determined by reinforcement learning, that is, a reinforcement learning framework is introduced to learn the optimal cutting plane selection strategy so that the cutting plane selection can be implemented accordingly in the future.

[0082] Specifically, the process includes adding cut planes to the search tree, updating the structure of the bipartite graph, and training and optimizing the policy network, thereby achieving the corresponding processing iteration and network optimization.

[0083] Furthermore, the cut plane selection decision is abstracted as a Markov Decision Process (MDP) in reinforcement learning. The current state represents the overall features output by the graph neural network and Transformer, and the action is the binary selection and sequential sorting of candidate cut planes. The reward function is designed as follows: the immediate reward is based on the improvement of the optimal solution gap, reflecting the magnitude of gap reduction brought about by the current cut plane; the termination reward is based on the solution time difference with the mixed-integer programming solver, reflecting the improvement in final solution time and the number of branch nodes. To train this decision policy, reinforcement learning algorithms such as Proximal Policy Optimization (PPO) can be used to jointly optimize the parameters of the policy network (i.e., the decision network) and the value network, repeatedly interacting with the actual MIP solver to obtain the optimal cut plane selection policy. The value network parameters are used to evaluate the "goodness" of the current state, providing a reference benchmark for policy optimization. The corresponding joint optimization process ensures that the value network provides a stable reference benchmark for policy network optimization, and the policy network provides effective sample data for value network learning, ultimately achieving a more efficient and robust cut plane selection policy learning, meeting the dual requirements of MIP solving for solution quality and efficiency. The repeated interaction with the actual MIP solver enables the reinforcement learning model (policy network + value network) to learn the optimal cutting plane selection strategy that fits the real solution scenario. That is, the repeated interaction with the actual MIP solver is the core link of the trial and error learning of reinforcement learning, which allows it to verify the effect of the strategy through feedback from the real environment, continuously iterate and optimize, and finally obtain a cutting plane selection strategy that can effectively improve the solution efficiency.

[0084] (5) Select the cutting plane based on the optimal cutting plane selection strategy;

[0085] This process is the application of the policy network obtained through the training and optimization process described above.

[0086] Specifically, during the actual reasoning process, at each branch node, the system automatically constructs a global graph structure of the current node and its historical states, dynamically calculating the feature representations of all variables, constraints, added cut planes, and candidate cut planes. Through a trained decision network, the system automatically decides at each node which cut planes should be selected and in what order, thereby driving the solver to efficiently perform LP (linear programming) relaxation and space compaction. This process forms a closed-loop system of data acquisition, feature encoding, intelligent decision-making, and feedback optimization, achieving efficient collaboration in cut plane selection across the entire branch and bound tree.

[0087] Through the above technical solutions, the embodiments of the present invention not only overcome the difficulties of traditional cut plane selection being limited to local information, redundant addition of cut planes, and low efficiency, but also adapt to different problem structures and scales. Experiments show that in typical large-scale MIP problems such as set covering, maximum independent set, and fixed-cost multi-commodity network flow, the technical solutions provided by the embodiments of the present invention can significantly improve cut plane utilization and reduce the number of nodes and solution time. The overall framework can be integrated into mainstream optimization solvers such as SCIP and Gurobi, and is adaptable to various industrial and scientific research scenarios.

[0088] This invention also provides a cutting plane selection system in mixed integer programming, which specifically includes:

[0089] (1) Feature vector acquisition unit, used to encode and obtain the global structure information of the entire branch and bound tree, and input it into the multi-layer graph neural network. The graph neural network uses the local and global connections of the node neighborhood to dynamically calculate the feature representation of each variable node, constraint node and cut plane node, and outputs the feature vector representing the global state of the entire branch and bound tree.

[0090] The process of encoding the global structural information of the entire branch-bound tree in the feature vector acquisition unit includes:

[0091] The entire branch and bound tree is encoded as a bipartite graph containing variable nodes, original constraint nodes, historical cut plane nodes, and candidate cut plane nodes. The variable nodes represent the decision variables of the MIP, the original constraint nodes describe the constraints of the original mathematical model, the historical cut plane nodes record all added cut planes and their attributes, and the candidate cut plane nodes represent the set of cut planes to be decided. The nodes are connected by directed edges that encode various structural relationships and features.

[0092] The characteristic representation of the variable node includes: the upper and lower bounds of the variable, the coefficients of the objective function, and the type;

[0093] The feature representation of the constraint node includes: the left and right ends of the constraint, the sparsity of the coefficients, and the type;

[0094] The features of the cutting plane node include: historical addition time, cutting plane effect, and parallelism with the objective function.

[0095] (2) The optimal cutting plane selection strategy determination unit is used to perform joint modeling and inference on the candidate cutting plane set composed of multiple candidate cutting plane nodes based on the feature vector, so as to obtain the interaction, order dependency and global cutting plane cooperative optimization effect between cutting planes; and to determine the optimal cutting plane selection strategy by using reinforcement learning based on the results of the joint modeling and inference.

[0096] The joint modeling and inference are achieved by pooling the input into the Transformer module;

[0097] Furthermore, the processing procedure executed by the optimal cutting plane selection strategy determination unit may specifically include:

[0098] Using the overall features output by the graph neural network and the Transformer module as the current state, the binary selection and sequential sorting of candidate cut planes are taken as actions. The immediate reward function and the termination reward function are used as reward functions. The reinforcement learning algorithm is used to jointly optimize the parameters of the policy network and the value network, and interacts with the actual MIP solver to obtain the optimal cut plane selection strategy.

[0099] (3) Cutting plane selection unit, used to select a cutting plane based on the optimal cutting plane selection strategy;

[0100] Specifically, the process of selecting a cutting plane based on the optimal cutting plane selection strategy may include:

[0101] Upon entering a branch node, the system automatically constructs a global graph structure of the current node and its historical states, and dynamically calculates the feature representations of all variable nodes, constraint nodes, added cut plane nodes, and candidate cut plane nodes.

[0102] Based on the feature representation obtained by dynamic calculation, the optimal cutting plane selection strategy is used to automatically decide the cutting plane to be selected and the order of addition at each node.

[0103] This invention achieves the selection of the cutting plane through branch-bound tree global information encoding, graph neural network feature fusion, and reinforcement learning decision optimization. Compared with traditional cutting plane selection methods, it has the following significant advantages and positive technical effects:

[0104] First, this invention employs a global branch-bound tree-cut plane joint modeling technique. By globally encoding each node's variables, constraints, added cut planes, and their historical effects, it accurately captures the cumulative and collaborative influence of cut planes across different nodes throughout the solution process. Compared to existing cut plane selection schemes based on single nodes or local features, the technical solution provided by this invention significantly improves the overall synergy of cut plane allocation, effectively avoiding common local selection problems such as repetitive cut decisions (i.e., cut plane decisions) and low cut utilization. The direct result is that the solver can achieve faster relaxation and tightening with fewer added cut planes, reducing the generation of redundant nodes.

[0105] Secondly, by employing graph neural network feature fusion and a Transformer structure, high-dimensional feature aggregation can be performed on complex constraint-variable-cut plane graph structures, enabling the modeling of long-distance dependencies between nodes. Through graph convolution and multi-layer information transfer, the system can learn the long-term effectiveness of cut plane combinations and their interaction with the original structure, improving the intelligence level of cut plane combination decision-making. Combined with reinforcement learning strategy optimization, the system can dynamically adjust the selection and arrangement of cut planes, adaptively making optimal decisions for different problem instances and different solution stages.

[0106] Furthermore, embodiments of the present invention can also achieve real-time and rapid construction and feature extraction of large-scale branch and bound trees and cut plane graphs through efficient incremental graph structure update design, significantly reducing the computational resources and time overhead required for high-dimensional information encoding and model inference. This feature is particularly suitable for the rapid response and online decision-making needs of large-scale complex MIP models in industrial scenarios.

[0107] More importantly, this invention also demonstrates significant performance improvements on multiple public benchmark sets through experimental comparisons with mainstream heuristics and single-node machine learning cut plane selection methods. The experiments included multiple comparison groups, encompassing native heuristics from open-source solvers, current mainstream machine learning cut plane selection methods (such as Score-BasedPolicy and Hierarchical Sequence Model), and the GCS (Global Cut Selection) method proposed in this invention. Experiments were conducted on six typical MIP problem sets: Set Covering, Maximum Independent Set (MIS), Fixed-Cost Multi-Item Network Flow (FCMCNF), and 0 / 1 Knapsack (MIK), covering various complex scenarios with varying numbers of variables and constraint sizes, ensuring the universality and reliability of the results. Using the global cut plane selection scheme of this invention, in the Set Cover (500 variables, 1000 constraints) problem, the average solution time decreased from 10.33 seconds to 5.79 seconds, a reduction of 43.9%; the number of nodes decreased from 160.6 to 109.5, a reduction of 31.8%. In the FCMCNF network flow problem, the time was reduced from 50.83 seconds to 37.18 seconds, and the number of nodes was reduced from 333.5 to 208.4, both showing significant improvements. Other datasets, such as the maximum independent set and MIK datasets, also showed efficiency improvements of 20%-40%. These effects are directly attributed to the global graph structure modeling and joint feature learning in the embodiments of this invention, which makes cut plane allocation more global and collaborative, significantly reducing redundant branches and invalid cut additions.

[0108] Experimental results show that the technical solution provided by the embodiments of the present invention can significantly reduce the solution time and number of nodes, and improve the utilization rate of cutting planes and the overall performance of the solver, compared with the existing local cutting plane selection technology, on multiple standard test sets.

[0109] The application effects of the embodiments of the present invention will be described below using two specific application examples.

[0110] Example 1: Deployment for the Largest Network Flow (FCMCNF) Problem

[0111] This invention focuses on the FCMCNF network flow problem, with 500–1500 variables and 500–2000 constraints. The underlying solver uses SCIP 8.0.4. The system runs on a GPU server (NVIDIA RTX 4090), and the graph neural network and Transformer modules are implemented using PyTorch. The decision network can have a 2–6 layer structure, preferably 4 layers. First, the MIP model is input into the system, and all variables, constraints, and initial model data are extracted. During the branch and bound (B&C) process, a new candidate cut plane is generated after each round of branching. The system automatically performs structured encoding on all historical nodes, variables, constraints, and cut planes of the branch and bound tree.

[0112] After encoding, a four-layer graph neural network (such as a Graph Convolutional Network) is used to recursively aggregate the bipartite graph features. The candidate cut plane pools are then fed into the Transformer module using mean pooling to capture the interactions between the cut planes. The model ultimately outputs the selection probability and addition order of each cut plane. The selected cut planes are added to their corresponding nodes in sequence, driving the next round of LP relaxation and branching operations. The entire process iterates until convergence conditions are met (e.g., gap convergence, less than 0.1%).

[0113] In the experiment, the PPO reinforcement learning algorithm was used to train the policy network offline, and the training data was automatically collected through interaction with the mixed-integer programming solver. The experiment shows that the solution provided by the embodiments of the present invention reduces the solution time for network flow problems from 50.83 seconds to 37.18 seconds and the number of nodes from 333.5 to 208.4, which is significantly better than traditional heuristic and local learning strategies.

[0114] Example 2: Intelligent Optimization of Practical Industrial Scheduling Problems

[0115] In real-world industrial scenarios, such as production scheduling and supply chain management, MIP models are typically larger in scale and more complex in structure. This invention integrates with an enterprise scheduling optimization platform, automatically constructing an MIP model from production order data input from ERP / MES systems. The system is deployed on a local server or private cloud platform, supporting large-scale models (1000+ variables, 2000+ constraints). Each time a solution task starts, it automatically performs global encoding of the branch and bound tree and extracts cut plane features, using a trained graph neural network and reinforcement learning strategy modules for cut plane selection and allocation. Due to the incremental graph structure and efficient update mechanism, even with complex structures involving thousands of nodes and hundreds of cut planes, it can still achieve fast solution and low-latency decision-making, meeting the real-time requirements of industrial scenarios.

[0116] Through the above description of the embodiments, those skilled in the art can clearly understand that the above embodiments can be implemented by software, or by using software plus necessary general-purpose hardware platforms. Based on this understanding, the technical solutions of the above embodiments can be embodied in the form of a software product, which can be stored in a non-volatile storage medium (such as a CD-ROM, USB flash drive, mobile hard drive, etc.), including several instructions to cause a computer device (such as a personal computer, server, or network device, etc.) to execute the methods described in the various embodiments of the present invention.

[0117] The above description is merely a preferred embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the scope of the technology disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims. The information disclosed in the background section is intended only to enhance the understanding of the overall background technology of the present invention and should not be construed as an admission or implication in any way that such information constitutes prior art known to those skilled in the art.

Claims

1. A method for selecting a cutting plane in mixed-integer programming, characterized in that, include: The global structural information of the entire branch and bounding tree is obtained by encoding and input into a multi-layer graph neural network. The graph neural network uses the local and global connections of the node neighborhood to dynamically calculate the feature representation of each variable node, constraint node, and cut plane node, and outputs a feature vector representing the global state of the entire branch and bounding tree. Based on the feature vectors, a set of candidate cut planes consisting of multiple candidate cut plane nodes is input into a neural network with sequential modeling capabilities for joint modeling and inference, so as to simultaneously characterize the interaction, sequential dependence and global collaborative optimization effect between cut planes. Based on the results of the joint modeling and inference, a reinforcement learning approach is used to determine the optimal cutting plane selection strategy, and the cutting plane is selected based on the optimal cutting plane selection strategy.

2. The method according to claim 1, characterized in that, The encoding process for obtaining the global structural information of the entire branch-bound tree includes: The entire branch and bound tree is encoded as a bipartite graph containing variable nodes, original constraint nodes, historical cut plane nodes, and candidate cut plane nodes. The variable nodes represent the decision variables of the mixed integer programming problem, the original constraint nodes describe the constraints of the original mathematical model, the historical cut plane nodes record all added cut planes and their attributes representing the generation time and usage effect, and the candidate cut plane nodes represent the set of cut planes to be decided. The nodes are connected by directed edges that encode various structural relationships and features.

3. The method according to claim 2, characterized in that, The feature representation of the variable node includes: upper and lower bounds of the variable, coefficients of the objective function, and type; the feature representation of the original constraint node includes: left-hand side of the constraint, right-hand side of the constraint, coefficient sparsity, and type; the feature representation of the historical cutting plane node includes: historical addition time, cutting plane effect, and parallelism with the objective function.

4. The method according to claim 1, 2 or 3, characterized in that, The process of encoding to obtain the global structural information of the entire branch-bound tree includes: The search tree is parsed to extract path and branch information, and the set of cut planes is parsed to distinguish between historical cuts and candidate cuts; Variable nodes, original constraint nodes, historical cut plane nodes and candidate cut plane nodes are constructed sequentially, and connections between variables and various constraint nodes are established. The coefficients on the edges are written, and branch information is added to the connections of the historical cut planes. Global structural information is obtained by encoding the edge features of the branch constraints.

5. The method according to claim 4, characterized in that, The joint modeling and inference are implemented through the input transformer module using pooling operations, and the process of determining the optimal cutting plane selection strategy using reinforcement learning includes: Using the overall features output by the graph neural network and the Transformer module as the current state, and taking the binary selection and sequential sorting of candidate cut planes as actions, an immediate reward function and a termination reward function are used as reward functions. The immediate reward is improved based on the gap between the optimal solution and the termination reward is set based on the solution time difference with the mixed integer programming solver. A reinforcement learning algorithm is used to jointly optimize the parameters of the policy network and the value network, and interacts with the actual solver to obtain the optimal cut plane selection strategy.

6. The method according to claim 5, characterized in that, The process of selecting the cutting plane based on the optimal cutting plane selection strategy includes: Upon entering a branch node, the system automatically constructs a global graph structure of the current node and its historical states, and dynamically calculates the feature representations of all variable nodes, constraint nodes, added cut plane nodes, and candidate cut plane nodes. Based on the calculated feature representation, the optimal cutting plane selection strategy is used to automatically decide the cutting plane to be selected and the order of addition at each node.

7. A cutting plane selection system in mixed integer programming, characterized in that, include: The feature vector acquisition unit is used to encode and obtain the global structural information of the entire branch and bound tree, and input it into the multi-layer graph neural network. The graph neural network uses the local and global connections of the node neighborhood to dynamically calculate the feature representation of each variable node, constraint node, and cut plane node, and outputs a feature vector representing the global state of the entire branch and bound tree. The optimal cutting plane selection strategy determination unit is used to jointly model and infer the set of candidate cutting planes composed of multiple candidate cutting plane nodes into a neural network with sequential modeling capabilities based on the feature vector, so as to simultaneously characterize the interaction, sequential dependence and global collaborative optimization effect between cutting planes. Based on the results of the joint modeling and inference, reinforcement learning is used to determine the optimal cutting plane selection strategy. The cutting plane selection unit is used to select the cutting plane based on the optimal cutting plane selection strategy.

8. The system according to claim 7, characterized in that, The process of encoding the global structural information of the entire branch and bound tree in the feature vector acquisition unit includes: The entire branch and bound tree is encoded as a bipartite graph containing variable nodes, original constraint nodes, historical cut plane nodes, and candidate cut plane nodes. The variable nodes represent the decision variables of the mixed integer programming problem, the original constraint nodes describe the constraints of the original mathematical model, the historical cut plane nodes record all added cut planes and their attributes representing the generation time and usage effect, and the candidate cut plane nodes represent the set of cut planes to be decided. The nodes are connected by directed edges that encode various structural relationships and features.

9. The system according to claim 7 or 8, characterized in that, The processing steps performed by the optimal cutting plane selection strategy determination unit include: Using the overall features output by the graph neural network and the Transformer module as the current state, and taking the binary selection and sequential sorting of candidate cut planes as actions, an immediate reward function and a termination reward function are used as reward functions. The immediate reward is based on the gap improvement setting, and the termination reward is based on the solution time difference with the baseline solver. A reinforcement learning algorithm is used to jointly optimize the parameters of the policy network and the value network, and interacts with the actual MIP solver to obtain the optimal cut plane selection strategy.

10. The system according to claim 9, characterized in that, The process of selecting a cutting plane based on the optimal cutting plane selection strategy in the cutting plane selection unit includes: Upon entering a branch node, the system automatically constructs a global graph structure of the current node and its historical states, and dynamically calculates the feature representations of all variable nodes, constraint nodes, added cut plane nodes, and candidate cut plane nodes. Based on the feature representation obtained by dynamic calculation, the optimal cutting plane selection strategy is used to automatically decide the cutting plane to be selected and the order of addition at each node.

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