Lagrange dual problem solving method for mixed integer linear programming
By transforming the Lagrange dual problem into a linear programming problem and constructing equivalent Lagrange functions and bounded dual functions, the convergence efficiency and accuracy problems caused by iteration in mixed-integer linear programming are solved, and the unit combination problem is solved efficiently and accurately.
Patent Information
- Application Number
- CN202511104741.9
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- Filing Date
- 2025-08-07
- Publication Date
- 2025-12-30
AI Technical Summary
In existing technologies, the Lagrange relaxation method for mixed integer linear programming suffers from convergence efficiency and progress issues due to iterative dependencies when solving unit combination problems, making it difficult to guarantee accuracy while avoiding iteration.
The Lagrange dual problem is transformed into a linear programming problem. By constructing the Lagrange function and the bounded dual function, they are merged into a single-layer problem. Using convex combinations and constraints, the Lagrange function is solved through equivalent transformations to obtain the optimal dual value.
It improves computational efficiency, ensures the accuracy of the solution, solves the convergence efficiency and convergence progress problems caused by iteration, and provides an efficient solution for solving large-scale unit combination problems.
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Figure CN121235166A_ABST
Abstract
Description
Technical Field
[0001] This application relates to the field of power system technology, and in particular to a method for solving the Lagrange duality problem of mixed integer linear programming. Background Technology
[0002] Unit commissioning is a crucial topic in power dispatching. Generally, unit commissioning optimizes decisions regarding unit start-up and shutdown states and output, meaning it is often formulated as a mixed-integer programming problem. Due to its NP-hard mathematical nature, mixed-integer programming problems are often difficult to solve directly. With advancements in optimization techniques, Lagrange relaxation has become a highly effective solution technique. It can relax complex constraints that are difficult to handle in the original mixed-integer linear programming problem, making it easier to solve. However, it's important to note that solving unit commissioning problems using Lagrange relaxation requires solving the Lagrange dual of a mixed-integer linear programming problem, often relying on numerous subgradient iterations. This limits the full potential of the Lagrange relaxation method.
[0003] While the topic of using Lagrange relaxation methods to solve unit combination problems is not new, research on improving its convergence efficiency and accuracy has been a hot topic. The aim is to fully leverage the advantages of Lagrange relaxation methods in solving large-scale problems by enhancing convergence efficiency and accuracy. Currently, related research focuses on this, but no effective method can guarantee accuracy while avoiding iteration; this has been a persistent challenge in the field. Summary of the Invention
[0004] This application provides a method for solving the Lagrange dual problem of mixed-integer linear programming. It transforms the Lagrange dual problem, which relies on iteration to solve the original mixed-integer linear programming problem, into solving a single linear programming problem. This avoids the problems of convergence efficiency and convergence progress caused by iteration, improving computational efficiency while ensuring the accuracy of the solution.
[0005] To address the aforementioned technical problems, this application provides a method for solving the Lagrange dual problem of mixed-integer linear programming, comprising the following steps: First, based on the model of the mixed-integer linear programming problem, the corresponding Lagrange dual function is obtained; then, linear programming is used to solve the Lagrange dual function, merging the two-level problem of the Lagrange dual function into a single-level problem, and constructing the Lagrange function; next, based on the Lagrange function and constraints, a bounded dual function is obtained; then, based on the bounded dual function, a convex combination is obtained; the feasible region defined by the convex combination is consistent with the convex hull of the mixed-integer linear programming set; finally, based on the convex combination, the Lagrange function is equivalently transformed, and the transformed Lagrange function is solved to obtain the optimal dual value of the Lagrange dual function.
[0006] In some exemplary embodiments, the Lagrangian dual function corresponding to the model of the mixed-integer linear programming problem is obtained based on the compact form of the model; the compact form of the mixed-integer linear programming problem is expressed as:
[0007] C1 min q T y (1)
[0008] stDy≥d (2)
[0009]
[0010] Where q and d represent coefficient vectors, D represents the coefficient matrix, and y represents the decision variable vector. Let q represent the set of mixed-integer linear programming problems with respect to y; in the model, q T y is a linear objective function, and Dy≥d is a system coupling constraint. These are the constraints of the generator set, represented by a solution space set.
[0011] In some exemplary embodiments, the Lagrange dual function obtained based on the compact form of the mixed-integer linear programming problem model is shown below:
[0012]
[0013] Where ω represents the dual variable corresponding to formula (2), Δ C1 (ω) represents the Lagrange dual function, Λ C1 (y,ω) represents the corresponding Lagrange function, in the following form:
[0014]
[0015] Among them, Λ C1 (y,ω) is a linear function of y; for a given dual variable ω, it is solved by the set of mixed-integer linear programming problems. Inner Lagrange function Λ C1 Finding the minimum value of (y,ω) yields the Lagrange dual function Δ. C1 (ω).
[0016] In some exemplary embodiments, it is assumed that For a set of mixed-integer linear programming problems The set of all vertices of the convex hull, v i for The Lagrange dual function of the i-th vertex in the array is expressed as:
[0017]
[0018] Where, Δ C1 (ω) is the Lagrange dual function, and η is an auxiliary variable representing Δ. C1 The optimal value of (ω), Λ C1 (v i ,ω) is about v i A linear function.
[0019] In some exemplary embodiments, the two-level problem of the Lagrange dual function is merged into a single-level problem, and the model of the single-level problem is shown below:
[0020]
[0021] The constructed Lagrange function is shown below:
[0022]
[0023] Where Δ(μ) denotes the dual function, Λ(η,ω,μ) denotes the corresponding Lagrangian function, and μ i Let be the dual variable corresponding to the constraint of the i-th vertex.
[0024] In some exemplary embodiments, the Lagrangian function Λ(η,ω,μ) corresponding to the dual function Δ(μ) is as follows:
[0025]
[0026] in,
[0027] In some exemplary embodiments, the constraints are:
[0028]
[0029] in, For a set of mixed-integer linear programming problems The set of all vertices of the convex hull, v i for The i-th vertex in, μ i Let be the dual variable corresponding to the constraint of the i-th vertex, where D represents the coefficient matrix and d represents the coefficient vector.
[0030] In some exemplary embodiments, the bounded dual function is as follows:
[0031]
[0032] In some exemplary embodiments, the dual variable will be the vertex v i for Weights are assigned to form convex combinations; the convex combinations are shown below:
[0033]
[0034] In some exemplary embodiments, the equivalent transformed Lagrangian function is as follows:
[0035] C2 min q T y (13)
[0036] st Dy≥d (14)
[0037]
[0038] Where q and d represent coefficient vectors, D represents the coefficient matrix, and y represents the decision variable vector. Let y represent the set of mixed-integer linear programming problems. This represents the operator for constructing the convex hull of the input set.
[0039] The technical solution provided in this application has at least the following advantages:
[0040] This application provides a method for solving the Lagrange dual problem of mixed-integer linear programming. The method includes the following steps: First, based on the model of the mixed-integer linear programming problem, the corresponding Lagrange dual function is obtained; then, the Lagrange dual function is solved using linear programming, merging the two-level problem of the Lagrange dual function into a single-level problem, and constructing the Lagrange function; next, based on the Lagrange function and constraints, a bounded dual function is obtained; then, based on the bounded dual function, a convex combination is obtained; the feasible region defined by the convex combination is consistent with the convex hull of the mixed-integer linear programming set; finally, based on the convex combination, the Lagrange function is equivalently transformed, and the transformed Lagrange function is solved to obtain the optimal dual value of the Lagrange dual function.
[0041] The method for solving the Lagrange dual problem of mixed-integer linear programming provided in this application designs an equivalent transformation method and constructs a linear programming problem based on it. The linear programming problem constructed in this way has an equivalent solution to the Lagrange dual problem of the mixed-integer linear programming problem. Therefore, the Lagrange dual problem of solving the original mixed-integer linear programming problem, which depends on iteration, is transformed into solving a linear programming problem. Because it avoids the convergence efficiency and convergence progress problems caused by iteration, it improves computational efficiency while ensuring the accuracy of the solution. Attached Figure Description
[0042] One or more embodiments are illustrated by way of example with reference to the accompanying drawings. These illustrations do not constitute a limitation on the embodiments, and unless otherwise stated, the figures in the drawings are not to be limited by scale.
[0043] Figure 1 This is a flowchart illustrating a method for solving the Lagrange duality problem of mixed integer linear programming, provided as an embodiment of this application. Detailed Implementation
[0044] As the background technology shows, there is currently no effective method that can guarantee accuracy while avoiding iteration, which has been a long-standing problem in the field.
[0045] Considering that improving convergence efficiency and accuracy is key to fully utilizing the Lagrange relaxation method, this application addresses this key issue by studying the Lagrange dual problem of mixed-integer linear programming problems and proposing an explicit linear programming form for solving the Lagrange dual problem of mixed-integer programming. This application provides a method for solving the Lagrange dual problem of mixed-integer linear programming, including the following steps: First, based on the model of the mixed-integer linear programming problem, the corresponding Lagrange dual function is obtained; then, linear programming is used to solve the Lagrange dual function, merging the two-level problem of the Lagrange dual function into a single-level problem, constructing the Lagrange function; next, based on the Lagrange function and constraints, a bounded dual function is obtained; then, based on the bounded dual function, a convex combination is obtained; the feasible region defined by the convex combination is consistent with the convex hull of the mixed-integer linear programming set; finally, based on the convex combination, the Lagrange function is equivalently transformed, and the transformed Lagrange function is solved to obtain the optimal dual value of the Lagrange dual function. The linear programming form provided in this application avoids the iterations required to solve the Lagrange dual problem of mixed-integer programming, and the dual solution obtained by solving the proposed form is the theoretically optimal solution. It eliminates iteration, improving solution efficiency while ensuring solution accuracy, thus solving a long-standing technical bottleneck. This application of Lagrange relaxation techniques provides an effective solution to large-scale unit combination problems, filling a technological gap. This application transforms the iterative solution of the original mixed-integer linear programming problem into solving a single linear programming problem, avoiding the convergence efficiency and convergence progress issues caused by iteration, improving computational efficiency while ensuring solution accuracy.
[0046] The embodiments of this application will now be described in detail with reference to the accompanying drawings. However, those skilled in the art will understand that many technical details have been provided in the embodiments of this application to facilitate a better understanding of the application. However, the technical solutions claimed in this application can be implemented even without these technical details and various variations and modifications based on the following embodiments.
[0047] See Figure 1 This application provides a method for solving the Lagrange dual problem of mixed-integer linear programming, including the following steps:
[0048] Step S1: Based on the model of the mixed integer linear programming problem, obtain the corresponding Lagrange dual function.
[0049] Step S2: Solve the Lagrange dual function using linear programming, merging the two-level problem of the Lagrange dual function into a single-level problem, and construct the Lagrange function.
[0050] Step S3: Based on the Lagrangian function and constraints, obtain the bounded dual function.
[0051] Step S4: Based on the bounded dual function, a convex combination is obtained; the feasible region defined by the convex combination is consistent with the convex hull of the mixed integer linear programming set.
[0052] Step S5: Based on convex combination, perform an equivalent transformation on the Lagrange function, solve for the transformed Lagrange function, and obtain the optimal dual value of the Lagrange dual function.
[0053] In some embodiments, the Lagrangian dual function corresponding to the model of the mixed-integer linear programming problem is obtained in step S1 based on the compact form of the model; the compact form of the mixed-integer linear programming problem is expressed as:
[0054] C1 min q T y (1)
[0055] st Dy≥d (2)
[0056]
[0057] Where q and d represent coefficient vectors, D represents the coefficient matrix, and y represents the decision variable vector. Let q represent the set of mixed-integer linear programming problems with respect to y; in the model, q T y is a linear objective function, and Dy≥d is a system coupling constraint. These are the constraints of the generator set, represented by a solution space set.
[0058] In some embodiments, the Lagrange dual function obtained in step S1 based on the compact form of the mixed-integer linear programming problem is shown below:
[0059]
[0060] Where ω represents the dual variable corresponding to formula (2), Δ C1 (ω) represents the Lagrange dual function, Λ C1 (y,ω) represents the corresponding Lagrange function, in the following form:
[0061]
[0062] Among them, Λ C1 (y,ω) is a linear function of y; for a given dual variable ω, it is solved by the set of mixed-integer linear programming problems. Inner Lagrange function Λ C1 Finding the minimum value of (y,ω) yields the Lagrange dual function Δ. C1(ω). Generally speaking, optimizing a linear function within a set of mixed-integer linear programming problems is equivalent to optimizing it on the convex hull of that set. Therefore, the dual function D... C1 The optimal solution of (w) must be located at some vertex on the convex hull of Y.
[0063] In some embodiments, a For a set of mixed-integer linear programming problems The set of all vertices of the convex hull, v i for The i-th vertex in the array, according to the above definition, can be computed using the following linear programming method to calculate the Lagrange dual function D. C1 (w):
[0064]
[0065] Where, Δ C1 (ω) is the Lagrange dual function, and η is an auxiliary variable representing Δ. C1 The optimal value of (ω), Λ C1 (v i ,ω) is about v i A linear function.
[0066] D C1 The solution to (w) can be rewritten as a maximization problem, which can combine the two-layer problem of the Lagrange dual function into a single-layer problem. The model of the single-layer problem is shown below:
[0067]
[0068] Of course, for the set of mixed-integer linear programming Enumerating all vertices of the convex hull is an Np-hard problem. Therefore, the resulting linear programming problem (Equation (7)) contains almost an infinite number of constraints. According to the strong duality theorem, linear programming does not have a duality difference, so the duality problem of linear programming can be used to alleviate the burden of these constraints. Here, let μ i The Lagrangian function constructed for the dual variable corresponding to the constraint of the i-th vertex is as follows:
[0069]
[0070] Here, Δ(μ) represents the dual function, and Λ(η,ω,μ) represents the corresponding Lagrange function.
[0071] In some embodiments, the Lagrangian function Λ(η,ω,μ) corresponding to the dual function Δ(μ) is as follows:
[0072]
[0073] in,
[0074] To ensure that D(m) is bounded, the model must satisfy constraints. In some embodiments, the constraints are:
[0075]
[0076] in, For a set of mixed-integer linear programming problems The set of all vertices of the convex hull, v i for The i-th vertex in, μ i Let be the dual variable corresponding to the constraint of the i-th vertex, where D represents the coefficient matrix and d represents the coefficient vector.
[0077] In some embodiments, the bounded dual function is as follows:
[0078]
[0079] Where, q T It is a linear objective function.
[0080] From formulas (10) and (11), it can be seen that the dual variable will be the vertex v i for Weights are assigned to form convex combinations; the convex combinations are shown below:
[0081]
[0082] It should be noted that the feasible region defined by formula (12) is the same as the set of mixed-integer linear programming problems. The convex hulls are completely identical.
[0083] In some embodiments, the equivalent transformed Lagrange function is as follows:
[0084] C2 min q T y (13)
[0085] st Dy≥d (14)
[0086]
[0087] Where q and d represent coefficient vectors, D represents the coefficient matrix, and y represents the decision variable vector. Let y represent the set of mixed-integer linear programming problems. This represents the operator for constructing the convex hull of the input set. The optimal dual of C1 can be obtained by solving only C2. Furthermore, the optimal dual of C1 can be directly derived based on C2.
[0088] This application proposes an equivalent transformation method for the Lagrange dual problem of mixed-integer linear programming by analyzing the mathematical essence of the Lagrange dual problem. Simultaneously, it constructs a linear programming form equivalent to the solution of the Lagrange dual problem of mixed-integer programming, avoiding the iterations required to solve the Lagrange dual problem of mixed-integer programming. Furthermore, the dual solution obtained by solving the proposed form is the theoretically optimal solution. This approach eliminates iterations, improves solution efficiency, and ensures solution accuracy, thus solving a long-standing technical bottleneck.
[0089] Based on the above technical solutions, this application provides a method for solving the Lagrange dual problem of mixed-integer linear programming. The method includes the following steps: First, based on the model of the mixed-integer linear programming problem, the corresponding Lagrange dual function is obtained; then, linear programming is used to solve the Lagrange dual function, merging the two-level problem of the Lagrange dual function into a single-level problem, and constructing the Lagrange function; next, based on the Lagrange function and constraints, a bounded dual function is obtained; then, based on the bounded dual function, a convex combination is obtained; the feasible region defined by the convex combination is consistent with the convex hull of the mixed-integer linear programming set; finally, based on the convex combination, the Lagrange function is equivalently transformed, and the transformed Lagrange function is solved to obtain the optimal dual value of the Lagrange dual function.
[0090] The method for solving the Lagrange dual problem of mixed-integer linear programming provided in this application designs an equivalent transformation method and constructs a linear programming problem based on it. The linear programming problem constructed in this way has an equivalent solution to the Lagrange dual problem of the mixed-integer linear programming problem. Therefore, the Lagrange dual problem of solving the original mixed-integer linear programming problem, which depends on iteration, is transformed into solving a linear programming problem. Because it avoids the convergence efficiency and convergence progress problems caused by iteration, it improves computational efficiency while ensuring the accuracy of the solution.
[0091] Those skilled in the art will understand that the above-described embodiments are specific examples of implementing this application, and in practical applications, various changes in form and detail may be made without departing from the spirit and scope of this application. Any person skilled in the art can make their own modifications and alterations without departing from the spirit and scope of this application; therefore, the scope of protection of this application should be determined by the scope defined in the claims.
Claims
1. A method of solving a Lagrangian dual problem of a mixed integer linear programming, characterized in that, The method comprises the following steps: a Lagrange dual function corresponding to the model of the mixed integer linear programming problem is obtained based on the model of the mixed integer linear programming problem; the Lagrange dual function is solved by using linear programming, the double-layer problem of the Lagrange dual function is combined into a single-layer problem, and a Lagrange function is constructed; a bounded dual function is obtained based on the Lagrange function and a constraint condition; a convex combination is obtained based on the bounded dual function; a feasible region defined by the convex combination is consistent with a convex hull of a mixed integer linear programming set; the Lagrange function is equivalently transformed based on the convex combination, the equivalently transformed Lagrange function is solved, and an optimal dual value of the Lagrange dual function is obtained.
2. The method of claim 1, wherein, a Lagrange dual function corresponding to the model of the mixed integer linear programming problem is obtained based on a compact form of the model of the mixed integer linear programming problem; the compact form of the model of the mixed integer linear programming problem is represented as: C1min q T y (1) s.t. Dy ≥ d (2) where q and d represent coefficient vectors, D represents a coefficient matrix, y represents a decision variable vector, represents a set of mixed integer linear programming problems with respect to y; in the model, q T y is a linear objective function, Dy ≥ d is a system coupling constraint condition, is a unit commitment constraint condition, expressed in a solution space set.
3. The method of claim 2, wherein, the Lagrange dual function obtained based on the compact form of the model of the mixed integer linear programming problem is as follows: where ω denotes a dual variable corresponding to equation (2), Δ C1 (ω) denotes a Lagrangian dual function, Λ C1 (y, ω) denotes a corresponding Lagrangian function, of the form where Λ C1 (y, ω) is a linear function in y; for a given dual variable ω, the Lagrangian function Λ (y, ω) is minimized over the set of mixed integer linear programs C1 (y, ω) to obtain the Lagrangian dual function Δ C1 (ω).
4. The method of claim 3, wherein, Let be the set of all vertices of the convex hull of the mixed integer linear programming set Let i be the i-th vertex in The Lagrangian dual function is given by where Δ C1 (ω) is the Lagrangian dual function, η is an auxiliary variable, represents the optimal value of Δ C1 (ω), and Λ C1 (v i , ω) is a linear function in v i .
5. The method of claim 4, wherein, the double-layer problem of the Lagrange dual function is combined into a single-layer problem, and a model of the single-layer problem is as follows: the constructed Lagrange function is as follows: where Δ(μ) denotes the dual function, Λ(η, ω, μ) denotes the corresponding Lagrangian function, μ i is the corresponding dual variable for the ith vertex constraint.
6. The method of claim 5, wherein, a Lagrange function Λ(η, ω, μ) corresponding to the dual function Δ(μ) is as follows: wherein 7. The method of claim 6, wherein, the constraint condition is as follows: and wherein, is a mixed integer linear programming set is a convex hull of all vertices of the mixed integer linear programming set i is the i-th vertex in is the i-th vertex in i is the i-th vertex constraint corresponding dual variable, D denotes the coefficient matrix, and d denotes the coefficient vector.
8. The method of claim 7, wherein, the bounded dual function is as follows:
9. The method of claim 8, wherein, The dual variable will be denoted by v i For Assigning weights, constructing convex combinations; the convex combinations are as follows: simultaneously satisfy and 10. The method of claim 1, wherein, the equivalently transformed Lagrange function is as follows: C2min q T y (13) s.t. Dy ≥ d (14) where q and d represent coefficient vectors, D represents a coefficient matrix, y represents a decision variable vector, represents a set of mixed integer linear programs with respect to y, represents an operator that constructs a convex hull of the input set.