Papermaking mixed integer nonlinear optimization method, device and equipment and storage medium
By integrating feasible pumps and external approximation algorithms, and combining behavior monitoring and adaptive switching mechanisms, the problems of low solution efficiency and difficulty in guaranteeing global optimality in MINLP problems in papermaking production are solved, and efficient and robust optimization results are output.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- SOUTH CHINA UNIV OF TECH
- Filing Date
- 2026-04-10
- Publication Date
- 2026-05-08
AI Technical Summary
Existing SCIP solvers and traditional external approximation algorithms suffer from several problems when dealing with the complex MINLP problem in papermaking production. These problems include difficulty in guaranteeing global optimality, difficulty in obtaining initial feasible solutions, low solution efficiency, difficulty in solving within feasible time due to the large model size, and a lack of adaptive solution strategies tailored to the characteristics of papermaking production.
By integrating the feasible pump and external approximation algorithms, an adaptive switching mechanism based on behavior monitoring is introduced. The feasible pump algorithm is used to quickly generate high-quality initial feasible solutions and linearized cut sets, and key state variables such as the stability of integer solutions and projection distance are monitored in real time. The algorithm dynamically switches to the external approximation stage, constructing an adaptive optimization framework to avoid invalid iterations and improve solution efficiency and robustness.
It significantly shortens the overall solution cycle, improves the solution efficiency and robustness of complex industrial MINLP problems, and the output optimization results are close to the optimal in terms of objective function value and process constraints, thus improving the optimization effect of paper production.
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Figure CN121996876A_ABST
Abstract
Description
Technical Field
[0001] This invention belongs to the field of papermaking process optimization technology, specifically relating to papermaking mixed integer nonlinear optimization methods, apparatus, equipment, and storage media. Background Technology
[0002] The SCIP solver is one of the world's leading non-commercial, general-purpose mixed-integer nonlinear programming (MINLP) solvers. Its core employs a branch-and-bound (B&B) framework, integrating cutting planes and various heuristic algorithms to accelerate the solution process. The typical workflow for solving MINLP problems includes: first, simplifying the original model through a pre-solution phase, including removing redundant constraints, fixing some variable values, and tightening the range of variable values through constraint propagation techniques, thereby reducing problem complexity. In the continuous relaxation phase, the algorithm ignores integer constraints to obtain a nonlinear programming relaxation problem, and solves this relaxation problem to obtain a lower bound estimate of the original problem. The branching operation is the core mechanism of SCIP. When integer variables in the relaxation solution are fractional, the algorithm selects a branch variable and creates two child nodes, adding corresponding boundary constraints to exclude the current fractional solution. To accelerate convergence, SCIP integrates cutting plane generation techniques, tightening the feasible region of the relaxation problem by adding effective linear or convex constraints. Meanwhile, the algorithm employs multiple heuristic strategies to find high-quality feasible solutions during the search process, providing upper bound references. The pruning mechanism utilizes upper and lower bound information to eliminate branches that cannot contain the optimal solution. The algorithm terminates when all nodes have been processed or the convergence condition is met. SCIP adopts a modular design architecture, supporting the integration of multiple external solvers (such as IPOPT for handling nonlinear subproblems and CPLEX for solving linear programming problems), exhibiting strong problem adaptability and scalability.
[0003] While SCIP offers comprehensive solution capabilities, it still exhibits significant limitations when handling complex MINLP problems specific to papermaking production. First, it struggles to guarantee global optimality for non-convex problems. While its branch-and-bound framework is theoretically applicable to global optimization, its efficiency hinges on the quality of relaxation techniques. For the prevalent strong non-convex and nonlinear constraints in papermaking processes, such as reaction kinetics in pulping and mass and heat transfer models in paper drying, constructing tight convex relaxations is challenging, leading to computational difficulties and hindering the timeliness requirements of actual production scheduling or real-time optimization. Second, papermaking optimization models are large-scale and involve numerous variables. When facing such high-dimensional problems, branch trees become exceptionally large, making it difficult to obtain satisfactory solutions within a feasible timeframe. This is particularly true when dealing with multi-time-segment production scheduling problems, where the problem size increases linearly with the time period, further complicating the solution process. The difficulty in obtaining initial feasible solutions is another significant drawback of SCIP. Its efficiency heavily relies on high-quality initial upper bounds for effective pruning, but the complex constraints of papermaking problems make obtaining initial feasible solutions challenging. Although it incorporates various heuristic methods, these methods have a low success rate when dealing with papermaking models with complex process constraints, resulting in a lack of effective upper bound information in the early stages of the search and low pruning efficiency. Finally, as a general solver, its algorithm includes a large number of general strategies and parameters designed to cope with various problem types, but it lacks stage division and adaptive solution strategies specific to the characteristics of papermaking production models, such as the identification and utilization of specific constraint structures and the adjustment of search strategies based on process characteristics. Therefore, its solution efficiency is not as good as customized algorithms designed specifically for the papermaking industry.
[0004] The traditional Outer-Approximation algorithm is a classic class of algorithms for solving convex MINLP problems. Its basic idea is to gradually approximate the optimal solution of the original problem by alternately solving nonlinear programming subproblems (NLP) and mixed-integer linear programming master problems (MILP). At the beginning of the algorithm, a suitable initial integer solution needs to be selected, usually obtained through relaxation problems or heuristic methods. Then, the integer variable is fixed, and the corresponding NLP subproblem is solved in each iteration. If the subproblem is feasible, the continuous optimal solution and objective function value at the fixed integer point are obtained, and a linearized cut (OA cut) of the objective function and constraints at that point is generated. If the subproblem is infeasible, a feasible recovery solution is obtained by solving the minimum constraint violation problem, and a corresponding feasible cut is generated. Subsequently, all linearized cuts collected in previous iterations are added to the master problem, constructing an continuously improving mixed-integer linear programming approximation problem. Solving this MILP problem yields new candidate integer solutions and a lower bound (LB). Finally, the difference between the upper and lower bounds is checked; if the convergence condition is met, the algorithm terminates; otherwise, it returns to solving the NLP subproblems. The OA algorithm theoretically guarantees convergence to the global optimal solution of the convex MINLP problem within a finite number of steps.
[0005] Traditional Open Algorithm (OA) algorithms face several major challenges when dealing with paper production problems. First, they heavily rely on the initial point. The algorithm's convergence speed and success rate depend heavily on the quality of the initial integer solution. Paper production involves numerous discrete decision variables (such as equipment start / stop status and production mode selection), making it extremely difficult to obtain a high-quality initial solution. If the initial point is poorly chosen, deviating from the optimal solution or rendering NLP subproblems infeasible, the algorithm may require many iterations to find a good solution, or even fail due to frequent encounters with infeasible subproblems. Second, the sheer number of cuts is also a problem. Each iteration generates new linearized cuts, leading to a continuous increase in the main problem size and solution time. For large-scale paper production problems, the main problem may become excessively large and difficult to solve efficiently. Furthermore, while the theoretical guarantee of the OA algorithm is based on the assumption of problem convexity, paper production models often contain non-convex constraints (such as nonlinear dynamic equations). In such cases, the cuts generated by the OA algorithm may be invalid, or even cut away the globally optimal solution, causing the algorithm to converge to a local optimum or fail completely. Finally, traditional OA algorithms typically employ a fixed-process solution mechanism, meaning they only enter the external approximation main loop after the pre-defined initialization or preprocessing stages are fully completed. This lack of ability to dynamically adjust the solution stages based on behavioral characteristics during the solution process is problematic. When the solution space contains multiple local extrema or complex non-convex structures, this fixed-stage approach easily leads to the accumulation of ineffective iterations, causing the search path to deviate from the potential global optimum, exhibiting a pronounced "myopia" search characteristic and making it difficult to balance solution efficiency and solution quality. Summary of the Invention
[0006] To address the shortcomings of existing technologies, the present invention aims to provide a method, apparatus, device, and storage medium for papermaking mixed integer nonlinear optimization.
[0007] According to one aspect of this application, a mixed integer nonlinear optimization method for papermaking is disclosed, the method comprising: Receive a mixed-integer nonlinear programming model to be optimized, wherein the mixed-integer nonlinear programming model to be optimized is determined based on the target optimization problem; The feasible pump is invoked to perform initial iterative optimization on the mixed-integer nonlinear programming model to be optimized, and preliminary optimization results are obtained. Obtain multiple behavioral state variables during the initialization iterative optimization process; Based on the comparison threshold of the state variables corresponding to each of the multiple behavioral state variables, determine whether the optimization switching conditions are met. When the optimization switching conditions are met, the external approximation main loop is invoked to perform iterative optimization of the preliminary optimization results until the convergence conditions are met, and then the target optimization results are output. Based on the optimization results of the target, the optimized operating parameters of the papermaking process are determined.
[0008] In some embodiments, the invocation of the feasible pump to perform initial iterative optimization on the mixed-integer nonlinear programming model to be optimized, and the obtaining of preliminary optimization results, includes: Determine the feasible pump configuration parameters; By combining the feasible pump configuration parameters, a continuous relaxation problem is solved for the mixed-integer nonlinear programming model to be optimized, and the relaxed solution is obtained. Based on the relaxation solution results, obtain the initial iterative input data for feasible pump initialization iterative optimization; Based on the initial iterative input data, the mixed integer nonlinear programming model to be optimized is initialized and iteratively optimized to obtain preliminary optimization results.
[0009] In some embodiments, the step of solving the continuous relaxation problem of the mixed-integer nonlinear programming model to be optimized by combining the feasible pump configuration parameters to obtain the relaxed solution results includes: Relax the integer constraints of the mixed-integer nonlinear programming model to be optimized, and change the integer variables. The feasible domain from relaxation This leads to a continuous relaxation nonlinear programming problem; in, for The original feasible region is The set of integers, After relaxation Constraints exist Take any real number within the interval. The lower realm For variables The lower bound vector, For the upper realm, For variables The upper bound vector; By combining the relaxation formula, the continuous relaxation nonlinear programming problem is solved to obtain the relaxed solution. The relaxation formula is as follows: ; in, For the objective function Minimize the optimization objective; The objective function is, i.e. , For continuous variables, For integer variables, For the corresponding variable and The transpose of the objective function coefficient vector; For nonlinear constraints, This is the index of which nonlinear constraint; For linear constraints, For corresponding continuous variables The coefficient matrix, For the corresponding integer variable The coefficient matrix, It is a constant vector.
[0010] In some embodiments, the initial iterative optimization of the mixed-integer nonlinear programming model to be optimized based on the initial iterative input data to obtain preliminary optimization results includes: Based on the initial iterative input data, a feasible pump iterative main loop is adopted for the mixed integer nonlinear programming model to be optimized using an inner and outer double-layer loop structure, wherein each inner iteration includes: Based on the projection point obtained in the previous iteration, an external approximation cut of the nonlinear constraint at that point is added to the current linearized model. A feasible pump nonlinear programming subproblem with the objective of minimizing the norm distance with the previous projection point is constructed and solved to obtain an integer solution. By fixing the initial integer solution, the feasible pump nonlinear subproblem with the objective of minimizing the square of the Euclidean distance to the previous projection point is solved to obtain a continuous solution; Determine the projection distance between the current continuous solution and the previous projection point; If the projected distance is less than the convergence tolerance, the iteration is terminated, and the current feasible solution is determined as the preliminary optimization result, wherein the current feasible solution is determined based on the integer solution and the continuous solution; If the projected distance is not less than the convergence tolerance, then the current feasible solution is determined as the initial iterative input data and the main loop step of the feasible pump iteration is returned to the mixed integer nonlinear programming model to be optimized.
[0011] In some embodiments, obtaining multiple behavioral state quantities during the initialization iterative optimization process includes: During the initialization iterative optimization process, behavior monitoring and state variable calculation are performed simultaneously to determine multiple behavior state vectors, including: Record the integer solution of each external iteration. If the integer solution remains unchanged in the external iteration for a preset number of consecutive iterations, it is marked as a stable state of integer solution. Record the projection distance of each internal iteration. If the current projection distance is less than the preset distance, mark it as a stable state of the projection process. Record the solution results of the nonlinear programming problem in each projection stage. If the nonlinear programming subproblem has a solution that satisfies all constraints in each consecutive number of solutions, it is marked as a continuously feasible state.
[0012] In some embodiments, when the optimization switching condition is met, calling the external approximation main loop to perform iterative optimization on the preliminary optimization result until the convergence condition is met and then outputting the target optimization result includes: Based on the preliminary optimization results, the upper bound of the objective function, and the initial linearized cut set, an initial mixed-integer linear programming master problem is constructed. The initial linearized cut set is determined based on the external approximation cut of adding nonlinear constraints at the corresponding point to the corresponding linearized model in each iteration. Based on the aforementioned mixed-integer linear programming master problem, candidate integer solutions and the current lower bound are determined; By fixing the candidate integer solutions, the nonlinear programming subproblem is solved to obtain the solution result; When the solution satisfies all constraints and the iteration conditions, the target optimization result is output. The main problem of the mixed-integer linear programming is: ; in, For variables Minimize the optimization objective; This is the upper bound variable for the objective function, used to linearize the objective function; For the objective function at point The function value at that location; For the objective function at point The gradient vector at point represents the objective function. The direction of the rate of change at that point; The dot product of the gradient vector and the difference vector is the objective function. exist The first-order Taylor expansion linear approximation term at; For variables With point The difference vector; These are the cut points generated during the FP phase; and These are continuous variables at the cutting point. and integer variables The value of , Which is the cutting point? For all indices in the cut collection ; The cut set generated for the FP stage contains the index set of all linearized cuts; For the first A nonlinear constraint function at point The gradient vector at point represents the constraint function. The direction of the rate of change at that point; The inner product of the gradient vector and the difference vector is the constraint function. exist The first-order Taylor expansion linear approximation term at; The nonlinear programming subproblem is: ; in, For fixed integer solutions The objective function after that; For fixed integer solutions The subsequent nonlinear constraints; For fixed integer solutions Linear constraints afterward.
[0013] In some embodiments, when the solution result contains unsatisfied constraints, the method includes: Solve the feasibility recovery problem and form a feasibility segmentation; The new cut is added to the linearized main problem to generate the optimal cut, which is then constrained by linear inequalities. The value of makes such that Always greater than or equal to the objective function A linear approximation is obtained to gradually approximate the true objective function; in, ; The feasibility segmentation is generated so that when a nonlinear programming subproblem is infeasible, the region that leads to infeasibility is identified and excluded through the feasibility segmentation. Among them, when there is When a nonlinear programming subproblem is determined to be infeasible, this infeasibility point is used as a new cutoff point. The corresponding nonlinear constraint function is then subjected to a first-order Taylor expansion at that point, transforming it into a linear approximation constraint, where the linear approximation constraint is: ; in, It is the negative of the constraint function value.
[0014] According to another aspect of this application, a papermaking hybrid integer nonlinear optimization device is also disclosed, the device comprising: The receiving module is used to receive a mixed-integer nonlinear programming model to be optimized, wherein the mixed-integer nonlinear programming model to be optimized is determined based on the target optimization problem; The preliminary optimization result determination module is used to call the feasible pump to perform initial iterative optimization on the mixed integer nonlinear programming model to be optimized, and obtain the preliminary optimization result. The behavior state quantity acquisition module is used to acquire multiple behavior state quantities during the initialization iterative optimization process; The switching condition determination module is used to determine whether the optimization switching conditions are met based on the comparison threshold of the state quantities corresponding to multiple behavioral state quantities. The target optimization result output module is used to call the external approximation main loop to perform iterative optimization of the preliminary optimization result when the optimization switching condition is met, until the convergence condition is met and the target optimization result is output. The optimized operating parameter determination module is used to determine the optimized operating parameters of the papermaking process based on the target optimization results.
[0015] According to another aspect of this application, an electronic device is also disclosed, the electronic device including a memory and at least one processor, the memory storing instructions; the at least one processor invokes the instructions in the memory to cause the electronic device to perform various steps of the papermaking hybrid integer nonlinear optimization method as described in any of the preceding claims.
[0016] According to another aspect of this application, a computer-readable storage medium is also disclosed, on which instructions are stored, which, when executed by a processor, implement the steps of the papermaking hybrid integer nonlinear optimization method as described in any of the preceding claims.
[0017] The present invention includes, but is not limited to, the following beneficial effects: (1) This scheme integrates two types of algorithms, namely feasible pump and external approximation, and introduces an adaptive switching mechanism based on behavior monitoring to form a new framework, which changes the rigid mode of fixed process of traditional algorithms. In the early stage of solving the problem, the feasible region is quickly explored by the feasible pump algorithm, and high-quality initial feasible solutions and linearized cut sets are efficiently generated, providing superior hot start conditions for subsequent accurate solutions. At the same time, the scheme monitors key behavioral state quantities such as the stability of integer solutions, projection distance, feasibility of subproblems and rate of change of objective function in real time during the feasible pump iteration process, and triggers the transition from the feasible pump stage to the external approximation stage based on these quantities. (1) Seamless switching of segments. This adaptive mechanism can effectively identify the algorithm running idle in invalid iterations, compress the overall solution cycle, and enhance the stability of the method when facing different working conditions and complex models; (2) Based on the iterative characteristics of the feasible pump stage, this scheme dynamically switches the start time. Through the global optimization solution concept of phased and collaborative optimization, it avoids invalid iteration caused by fixed stage division, improves the efficiency and robustness of solving complex industrial MINLP problems, and is used to solve complex optimization problems in papermaking production; (3) By constructing a problem transformation framework, this scheme temporarily relaxes integer constraints into continuous constraints, thereby transforming a mixed integer problem that is difficult to solve directly into a solution. The nonlinear programming model is transformed into a relatively easy-to-handle continuous nonlinear programming problem. This transformation is not a simple neglect, but rather a process that preserves all nonlinear and linear constraints of the original problem, as well as the complete structure of the objective function. A relaxed solution, which is usually very close to the optimal region in terms of objective function value and satisfaction of process constraints, can be obtained using a continuous optimization solver. This provides a well-guided search starting point for subsequent feasible pump iterations, avoiding the waste of resources from blindly searching from scratch, shortening the initialization time, and enabling the algorithm to generate effective linearization cuts and guide the rapid convergence of the external approximation stage. The original materials improved the overall solution efficiency; (4) In each external iteration of the feasible pump, this scheme calculates and records four key behavioral state quantities in real time: stability of integer solutions, projection distance, feasibility of continuous subproblems and rate of change of objective function. It builds a dynamic and quantitative health diagnosis index system for the algorithm, reduces the waste of computing resources caused by meaningless iteration after the performance of the feasible pump stage is saturated, and can also trigger external approximation intervention in time when the quality improvement of the solution tends to be moderate or the search falls into local stability, thereby significantly improving the intelligence level, running efficiency and quality of the final solution of the solution process.(5) This scheme, through the constructed external approximation main loop mechanism, achieves the transformation from the preliminary solution of the feasible pump to the global optimal solution through the alternating iteration of the mixed integer linear main problem and nonlinear subproblems, the dynamic accumulation of linearization cuts, and precise convergence control. Based on the preliminary optimization results of the feasible pump, the upper bound of the objective function, and the initial linearization cut set, it constructs a mixed integer linear programming main problem. It uses the external approximation cut to linearize the objective function and constraints using a first-order Taylor expansion, which not only preserves the nonlinear characteristics of the original problem but also uses a mature linear programming solver to quickly obtain candidate integer solutions and the current lower bound. At the same time, it updates the upper bound by solving nonlinear programming subproblems with fixed integer solutions, forming a two-way approximation between the upper and lower bounds. The cut management continuously tightens the feasible region of the linearization model by accumulating each round of external approximation cuts, gradually approaching the true optimal solution of the original problem, and outputs the final optimization result that can be directly used for paper production scheduling. It makes full use of the computational results of the preceding stages and improves the accuracy of the final solution through a rigorous iterative mechanism and resource management. Attached Figure Description
[0018] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the accompanying drawings used in the description of the embodiments or the prior art will be briefly introduced below.
[0019] Figure 1 This is a flowchart of the papermaking hybrid integer nonlinear optimization method according to an embodiment of this application; Figure 2 This is a structural block diagram of the papermaking hybrid integer nonlinear optimization device according to an embodiment of this application; Figure 3 This is a schematic diagram of the structure of the electronic device provided in the embodiments of this application; Figure 4 This is a comparison diagram of the case model solutions in the embodiments of this application. Detailed Implementation
[0020] The terms "first," "second," "third," "fourth," etc. (if present) in the specification, claims, and accompanying drawings of this invention are used to distinguish similar objects and are not necessarily used to describe a specific order or sequence. It should be understood that such data can be interchanged where appropriate so that the embodiments described herein can be implemented in orders other than those illustrated or described herein. Furthermore, the terms "comprising" or "having" and any variations thereof are intended to cover a non-exclusive inclusion; for example, a process, method, system, product, or apparatus that comprises a series of steps or units is not necessarily limited to those steps or units explicitly listed, but may include other steps or units not explicitly listed or inherent to such processes, methods, products, or apparatus.
[0021] For ease of understanding, the specific process of the embodiments of the present invention will be described below. Figure 1 This is a flowchart of the papermaking hybrid integer nonlinear optimization process according to an embodiment of this application. Figure 1 It includes the following steps: S100, Receive the mixed-integer nonlinear programming model to be optimized.
[0022] Specifically, the MINLP model describing the paper production optimization problem is received and formally represented as follows: ; in, For continuous variables; It is an integer variable; It is the set of real numbers; It is a set of integers; For continuous variable dimensions; Dimensions are integer variables; for 3D real space; for 3D real space; For nonlinear constraints, This is the index of which nonlinear constraint; For linear constraints, For corresponding continuous variables The coefficient matrix, For the corresponding integer variable The coefficient matrix, It is a constant vector; For the corresponding variable and The transpose of the objective function coefficient vector.
[0023] The mixed-integer nonlinear programming model to be optimized is determined based on the objective problem to be optimized.
[0024] S101. Call the feasible pump to perform initial iterative optimization on the mixed integer nonlinear programming model to be optimized, and obtain preliminary optimization results.
[0025] Specifically, this step may include the following steps: S1.1 Determine feasible pump configuration parameters.
[0026] Specifically, relevant parameters include the maximum number of iterations, such as the maximum number of external iterations. Maximum number of internal iterations Maximum iteration time and convergence tolerance Etc.; Number of iterations N for determining the stability of integer solutions, number of iterations M for determining the feasibility of continuous NLP subproblems, and projection distance threshold. Threshold for the rate of change of the objective function Initialize and set the external iteration counter. Internal Iteration Counter Gap setting for MILP subproblem solving accuracy .
[0027] S1.2. Based on the feasible pump configuration parameters, the continuous relaxation problem of the mixed integer nonlinear programming model to be optimized is solved to obtain the relaxation solution.
[0028] Specifically, this involves relaxing the integer constraints of the mixed-integer nonlinear programming model to be optimized, and reducing the integer variables... The feasible domain from relaxation This leads to a continuous relaxation nonlinear programming problem; Right now: ; in, for The original feasible region is The set of integers; After relaxation Constraints Available Take any real number within the interval. The lower realm For variables The lower bound vector, For the upper realm, For variables The upper bound vector.
[0029] Furthermore, by combining the relaxation formula, the continuous relaxation nonlinear programming problem is solved to obtain the relaxed solution.
[0030] The relaxation formula is as follows: ; in, For the objective function Minimize the optimization objective; The objective function is, i.e. , For continuous variables, For integer variables, For the corresponding variable and The transpose of the objective function coefficient vector; For nonlinear constraints, This is the index of which nonlinear constraint; For linear constraints, For corresponding continuous variables The coefficient matrix, For the corresponding integer variable The coefficient matrix, It is a constant vector.
[0031] S1.3 Based on the relaxation solution results, obtain the initial iterative input data for feasible pump initialization iterative optimization.
[0032] If relaxation solution If the integer constraints are satisfied, then the relaxation problem is feasible. In this case, the relaxation solution is determined as the initial iteration input data. If the relaxation solution... If the integer constraints are not satisfied, the relaxation problem becomes infeasible. In this case, a feasibility repair strategy is employed to solve a feasibility recovery problem, which is typically obtained by minimizing constraint violations. In order to obtain a solution point that satisfies the constraints as much as possible, for subsequent cutting generation, the results obtained from the determined feasibility recovery problem are further used as the initial iterative input data.
[0033] S1.4. Based on the initial iterative input data, perform initial iterative optimization on the mixed integer nonlinear programming model to be optimized, and obtain preliminary optimization results.
[0034] Specifically, based on the initial iterative input data, a feasible pump iterative main loop is adopted using an inner and outer double-layer loop structure for the mixed-integer nonlinear programming model to be optimized, including an outer loop and an inner loop. Controlling the global search process, inner loop Perform a rounding-projection iteration. Each internal iteration includes: S1.4.1 Based on the projection point obtained in the previous iteration, add an external approximation cut of the nonlinear constraint at that point to the current linearized model, construct and solve the feasible pump nonlinear programming subproblem with the objective of minimizing the norm distance with the previous projection point, and obtain an integer solution.
[0035] Specifically, in the inner loop, the projection points are based on the previous iteration. Add projection points OA cutting: ; in, For point The gradient vector at point represents the constraint function. The direction of the rate of change at that point; For variables With point The difference vector; The dot product of the gradient vector and the difference vector is the constraint function. At point Linear approximation term at; It is a constraint function At point The function value at that location; For the first In the next outer iteration The projection points obtained from the next inner iteration.
[0036] Used to approximate nonlinear constraint sets and linearize nonlinear constraints. Form a cut set.
[0037] For each nonlinear constraint At point The OA cutting structure at the location is as follows: ; in, For point The gradient vector at point represents the constraint function. Direction of the rate of change at that point For variables With point The difference vector; The dot product of the gradient vector and the difference vector is the constraint function. At point Linear approximation term at; It is a constraint function At point The function value at that location; For the first Linearized points used in the next outer iteration to construct the external approximation cut.
[0038] Construct and solve for minimizing the previous projection point FP-MILP problem with distance as the objective function: ; in, for Norm, Used to measure variables With projection point The distance; Solving for new integer solutions yields new solutions. This is the new integer iteration point.
[0039] S1.4.2. With a fixed initial integer solution, solve the feasible pump nonlinear subproblem with the objective of minimizing the square of the Euclidean distance to the previous projection point, and obtain a continuous solution.
[0040] S1.4.3 Determine the projection distance between the current continuous solution and the previous projection point.
[0041] S1.4.4 If the projected distance is less than the convergence tolerance, the iteration is terminated, and the current feasible solution is determined as the preliminary optimization result.
[0042] Among them, the current feasible solution is determined based on the integer solution kernel continuous solution.
[0043] S1.4.5 If the projection distance is not less than the convergence tolerance, then the current feasible solution is determined as the initial iterative input data and the main loop step of the feasible pump iteration is returned to the mixed integer nonlinear programming model to be optimized.
[0044] Specifically, fixed integer variables By minimizing the previous projection point Using the Euclidean distance, solve the NLP subproblem with a distance target: ; in, The square of the Euclidean norm, used to measure the value of a point. Compared with the previous projection point The distance between two points; The current iteration point is a fixed discrete variable. in For external iteration counter, This is an internal iteration counter.
[0045] Solving yields continuous solutions And calculate the current continuous solution. Compared with the previous projection point Euclidean distance This is used to subsequently determine the convergence of the iteration. It checks the distance between continuous solutions and integer solutions; if the distance is less than the set convergence tolerance, then... Then record the current feasible solution. ,Right now otherwise, Return to step S1.4.1.
[0046] S102. Obtain multiple behavioral state variables during the initialization iterative optimization process.
[0047] Specifically, during the feasible pump iteration process, in addition to performing the above-mentioned rounding-projection iteration, the following behavioral state variables are simultaneously calculated and recorded: Integer solution steady state: Record the integer solution for each external iteration. If integer solutions are found in N consecutive external iterations If it remains unchanged, it is marked as a stable state with integer solutions; Projection distance change: Records the projection distance for each internal iteration. If the current projection distance Then it is determined that the projection process tends to be stable; Feasible state of continuous subproblems: Record the solution results of NLP subproblems in each projection stage. If the NLP subproblems are feasible for M consecutive times, that is, there are no constraints violated, then it is marked as a continuous feasible state. Rate of change of objective function: Calculate the objective function values of the three most recent external iterations. Calculate the relative rate of change using the following formula: ; like Then it is determined that the improvement of the objective function tends to be moderate.
[0048] S103. Based on the comparison threshold of the state variables corresponding to each of the multiple behavioral state variables, determine whether the optimization switching conditions are met.
[0049] Specifically, after each feasible pump external iteration (i.e., k increments), the following stage switching logic is executed: A phase switching signal is triggered if any of the following conditions are met: The integer solution remains unchanged for N consecutive iterations, i.e. ; Projection distance less than threshold ; It is feasible to solve the NLP subproblem M times in a row; The rate of change of the objective function is less than the threshold ; If none of the above conditions are met, continue with the feasible pump external iteration. Return to step S2.3.1.
[0050] When the phase switching signal is triggered, the feasible pump phase is terminated early, and the initial feasible solution is output. Upper Realm and the initial cut set The local linearization cut (OA cut) of the model in the current iteration is passed to the external approximation main loop stage as the initial cut of the OA main problem, providing it with a preheated, high-quality initial solution.
[0051] S104. When the optimization switching conditions are met, the external approximation main loop is called to perform iterative optimization of the preliminary optimization results until the convergence conditions are met, and then the target optimization results are output.
[0052] Specifically, in the initialization phase of the iterative optimization of the external approximation main loop, the preliminary optimization results based on the output of the feasible pump phase termination are used, i.e., the initial feasible solution. As an initial integer solution, the cut set generated during the feasible pump stage is added. The local linear approximation information of the model at the switching moment (including the gradient at the cut-off point, function values, etc.) is used as a warm start. The process of initiating precise optimization is then performed, constructing the MILP master problem for the initialization OA phase: ; in, For variables Minimize the optimization objective; This is the upper bound variable for the objective function, used to linearize the objective function; For the objective function at point The function value at that location; For the objective function at point The gradient vector at point represents the objective function. The direction of the rate of change at that point; The dot product of the gradient vector and the difference vector is the objective function. exist The first-order Taylor expansion linear approximation term at; For variables With point The difference vector; These are the cut points generated during the FP phase; and These are continuous variables at the cutting point. and integer variables The value of , Which is the cutting point? For all indices in the cut collection ; The cut set generated for the FP stage contains the index set of all linearized cuts; For the first A nonlinear constraint function at point The gradient vector at point represents the constraint function. The direction of the rate of change at that point; The inner product of the gradient vector and the difference vector is the constraint function. exist The first-order Taylor expansion linear approximation term at the given location.
[0053] Further solving the MILP master problem of OA yields new integer solutions. and the lower realm .
[0054] Proceed to the next step of the judgment: if If the convergence is determined, proceed to step S4.3.
[0055] Furthermore, fixed integer solutions Solve the NLP subproblem: ; in, For fixed integer solutions The objective function after that; For fixed integer solutions The subsequent nonlinear constraints; For fixed integer solutions Linear constraints afterward.
[0056] If the NLP solution is feasible, then update the upper bound. This generates new, precise OA cuts.
[0057] If the NLP solution is not feasible, then solve the feasibility recovery problem and form a feasibility cut.
[0058] Furthermore, the new cut is added to the main OA problem to generate an optimal cut, which is then constrained by linear inequalities. The value of makes such that Always greater than or equal to the objective function The linear approximation gradually approximates the true objective function; in,
[0059] .
[0060] Generate feasible cuts when the NLP subproblem is infeasible, i.e., exists. At this time, this cut eliminates infeasible regions, guiding subsequent iterations to move closer to the feasible region: ; in, It is the negative of the constraint function value.
[0061] Implement a segmentation and filtering strategy: By adding active segments As a threshold for determining active cuts, the absolute value of the dual variable is greater than... Cuts are considered "active," meaning they have a significant constraint effect on the current solution and should be retained and added to the OA main problem; otherwise, they can be removed to reduce the problem size and improve solution efficiency.
[0062] The termination conditions include: To achieve convergence accuracy, it employs a dual convergence criterion: (1) Absolute gap convergence: (default is) ); (2) Relative gap convergence: (default is) ); Or it may reach resource limits such as the maximum number of iterations or the maximum solution time limit.
[0063] If convergence fails (i.e., none of the above convergence conditions are met, and the resource limit is not reached), then return to step S4.2.1 (solving the MILP main problem of OA) and begin the next round of OA iteration. In the next round of iteration, the OA main problem will include newly added cuts, which may generate better integer solutions and further tighten the upper and lower bounds until final convergence.
[0064] Furthermore, output the final optimal solution. and the optimal target value This represents the final result of the paper production optimization model.
[0065] To verify the solution effect and practicality of the present invention, such as Figure 4 As shown, experiments were conducted using the optimization model problem of the cold-end system of a papermaking self-owned power plant as the verification object. Traditional external approximation algorithms are classic solutions for this type of problem, and are therefore used as a benchmark for comparison. This invention provides a solution method for a mixed-integer nonlinear model of papermaking that integrates adaptive feasible pumps and external approximation. By introducing an adaptive step switching mechanism, an FP-OA fusion framework with behavior monitoring and dynamic switching capabilities is constructed. First, the feasible pump algorithm quickly generates a high-quality initial feasible solution and a linearized cut set. Then, adaptive stage switching is triggered by monitoring the behavior state variables to avoid invalid iterations. Finally, the external approximation algorithm completes the accurate solution, ultimately obtaining the globally optimal solution.
[0066] Experimental results show that, under 359 sets of typical operating conditions, the traditional external approximation algorithm takes 178.74 seconds to solve all 359 conditions. In contrast, the framework of this invention, relying on a feasible pump algorithm for hot start-up, compresses the search cycle and reduces the overall solution time to 135.26 seconds, a reduction of 24.3%. Furthermore, compared to the actual power generation per ton of steam in a self-owned power plant, the optimization of this invention results in an average increase of 3.275 MW in power generation per ton of steam, representing an average optimization ratio of 1.3%.
[0067] S105. Based on the target optimization results, determine the optimized operating parameters for the papermaking process.
[0068] Furthermore, Figure 2 This is a structural block diagram of the papermaking hybrid integer nonlinear optimization device according to an embodiment of this application, as shown below. Figure 2 As shown, the device includes: The receiving module is used to receive the mixed-integer nonlinear programming model to be optimized, which is determined based on the objective problem to be optimized. The preliminary optimization result determination module is used to call the feasible pump to perform initial iterative optimization of the mixed integer nonlinear programming model to be optimized, and obtain the preliminary optimization results. The behavior state quantity acquisition module is used to acquire multiple behavior state quantities during the initialization iterative optimization process; The switching condition determination module is used to determine whether the optimization switching conditions are met based on the comparison threshold of the state quantities corresponding to multiple behavioral state quantities. The target optimization result output module is used to call the external approximation main loop to perform iterative optimization of the preliminary optimization result when the optimization switching condition is met, until the convergence condition is met and the target optimization result is output. The optimized operating parameter determination module is used to determine the optimized operating parameters of the papermaking process based on the target optimization results.
[0069] The application of the relevant modules of the device in this example can be referred to the relevant introduction of the method principle above, and will not be repeated here.
[0070] above Figure 2 The papermaking hybrid integer nonlinear optimization device in this embodiment of the invention is described in detail from the perspective of modular functional entities. The electronic device in this embodiment of the invention is described in detail from the perspective of hardware processing.
[0071] Figure 3 This is a schematic diagram of the structure of an electronic device 300 provided in an embodiment of the present invention. The electronic device 300 can vary significantly due to different configurations or performance characteristics. It may include one or more central processing units (CPUs) 310 (e.g., one or more processors) and a memory 320, and one or more storage media 330 (e.g., one or more mass storage devices) for storing application programs 333 or data 332. The memory 320 and storage media 330 can be temporary or persistent storage. The program stored in the storage media 330 may include one or more modules (not shown in the diagram), each module including a series of instruction operations on the electronic device 300. Furthermore, the processor 310 may be configured to communicate with the storage media 330 and execute the series of instruction operations in the storage media 330 on the electronic device 300.
[0072] Electronic device 300 may also include one or more power supplies 340, one or more wired or wireless network interfaces 350, one or more input / output interfaces 360, and / or one or more operating systems 331, such as Windows Server, MacOSX, Unix, Linux, FreeBSD, etc. Those skilled in the art will understand that... Figure 3 The illustrated electronic device structure does not constitute a limitation on electronic devices and may include more or fewer components than illustrated, or combine certain components, or have different component arrangements.
[0073] The present invention also provides a computer-readable storage medium, which can be a non-volatile computer-readable storage medium or a volatile computer-readable storage medium, wherein the computer-readable storage medium stores instructions that, when executed on a computer, cause the computer to perform the steps of any of the above-described papermaking hybrid integer nonlinear optimization methods.
[0074] Those skilled in the art will clearly understand that, for the sake of convenience and brevity, the specific working process of the system, device, or unit described above can be referred to the corresponding process in the foregoing method embodiments, and will not be repeated here.
[0075] If the integrated unit is implemented as a software functional unit and sold or used as an independent product, it can be stored in a computer-readable storage medium. Based on this understanding, the technical solution of the present invention, in essence, or the part that contributes to the prior art, or all or part of the technical solution, can be embodied in the form of a software product. This computer software product is stored in a storage medium and includes several instructions to cause a computer device (which may be a personal computer, server, or network device, etc.) to execute all or part of the steps of the methods of the various embodiments of the present invention. The aforementioned storage medium includes various media capable of storing program code, such as USB flash drives, portable hard drives, read-only memory (ROM), random access memory (RAM), disks, or optical discs.
[0076] The above embodiments are only used to illustrate the technical solutions of the present invention, and are not intended to limit it. 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 of the technical features. Such modifications or substitutions do not cause the essence of the corresponding technical solutions to deviate from the spirit and scope of the technical solutions of the embodiments of the present invention.
Claims
1. A mixed integer nonlinear optimization method for papermaking, characterized in that, The method includes: Receive a mixed-integer nonlinear programming model to be optimized, wherein the mixed-integer nonlinear programming model to be optimized is determined based on the target optimization problem; The feasible pump is invoked to perform initial iterative optimization on the mixed-integer nonlinear programming model to be optimized, and preliminary optimization results are obtained. Obtain multiple behavioral state variables during the initialization iterative optimization process; Based on the comparison threshold of the state variables corresponding to each of the multiple behavioral state variables, determine whether the optimization switching conditions are met. When the optimization switching conditions are met, the external approximation main loop is invoked to perform iterative optimization of the preliminary optimization results until the convergence conditions are met, and then the target optimization results are output. Based on the optimization results of the target, the optimized operating parameters of the papermaking process are determined.
2. The papermaking mixed integer nonlinear optimization method according to claim 1, characterized in that, The invocation of the feasible pump to perform initialization and iterative optimization of the mixed-integer nonlinear programming model to be optimized yields preliminary optimization results including: Determine the feasible pump configuration parameters; By combining the feasible pump configuration parameters, a continuous relaxation problem is solved for the mixed-integer nonlinear programming model to be optimized, and the relaxed solution is obtained. Based on the relaxation solution results, obtain the initial iterative input data for feasible pump initialization iterative optimization; Based on the initial iterative input data, the mixed integer nonlinear programming model to be optimized is initialized and iteratively optimized to obtain preliminary optimization results.
3. The papermaking mixed integer nonlinear optimization method according to claim 1, characterized in that, The continuous relaxation problem solution obtained by combining the feasible pump configuration parameters with the mixed integer nonlinear programming model to be optimized includes: Relax the integer constraints of the mixed-integer nonlinear programming model to be optimized, and change the integer variables. The feasible domain from relaxation This leads to a continuous relaxation nonlinear programming problem; in, for The original feasible region is The set of integers, After relaxation Constraints exist Take any real number within the interval. The lower realm For variables The lower bound vector, For the upper realm, For variables The upper bound vector; By combining the relaxation formula, the continuous relaxation nonlinear programming problem is solved to obtain the relaxed solution. The relaxation formula is as follows: ； in, For the objective function Minimize the optimization objective; The objective function is, i.e. , For continuous variables, For integer variables, For the corresponding variable and The transpose of the objective function coefficient vector; For nonlinear constraints, This is the index of which nonlinear constraint; For linear constraints, For corresponding continuous variables The coefficient matrix, For the corresponding integer variable The coefficient matrix, It is a constant vector.
4. The papermaking mixed integer nonlinear optimization method according to claim 3, characterized in that, The initial iteration optimization of the mixed-integer nonlinear programming model to be optimized based on the initial iteration input data yields preliminary optimization results including: Based on the initial iterative input data, a feasible pump iterative main loop is adopted for the mixed integer nonlinear programming model to be optimized using an inner and outer double-layer loop structure, wherein each inner iteration includes: Based on the projection point obtained in the previous iteration, an external approximation cut of the nonlinear constraint at that point is added to the current linearized model. A feasible pump nonlinear programming subproblem with the objective of minimizing the norm distance with the previous projection point is constructed and solved to obtain an integer solution. By fixing the initial integer solution, the feasible pump nonlinear subproblem with the objective of minimizing the square of the Euclidean distance to the previous projection point is solved to obtain a continuous solution; Determine the projection distance between the current continuous solution and the previous projection point; If the projected distance is less than the convergence tolerance, the iteration is terminated, and the current feasible solution is determined as the preliminary optimization result, wherein the current feasible solution is determined based on the integer solution and the continuous solution; If the projected distance is not less than the convergence tolerance, then the current feasible solution is determined as the initial iterative input data and the main loop step of the feasible pump iteration is returned to the mixed integer nonlinear programming model to be optimized.
5. The papermaking mixed integer nonlinear optimization method according to claim 4, characterized in that, The acquisition of multiple behavioral state variables during the initialization iterative optimization process includes: During the initialization iterative optimization process, behavior monitoring and state variable calculation are performed simultaneously to determine multiple behavior state vectors, including: Record the integer solution of each external iteration. If the integer solution remains unchanged in the external iteration for a preset number of consecutive iterations, it is marked as a stable state of integer solution. Record the projection distance of each internal iteration. If the current projection distance is less than the preset distance, mark it as a stable state of the projection process. Record the solution results of the nonlinear programming problem in each projection stage. If the nonlinear programming subproblem has a solution that satisfies all constraints in each consecutive number of solutions, it is marked as a continuously feasible state.
6. The papermaking mixed integer nonlinear optimization method according to claim 4, characterized in that, When the optimization switching condition is met, the external approximation main loop is invoked to perform iterative optimization of the preliminary optimization result until the convergence condition is met, and then the target optimization result is output. This includes: Based on the preliminary optimization results, the upper bound of the objective function, and the initial linearized cut set, an initial mixed-integer linear programming master problem is constructed. The initial linearized cut set is determined based on the external approximation cut of adding nonlinear constraints at the corresponding point to the corresponding linearized model in each iteration. Based on the aforementioned mixed-integer linear programming master problem, candidate integer solutions and the current lower bound are determined; By fixing the candidate integer solutions, the nonlinear programming subproblem is solved to obtain the solution result; When the solution satisfies all constraints and the iteration conditions, the target optimization result is output. The main problem of the mixed-integer linear programming is: ； in, For variables Minimize the optimization objective; This is the upper bound variable for the objective function, used to linearize the objective function; For the objective function at point The function value at that location; For the objective function at point The gradient vector at point represents the objective function. The direction of the rate of change at that point; The dot product of the gradient vector and the difference vector is the objective function. exist The first-order Taylor expansion linear approximation term at; For variables With point The difference vector; These are the cut points generated during the FP phase; and These are continuous variables at the cutting point. and integer variables The value of , Which is the cutting point? For all indices in the cut collection ; The cut set generated for the FP stage contains the index set of all linearized cuts; For the first A nonlinear constraint function at point The gradient vector at point represents the constraint function. The direction of the rate of change at that point; The inner product of the gradient vector and the difference vector is the constraint function. exist The first-order Taylor expansion linear approximation term at; The nonlinear programming subproblem is: ； in, For fixed integer solutions The objective function after that; For fixed integer solutions The subsequent nonlinear constraints; For fixed integer solutions Linear constraints afterward.
7. The papermaking mixed integer nonlinear optimization method according to claim 6, characterized in that, When the solution result contains unsatisfied constraints, the method includes: Solve the feasibility recovery problem and form a feasibility segmentation; The new cut is added to the linearized main problem to generate the optimal cut, which is then constrained by linear inequalities. The value of makes such that Always greater than or equal to the objective function A linear approximation is obtained to gradually approximate the true objective function; in, ; The feasibility segmentation is generated so that when a nonlinear programming subproblem is infeasible, the region that leads to infeasibility is identified and excluded through the feasibility segmentation. Among them, when there is When a nonlinear programming subproblem is determined to be infeasible, this infeasibility point is used as a new cutoff point. The corresponding nonlinear constraint function is then subjected to a first-order Taylor expansion at that point, transforming it into a linear approximation constraint, where the linear approximation constraint is: ； in, It is the negative of the constraint function value.
8. A papermaking hybrid integer nonlinear optimization device, characterized in that, The device includes: The receiving module is used to receive a mixed-integer nonlinear programming model to be optimized, wherein the mixed-integer nonlinear programming model to be optimized is determined based on the target optimization problem; The preliminary optimization result determination module is used to call the feasible pump to perform initial iterative optimization on the mixed integer nonlinear programming model to be optimized, and obtain the preliminary optimization result. The behavior state quantity acquisition module is used to acquire multiple behavior state quantities during the initialization iterative optimization process; The switching condition determination module is used to determine whether the optimization switching conditions are met based on the comparison threshold of the state quantities corresponding to multiple behavioral state quantities. The target optimization result output module is used to call the external approximation main loop to perform iterative optimization of the preliminary optimization result when the optimization switching condition is met, until the convergence condition is met and the target optimization result is output. The optimized operating parameter determination module is used to determine the optimized operating parameters of the papermaking process based on the target optimization results.
9. An electronic device, characterized in that, The electronic device includes a memory and at least one processor, the memory storing instructions; the at least one processor invokes the instructions in the memory to cause the electronic device to perform the steps of the papermaking hybrid integer nonlinear optimization method as described in any one of claims 1-7.
10. A computer-readable storage medium storing instructions thereon, characterized in that, When the instructions are executed by the processor, they implement the various steps of the papermaking hybrid integer nonlinear optimization method as described in any one of claims 1-7.