A method for generating multi-objective non-inferior solution sets for inter-basin water transfer based on convex set feasible region

Abstract

The present invention discloses a method for generating a multi-objective non-inferior solution set for inter-basin water transfer based on a convex set feasible domain. The method comprises the following steps: collecting hydrological data from the study area, selecting a primary objective function from pre-constructed objective functions, constructing a plane set covering the entire spatial angle, and searching for boundary points of the target space corresponding to other objective functions; pre-processing the boundary points of the target space, and constructing a convex set boundary range of the target space using a spatial boundary convex hull algorithm; generating an ε-constrained threshold combination within the convex set boundary range of the target space; and for each ε-constrained threshold combination, converting the multi-objective optimization problem into multiple single-objective optimization sub-problems, solving each of these sub-problems using a sparse optimization interior point method, to generate a non-inferior solution set for the inter-basin water transfer scheme. The present invention effectively reduces the possibility of generating infeasible solutions and duplicate solutions in traditional ε-constrained algorithms, and improves the efficiency and quality of generating non-inferior solution sets.

Description

Technical Field

[0001] The present invention relates to inter-basin water transfer technology in the field of water conservancy projects, and in particular to a method for efficiently generating a multi-objective non-inferior solution set for inter-basin water transfer based on a convex set feasible domain. Background Art

[0002] Inter-basin water transfer projects coordinate the allocation of local and external water through real-time scheduling of multiple water sources, comprehensively optimizing water shortage losses, water transfer costs, and ecological benefits. Inter-basin water transfer involves a wide range of regions, long water transfer distances, and a large number of projects. Especially in open water systems, inter-basin water transfers must integrate the connectivity of complex water networks and comprehensively consider the coordination of multiple regions, projects, water sources, users, and departments. The complex relationship structure and the large number of decision variables constitute a typical complex system multi-objective optimization problem.

[0003] Finding a set of non-inferior solutions is key to multi-objective optimization problems. In inter-basin water transfer decisions, it provides decision-makers with a set of alternative options that balance multiple objectives, such as water supply security, ecological balance, and economic costs. High-quality non-inferior solutions have the following characteristics: first, they should have broad coverage—the solution set should fully cover the target space, allowing decision-makers to make full choices based on actual scheduling needs; second, they should have high precision—the solution set should be as close as possible to the true Pareto frontier.

[0004] The core idea of ​​the ε-constraint method is to reduce the dimensionality of a multi-objective problem by transforming it into a series of single-objective optimization problems. This method selects a primary objective function for optimization and transforms the remaining objective functions into constraints, assigning them an allowable threshold range (ε value). By gradually adjusting the range of these ε values, different solutions can be obtained. However, traditional ε-constraint algorithms generally use a grid search method to set a combination of constraint thresholds, rejecting only feasible solutions. This method has the following shortcomings: First, the generation of non-inferior solution sets is inefficient, and the resulting candidate sets often contain infeasible solutions. Second, the quality of non-inferior solution sets is low, and the grid-based constraint threshold setting may lead to duplicate solutions in the solution set. This is primarily because the setting of the ε-constraint threshold combination fails to consider the boundary range of the constraint space, placing grid points with ε values ​​outside the feasible range, resulting in infeasible or duplicate solutions.

[0005] The multi-objective optimization of inter-basin water transfer in the medium and long term mainly involves the coordinated dispatch of local water and external water. After being transformed by the ε constraint algorithm, this type of problem is presented as a convex programming problem: (1) The constraints of water dispatch are all linear constraints except the discharge capacity constraint, and the discharge capacity does not constitute a substantial constraint on water dispatch in the medium and long term dispatch process; (2) The multi-objective optimization of the water transfer system usually focuses on water supply security, economic benefits, ecological protection and other goals. The objective function is often constructed by the sum of squares of water shortage rate, water transfer volume or cost, ecological water shortage or sum of squares of water shortage rate, etc., which is a convex function. After being transformed by the ε constraint, it becomes a convex function constraint. The objective space of this problem is proved to be a convex set. Under the above conditions, the boundary of the objective space of the corresponding multi-objective optimization dispatch can be directly determined by convex optimization technology.

[0006] Traditional methods for finding the target space for multi-objective optimization problems include constraint enumeration and combination methods, as well as the simplex method. However, these methods are only applicable to optimization problems with a relatively low dimensionality of decision variables. Multi-objective water scheduling models for inter-basin water transfers involve numerous decision variables and complex constraints, making traditional methods impractical for searching the target space.

[0007] Therefore, the present invention proposes a method for generating multi-objective non-inferior solution sets for cross-basin water transfer based on the convex set feasible domain. By exploring the boundary range of the convex set target space and optimizing the constraint condition threshold combination setting, the generation efficiency and quality of the non-inferior solution sets are improved. Summary of the Invention

[0008] Purpose of the invention: The present invention aims to provide a method for generating multi-objective non-inferior solution sets for cross-basin water transfer based on convex set feasible domain. By exploring the boundary range of the target space, optimizing the combination setting of constraint thresholds, and improving the traditional ε constraint algorithm, the possibility of generating infeasible solutions and duplicate solutions can be effectively reduced, the efficiency and quality of generating non-inferior solution sets can be improved, and the multi-objective water scheduling needs of cross-basin water transfer projects can be better adapted.

[0009] Technical solution: The method for generating a multi-objective non-inferior solution set for inter-basin water transfer based on a convex set feasible region of the present invention comprises the following steps:

[0010] Step S1: Collect hydrological data of the study area, select a main objective function from the pre-built objective functions, and search for the boundary points of the target space corresponding to other objective functions. Step S1 includes:

[0011] Step S11: Collect hydrological data for the study area and build a multi-objective optimization model for inter-basin water transfer, which includes at least two objective functions and determines one of them as the main objective function. Specifically:

[0012] Determine the study area and collect hydrological data: Select the study area of ​​the inter-basin water transfer project and clarify the geographical scope of each water source, water receiving area, and water transfer line; collect relevant hydrological data including runoff, initial storage capacity of each water source, upper and lower limits of water storage of each water source, suitable ecological water level, and water demand of each water receiving user as the basic input for model construction; set objective functions f1, f2, ..., f2 for the multiple optimization goals involved in the inter-basin water transfer project, such as water supply guarantee, economic benefits, and ecological protection. n ; Set constraints, including: water balance constraints, upper and lower limit constraints on the water storage capacity of regulating reservoirs, total water transfer constraints from the source, hydraulic connection constraints and other constraints.

[0013] Step S12: Construct an n-1 dimensional plane set covering all angles in the space corresponding to the n objective functions other than the main objective function. The plane set is specifically:

[0014] In the n-dimensional target space, the equation of the line is y(x)=w1f1+w2f2+…+w n f n , set the normal vector weight combination w=(sinθ1sinθ2…sinθ n-2 cosθ n-1 , sinθ1sinθ2…sinθ n-2 sinθ n-1 , sinθ1sinθ2…cosθ n-2 ,…,sinθ1cosθ2,cosθ1),θ1,θ2,…,θ n-2 ∈[0,π],θ n-1 ∈[0, 2π], by changing θ 1， θ 2， …, the value of θn-1 enables the plane to rotate freely in all directions in space.

[0015] Step S12 includes the following steps:

[0016] Step S121: Determine the target space dimension n based on the number of target functions other than the main target function, construct a general formula for the n-1-dimensional plane set covering all angles, and set the normal vector weight combination. Specifically:

[0017] According to the number of objective functions other than the main objective function, the dimension of the target space is determined, and the general formula of the n-1 dimensional plane set covering all angles is constructed, and the normal vector weight combination is set. For the n-dimensional target space, the direction of the normal vector is composed of n-1 angles θ1, θ2, ..., θ n-1 Determine. Among them θ1, θ2, ..., θ n-2 ∈[0,π],θ n-1 ∈[0, 2π].

[0018] Step S122: Divide the plane set normal vector angle intervals for different objective functions to obtain the normal vector angle intervals corresponding to each objective function. For resource-related, economic-related, and ecological-related objective functions, the division accuracy is reduced successively. The angle interval corresponding to [0, π] is divided into N1 subintervals, and the angle interval corresponding to [0, 2π] is divided into N2 subintervals, where N1 and N2 are set according to different calculation accuracy requirements.

[0019] Step S123: In each angle interval of the normal vector corresponding to each type of objective function, uniformly select an angle and combine them to form an n-1 dimensional plane set covering all angles.

[0020] Step S13: Solve the extreme values ​​of the n-1 dimensional plane set covering the full angle respectively, and find the corresponding boundary point of the target space for each extreme value. Specifically:

[0021] The extreme values ​​of the n-1 dimensional plane set covering the entire angle are solved respectively. Each solution process is the process of translating the plane along the normal vector direction. Under the original constraints, the optimal value is obtained when the plane is tangent to the target space. Each solution result corresponds to finding a boundary point in the target space of the multi-objective optimization problem of cross-basin water transfer.

[0022] miny(x)=w1f1+w2f2+…+w n f n ; maxy(x)=w1f1+w2f2+…+w n f n ;

[0023] Step S2: pre-process the boundary points of the target space, and then use the spatial boundary convex hull algorithm to construct the convex set boundary range of the target space. Step S2 includes:

[0024] Step S21: Based on the actual scheduling requirements, outliers are eliminated and similar values ​​are deduplicated for the boundary points of the target space to obtain the optimized boundary points of the target space.

[0025] Step S22: Using the spatial boundary convex hull algorithm for the optimization boundary points of the target space, the convex set boundary range of the target space of the multi-objective optimization problem of cross-basin water transfer is obtained. Step S22 includes:

[0026] Step S221: Select an initial reference point from the optimized boundary points, and for each point, calculate its direction and angle relative to the reference point to determine its relative position in space.

[0027] Step S222: sort each optimized boundary point in ascending order according to its relative position in space.

[0028] Step S223: Select the first n points after sorting and construct an initial convex hull to form a minimum n-1-dimensional convex structure. Process the remaining points one by one to determine whether they are outside the convex structure. If so, update the convex hull structure to obtain the convex set boundary range of the target space corresponding to the multi-objective optimization problem of cross-basin water transfer.

[0029] The specific ordering method of optimizing the boundary points in space in step S222 is:

[0030] In two-dimensional space, a counterclockwise sorting method is used to arrange all points in ascending order of angle. In three-dimensional or higher-dimensional space, the azimuth and elevation of each point relative to the reference point are calculated, the spatial position of the point is converted into angle information, and all points are sorted by azimuth. If multiple points have the same azimuth, they are further sorted according to the elevation to form an ordered point set.

[0031] Step S3: Generate an ε-constrained threshold combination within the convex set boundary of the target space; for each ε-constrained threshold combination, transform the multi-objective optimization problem into multiple single-objective optimization sub-problems, and use the sparse optimization interior point method to solve them separately to generate a non-inferior solution set for the inter-basin water transfer scheme. Step S3 includes:

[0032] Step S31: Set the gridding fraction q within the convex set boundary of the target space, and calculate the ε constraint threshold of the objective function for each gridding fraction to form an ε constraint threshold combination. Specifically:

[0033] Calculate the optimal solution f of the remaining objective functions except the main objective function U i and the worst solution f S i For the objective function f i , set the gridding fraction q, and adjust the value of q according to the accuracy requirements. The ε constraint threshold of the objective function is calculated as follows:

[0034] ε i，j= f U i -j(f U i -f S i ) / q, j=1, 2, …, q;

[0035] Where i represents the objective function number and j represents the gridding score number of the objective function. If the upper bound ε of the constraint for all objective functions except the primary objective function is within the target space, the constraint threshold combination is retained for solution.

[0036] Step S32: Convert each ε constraint threshold combination into a constraint situation, and combine it with the main objective function to convert the multi-objective optimization problem into multiple single-objective optimization sub-problems. Specifically:

[0037] minf(x); f1≤ε 1，j ,…,f n ≤ε n，j ;h(x)=0;g(x)≤0;

[0038] Step S33: Use the sparse optimization interior point method to solve the above single-objective optimization sub-problems respectively to obtain a non-inferior solution set for the inter-basin water transfer scheme. Step S33 includes:

[0039] Step S331: For the single-objective optimization sub-problem of inter-basin water transfer, a Lagrangian function is constructed to combine the objective function and the constraints:

[0040] L(x,λ,v)=f(x)+λ T (Ax-b)+v T (Cx-d);

[0041] where λ is the Lagrange multiplier for the equality constraints; v is the Lagrange multiplier for the inequality constraints; A is the coefficient matrix for the equality constraints, and C is the coefficient matrix for the inequality constraints. In inter-basin water transfer problems, A and C are often sparse matrices with most elements being zero.

[0042] Step S332: construct a Hessian matrix for the above Lagrangian function and perform sparsification to obtain a sparse matrix.

[0043] Construct the Hessian matrix:

[0044] HΔx=-g;

[0045] Where H is the Hessian matrix, the matrix of second-order derivatives of the objective and constraint functions, and g is the gradient vector. Due to the sparsity of the coefficient matrices A and C, the Hessian matrix H for the entire problem is also sparse. Using the Compressed Sparse Row (CSR) sparse matrix storage format, only nonzero elements and their positions are stored, reducing memory requirements and optimizing the efficiency of matrix operations.

[0046] Step S333: Use the conjugate gradient method to iteratively solve the sparse matrix and obtain the solution of the single-objective optimization subproblem, which is the non-inferior solution set of the multi-objective optimization problem of cross-basin water transfer. The update formula is:

[0047] r k+1 =r k -α k Hd k ;d k+1 =r k+1 +β k d k ;

[0048] Among them, r k is the residual, α k and β k is the step size factor, d k is the search direction.

[0049] Solve all single-objective optimization sub-problems and obtain a set of non-inferior solutions to the multi-objective optimization problem of inter-basin water transfer.

[0050] According to one aspect of the present application, finding a boundary point of a corresponding target space for each extreme value includes:

[0051] Perform an initial coarse-grained scan to generate an initial set of boundary points;

[0052] Based on the initial boundary point set, determine the angle interval to be encrypted;

[0053] The angle interval to be encrypted is iteratively encrypted until the preset termination condition is met to generate the boundary points of the final target space.

[0054] Beneficial effects: By exploring the boundary range of the target space, optimizing the constraint threshold combination setting, and improving the traditional ε constraint algorithm, the possibility of generating infeasible solutions and repeated solutions in the traditional method is reduced, and the generation efficiency and quality of the non-inferior solution set are improved; a quantitative description of the target space is achieved, and the boundary points of the target space are searched by a plane set covering all angles, and its internal structure is mapped by combining the gridding method, which reveals the conflicts and connections between multiple objectives and enhances the interpretability of the decision; the present invention combines the characteristics of the multi-objective optimization problem of cross-basin water transfer, uses the sparse matrix optimization interior point method to solve the single-objective optimization subproblem, and improves the solution efficiency.

[0055] In summary, by applying the method for solving the multi-objective non-inferior solution set for cross-basin water transfer based on the convex set feasible domain proposed in this application, it is possible to explore the boundary range of the target space, optimize the combination setting of constraint thresholds, and improve the traditional ε constraint algorithm to effectively reduce the possibility of generating infeasible solutions and duplicate solutions, improve the efficiency and quality of generating non-inferior solution sets, and better adapt to the multi-objective water scheduling needs of cross-basin water transfer projects. According to one aspect of this application, the specific output of the present invention is a non-inferior solution set for the multi-objective optimization problem of cross-basin water transfer. BRIEF DESCRIPTION OF THE DRAWINGS

[0056] Figure 1 It is a flow chart of the method of the present invention.

[0057] Figure 2 This is a schematic diagram of the inter-basin water transfer system.

[0058] Figure 3 It is a schematic diagram of the boundary points of the target space.

[0059] Figure 4 It is a schematic diagram of the target space boundary range.

[0060] Figure 5 It is a schematic diagram of the position generated by the combination of constraint thresholds within the boundary of the target space.

[0061] Figure 6 This is a schematic diagram of the non-inferior solution set for multi-objective optimization of inter-basin water transfer. DETAILED DESCRIPTION

[0062] The following is combined with Figures 1 to 6 The present invention is described in detail with specific examples.

[0063] The method for generating a multi-objective non-inferior solution set for inter-basin water transfer based on a convex set feasible region includes the following steps:

[0064] Step S1: Collect hydrological data of the study area, select a main objective function from the pre-constructed objective functions, and search for boundary points of the target space corresponding to other objective functions.

[0065] Step S2: pre-process the boundary points of the target space, and then use the space boundary convex hull algorithm to construct the convex set boundary range of the target space.

[0066] Step S3: Generate an ε-constrained threshold combination within the convex set boundary range of the target space; for each ε-constrained threshold combination, transform the multi-objective optimization problem into multiple single-objective optimization sub-problems, and use the sparse optimization interior point method to solve them separately to generate a non-inferior solution set of the inter-basin water transfer scheme.

[0067] This embodiment can effectively explore the target space boundary range of the cross-basin water transfer problem, optimize the combination setting of constraint thresholds, improve the traditional ε constraint algorithm, effectively reduce the possibility of generating infeasible solutions and duplicate solutions, improve the efficiency and quality of generating non-inferior solution sets, and better adapt to the multi-objective water scheduling needs of cross-basin water transfer projects. In other words, before calling the ε constraint method, the convex set feasible domain boundary of the target space is actively explored and constructed through the weighted method and convex hull algorithm. Then, the generation (gridding) of the ε constraint threshold is strictly restricted to within this known feasible domain. This transforms the blind search into a bounded search, solving the problems of low generation efficiency and low solution set quality mentioned in the background art, and effectively reducing the possibility of generating infeasible solutions and duplicate solutions. The traditional ε constraint method uses a grid search method to set the constraint threshold, which is blind and does not consider the boundary of the feasible domain, resulting in a large number of infeasible solutions and duplicate solutions.

[0068] According to one aspect of the present application, step S1 includes:

[0069] Step S11: Collect hydrological data for the study area and build a multi-objective optimization model for inter-basin water transfer, which includes at least two objective functions and determines one of them as the main objective function. Specifically:

[0070] Determine the study area and collect hydrological data: Select the study area of ​​the inter-basin water transfer project and clarify the geographical scope of each water source, water receiving area, and water transfer line; collect relevant hydrological data including runoff, initial storage capacity of each water source, upper and lower limits of water storage of each water source, suitable ecological water level, and water demand of each water receiving user as the basic input for model construction; set objective functions f1, f2, ..., f2 for the multiple optimization goals involved in the inter-basin water transfer project, such as water supply guarantee, economic benefits, and ecological protection. n ; Set constraints, including: water balance constraints, upper and lower limit constraints on the water storage capacity of regulating reservoirs, total water transfer constraints from the source, hydraulic connection constraints and other constraints.

[0071] In the implementation case, the objective function is set with the optimization goals of minimizing the comprehensive water shortage rate, minimizing the amount of water diverted from the source, and minimizing the ecological water replenishment as an example, and minimizing the ecological water replenishment is determined as the main objective function.

[0072] Objective function 1: Minimize the sum of the weighted squares of water shortage rates of users in each water-receiving area of ​​the system: Using the weighted square sum of water shortage rates as the expression of water shortage losses reflects the marginal growth relationship between water shortage losses and water shortage rates, that is, the greater the water shortage, the higher the marginal loss.

[0073] Objective function 2, the total amount of water transferred from the source is the least: the total amount of water transferred from the source is an approximate term of the water transfer cost. Generally speaking, the smaller the total amount of water transferred from the source, the lower the water transfer cost.

[0074] Objective function 3, minimum ecological water replenishment: By quantifying the ecological water replenishment, it helps to protect the stability of the reservoir ecosystem, maintain biodiversity and ecological functions, and provide a scientific basis for water resources management.

[0075] Set constraints, including: water balance constraints, upper and lower limit constraints on the water storage capacity of regulating reservoirs, total water transfer constraints from the source, hydraulic connection constraints, and other constraints.

[0076] The water balance constraint reflects the relationship between the water input, output and storage of each water body in the basin during the water resources scheduling process, ensuring that the water volume in the system is always balanced during the scheduling process; the supply and demand constraint reflects the matching relationship between water resources supply and demand, ensuring that the water supply to water users does not exceed the water demand during the scheduling process; the upper and lower limit constraints of the regulating reservoir water storage stipulate that the water storage capacity of the regulating reservoir must be between a certain upper and lower limit to prevent the reservoir from storing too much or too little water and ensure the safe operation of the reservoir; the water storage capacity constraint at the beginning and end of the scheduling period requires that the water storage capacity of the reservoir must meet the preset conditions at the beginning and end of the scheduling period to ensure The water storage capacity of the reservoir changes steadily throughout the entire scheduling cycle; the constraint on local water withdrawal limits the amount of water that can be withdrawn from local water sources during the water resources scheduling process. Its significance is to prevent over-reliance on local water sources and ensure the sustainable use of local water resources; the water transfer capacity constraint limits the water transfer volume of water resources within a specific time and space range, and cannot exceed the water transfer capacity of the system; the water transfer constraint describes the relationship between water transfer between different water sources, that is, the water outflow of the upper-level reservoir minus the water transmission loss is equal to the water inflow of the lower-level reservoir; the total water transfer constraint at the source stipulates the upper limit of the total amount of water transferred from the source to ensure that the water transfer volume at the source is within a reasonable range.

[0077] Figure 2 This paper demonstrates a simplified inter-basin water transfer system consisting of two water sources: local water and externally transferred water. The model employs a multi-stage water transfer model, with water first transferred to the reservoir at the corresponding level and then transferred to the next level after storage. Thus, user water needs are met by both local runoff and externally transferred water.

[0078] By accurately collecting hydrological data and building a multi-objective optimization model, we can find a balance between ensuring water supply security, economic benefits, and ecological protection. Clarifying the primary objective function helps optimize resource allocation, reduce water shortages and water transfer costs in inter-basin water transfer projects, while maximizing ecological protection and achieving sustainable water resource management.

[0079] Step S12: Construct an n-1 dimensional plane set covering all angles in the space corresponding to the n objective functions other than the main objective function. The plane set is specifically:

[0080] In the n-dimensional target space, the equation of the line is y(x)=w1f1+w2f2+…+w n f n , set the normal vector weight combination w=(sinθ1sinθ2…sinθ n-2 cosθ n-1 , sinθ1sinθ2…sinθ n-2 sinθ n-1 , sinθ1sinθ2…cosθ n-2,…,sinθ1cosθ2,cosθ1),

[0081] Among them, θ1, θ2, ..., θ n-2 ∈[0,π],θ n-1 ∈[0, 2π], by changing θ 1， θ 2， …, the value of θn-1 enables the plane to rotate freely in all directions in space.

[0082] In the implementation case, the plane equation is y(x)=w1f1+w2f2, and the normal vector weight combination w=(cosθ, sinθ) is set, θ∈[0,π]. By changing the value of θ, the plane can be freely rotated in all directions in space.

[0083] Step S12 includes the following steps:

[0084] Step S121: Determine the target space dimension n based on the number of target functions other than the main target function, construct a general formula for the n-1-dimensional plane set covering all angles, and set the normal vector weight combination. Specifically:

[0085] According to the number of objective functions other than the main objective function, the dimension of the target space is determined, and the general formula of the n-1 dimensional plane set covering all angles is constructed, and the normal vector weight combination is set. For the n-dimensional target space, the direction of the normal vector is composed of n-1 angles θ1, θ2, ..., θ n-1 Determine. Among them θ1, θ2, ..., θ n-2 ∈[0,π],θ n-1 ∈[0, 2π].

[0086] Step S122: Divide the plane set normal vector angle intervals for different objective functions to obtain the normal vector angle intervals corresponding to each objective function. For resource-related, economic-related, and ecological-related objective functions, the division accuracy is reduced successively. The angle interval corresponding to [0, π] is divided into N1 subintervals, and the angle interval corresponding to [0, 2π] is divided into N2 subintervals, where N1 and N2 are set according to different calculation accuracy requirements.

[0087] Step S123: In each angle interval of the normal vector corresponding to each type of objective function, uniformly select an angle and combine them to form an n-1 dimensional plane set covering all angles.

[0088] In this implementation, the number of objective functions other than the primary objective function is two, and the dimension of the target space is two. A one-dimensional plane set is constructed using the formula: y(x) = w1f1 + w2f2. The direction of the normal vector is determined by an angle θ, where θ∈[0,π]. The interval containing θ is divided into 179 intervals at 1° intervals. The median value of each interval is used as the angle corresponding to the normal vector, forming a one-dimensional plane set covering all angles.

[0089] By constructing a plane set that covers all angles and rotates evenly, each objective function can be fully evaluated, ensuring the comprehensiveness of the target space subsequently searched and the diversity of non-inferior solution sets in multi-objective optimization, thereby achieving more accurate and scientific optimization results.

[0090] It is important to note that the normal vector weight is constructed by combining n-1 angles θ to achieve a full-angle search. In addition, the idea of ​​setting a gradually reduced partitioning accuracy for different objective functions (resource, economic, ecological) is proposed.

[0091] Step S13: Solve the extreme values ​​of the n-1 dimensional plane set covering the full angle respectively, and find the corresponding boundary point of the target space for each extreme value. Specifically:

[0092] The extreme values ​​of the n-1 dimensional plane set covering the entire angle are solved respectively. Each solution process is the process of translating the plane along the normal vector direction. Under the original constraints, the optimal value is obtained when the plane is tangent to the target space. Each solution result corresponds to finding a boundary point in the target space of the multi-objective optimization problem of cross-basin water transfer.

[0093] miny(x)=w1f1+w2f2+…+w n f n ; maxy(x)=w1f1+w2f2+…+w n f n ;

[0094] In the implementation case, the extreme values ​​of the set of 1-dimensional planes covering all angles are solved separately:

[0095] miny(x)=w1f1+w2f2; maxy(x)=w1f1+w2f2;

[0096] Based on the solution results, the boundary points of the target space corresponding to the two-dimensional objective function of comprehensive water shortage rate and source water diversion volume are obtained. Figure 3 shown.

[0097] Compared with the traditional simplex method of finding the vertices of the target space, the method of solving the plane extreme value is more suitable for problems with many decision variables and high target space dimensions, avoiding the complex calculations caused by the dimensionality curse.

[0098] Step S2: pre-process the boundary points of the target space, and then use the spatial boundary convex hull algorithm to construct the convex set boundary range of the target space. Step S2 includes:

[0099] Step S21: Based on the actual scheduling requirements, outliers are eliminated and similar values ​​are deduplicated for the boundary points of the target space to obtain the optimized boundary points of the target space.

[0100] Step S22: Using the spatial boundary convex hull algorithm for the optimization boundary points of the target space, the convex set boundary range of the target space of the multi-objective optimization problem of cross-basin water transfer is obtained. Step S22 includes:

[0101] Step S221: Select an initial reference point from the optimized boundary points, and for each point, calculate its direction and angle relative to the reference point to determine its relative position in space.

[0102] Step S222: sort each optimized boundary point in ascending order according to its relative position in space.

[0103] Step S223: Select the first n points after sorting and construct an initial convex hull to form a minimum n-1-dimensional convex structure. Process the remaining points one by one to determine whether they are outside the convex structure. If so, update the convex hull structure to obtain the convex set boundary range of the target space corresponding to the multi-objective optimization problem of cross-basin water transfer.

[0104] The specific ordering method of optimizing the boundary points in space in step S222 is:

[0105] In two-dimensional space, a counterclockwise sorting method is used to arrange all points in ascending order of angle. In three-dimensional or higher-dimensional space, the azimuth and elevation of each point relative to the reference point are calculated, the spatial position of the point is converted into angle information, and all points are sorted by azimuth. If multiple points have the same azimuth, they are further sorted according to the elevation to form an ordered point set.

[0106] In the implementation case, the boundary points of the two-dimensional target space found in step S13 are subjected to outlier elimination and similarity deduplication. The vertex on the lower left is selected as the reference point. The directions and angles of the other optimized boundary points relative to the reference point in the two-dimensional space are calculated and sorted in a counterclockwise order from small to large. The first two points are selected to construct the initial convex hull, and the remaining points are processed one by one to finally obtain the boundary range of the target space. Figure 4 shown.

[0107] The quantitative description of the target space boundary range is achieved, and its internal structure is mapped through the spatial boundary convex hull algorithm, which reveals the conflicts and connections between multiple targets and enhances the interpretability of decision-making.

[0108] Step S3: Generate an ε-constrained threshold combination within the convex set boundary of the target space; for each ε-constrained threshold combination, transform the multi-objective optimization problem into multiple single-objective optimization sub-problems, and use the sparse optimization interior point method to solve them separately to generate a non-inferior solution set for the inter-basin water transfer scheme. Step S3 includes:

[0109] Step S31: Set the gridding fraction q within the convex set boundary of the target space, and calculate the ε constraint threshold of the objective function for each gridding fraction to form an ε constraint threshold combination. Specifically:

[0110] Calculate the optimal solution f of the remaining objective functions except the main objective function U i and the worst solution f S i For the objective function f i , set the gridding fraction q, and adjust the value of q according to the accuracy requirements. The ε constraint threshold of the objective function is calculated as follows:

[0111] ε i，j= f U i -j(f U i -f S i ) / q, j=1, 2, …, q;

[0112] Where i represents the objective function number and j represents the gridding score number of the objective function. If the upper bound ε of the constraint for all objective functions except the primary objective function is within the target space, the constraint threshold combination is retained for solution.

[0113] In the implementation case, the grid fraction q is set to 25, and f is calculated U 1=0, f S 1=40;f U 2=1.2,f S 1=80. Calculate the ε constraint threshold combination. Figure 5 shown.

[0114] By setting the gridding fraction within the convex set boundary of the target space and calculating the ε constraint threshold of the objective function, we can refine the exploration of the target space, flexibly adjust the accuracy to generate ε constraint threshold combinations, and provide a more uniform and comprehensive constraint range for multi-objective optimization problems.

[0115] Step S32: Convert each ε constraint threshold combination into a constraint situation, and combine it with the main objective function to convert the multi-objective optimization problem into multiple single-objective optimization sub-problems. Specifically:

[0116] minf(x); f1≤ε 1，j,…,f n ≤ε n，j ;h(x)=0;g(x)≤0;

[0117] Converting each constraint threshold combination into the constraint conditions of a single-objective optimization sub-problem can effectively decompose the complex multi-objective optimization problem into multiple easy-to-solve sub-problems, reducing the computational complexity, ensuring that the objective function and constraint conditions of each single-objective optimization problem are clear and independent, and improving the solution efficiency and accuracy.

[0118] Step S33: Use the sparse optimization interior point method to solve the above single-objective optimization sub-problems respectively to obtain a non-inferior solution set for the inter-basin water transfer scheme. Step S33 includes:

[0119] Step S331: For the single-objective optimization sub-problem of inter-basin water transfer, construct a Lagrangian function combining the objective function and the constraints: L(x, λ, v) = f(x) + λ T (Ax-b)+v T (Cx-d);

[0120] where λ is the Lagrange multiplier for the equality constraints; v is the Lagrange multiplier for the inequality constraints; A is the coefficient matrix for the equality constraints, and C is the coefficient matrix for the inequality constraints. In inter-basin water transfer problems, A and C are often sparse matrices with most elements being zero.

[0121] Step S332: construct a Hessian matrix for the above Lagrangian function and perform sparsification to obtain a sparse matrix.

[0122] Construct the Hessian matrix: HΔx=-g;

[0123] Where H is the Hessian matrix, the matrix of second-order derivatives of the objective and constraint functions, and g is the gradient vector. Due to the sparsity of the coefficient matrices A and C, the Hessian matrix H for the entire problem is also sparse. Using the Compressed Sparse Row (CSR) sparse matrix storage format, only nonzero elements and their positions are stored, reducing memory requirements and optimizing the efficiency of matrix operations.

[0124] Step S333: Use the conjugate gradient method to iteratively solve the sparse matrix and obtain the solution of the single-objective optimization subproblem, which is the non-inferior solution set of the multi-objective optimization problem of cross-basin water transfer. The update formula is:

[0125] r k+1 =r k -α k Hd k ;d k+1 =r k+1 +β k d k ;

[0126] Among them, rk is the residual, α k and β k is the step size factor, d k is the search direction.

[0127] In the implementation case, set α k =0.01,β k =0.9. Solve all the single-objective optimization subproblems and obtain the non-inferior solution set of the multi-objective optimization problem of inter-basin water transfer.

[0128] Using a sparse optimization interior point method to solve the single-objective optimization subproblem significantly reduces memory consumption and speeds up matrix operations. The conjugate gradient method ensures good convergence and stability, enabling rapid solutions to the single-objective optimization subproblem, thus providing a set of non-inferior solutions to the entire inter-basin water transfer problem.

[0129] According to one aspect of this application, a cross-basin water transfer problem with three optimization objectives, namely, water supply security, economic benefits, and ecological protection, is used as an example. Minimizing ecological water replenishment is selected as the primary objective function. The remaining two objective functions (e.g., minimizing the system's overall water shortage rate and minimizing the total amount of water transferred from the source) form a two-dimensional (n=2) objective space for analysis. It should be noted that the processes already described in the above embodiments will not be repeated here.

[0130] Step S1: Collect hydrological data of the study area, select a main objective function from the pre-constructed objective functions, and search for boundary points of the target space corresponding to other objective functions.

[0131] In this embodiment, this step is used to construct an optimization model and systematically search for the boundary of the target space using the weighted sum method. This step is the basis for subsequently determining the scope of the feasible region.

[0132] Step S11: Collect hydrological data for the study area, build a multi-objective optimization model for inter-basin water transfer, including at least two objective functions, and determine one of them as the main objective function.

[0133] In this example, the hydrological data for the study area includes the multi-year average daily runoff of each water source, initial reservoir storage, storage capacity limitations, and the hierarchical water demand of each water user. The objective functions are set as: f1 (system comprehensive water shortage rate, resource-related) and f2 (total water diversion from the source, economically relevant). The primary objective function is f3 (ecological water replenishment, ecologically relevant).

[0134] Step S12: Construct a set of n-1 dimensional planes covering all angles in the space corresponding to the n objective functions other than the primary objective function. The set of n-1 dimensional planes covering all angles is a set of hyperplanes whose normal vectors can point uniformly and systematically in all directions in the n-dimensional target space, thereby ensuring that the boundary of the target space is fully explored without omission.

[0135] In this example, n = 2, so a one-dimensional plane set (i.e., a set of lines) is constructed within a two-dimensional space. The general formula is y(x) = w1*f1+w2*f2. The normal vector weight combination is w = (cosθ, sinθ), where the angle θ ∈ [0, π].

[0136] Using angle parameterization to construct weights ensures their non-negativity and normalization. By uniformly varying the angle θ, the entire target space can be scanned without blind spots. To balance solution accuracy and computational efficiency, different partitioning accuracies are set for objective functions of different properties.

[0137] Preferably, the angle intervals are non-uniformly divided according to the importance or sensitivity of the objective function. For example, according to the requirements of step S122, the division accuracy can be set to:

[0138] Resource-related objective functions (such as water scarcity rate f1): These objectives are closely related to people's livelihoods, and decision makers are extremely sensitive to their changes, requiring detailed exploration. The corresponding angle interval division has the highest accuracy. For example, the angle step size is set to 1.

[0139] Economic-related objective functions (such as water transfer cost f2): This type of objective is less important, and the division accuracy can be moderately reduced. For example, the angle step size is set to 3.

[0140] Ecologically relevant objective function: If used as a non-primary objective function, its constraints are relatively loose and the division accuracy can be further reduced. For example, the angle step size is set to 5.

[0141] In this example, n=2, and there are only two non-primary objective functions f1 and f2. The interval [0,π] can be comprehensively divided into about 100 subintervals. An angle θ is uniformly selected in each subinterval to form a plane set covering all angles.

[0142] Step S13: respectively solve the extreme values ​​of the n-1 dimensional plane set covering the full angle, and find the corresponding boundary point of the target space for each extreme value.

[0143] For each normal vector w generated in step S12, solve two single-objective optimization problems: min(y(x)) and max(y(x)). The objective function value (f1, f2) corresponding to the optimal solution obtained each time is a boundary point in the target space. By solving the extreme values ​​of all plane sets, a series of boundary points are obtained, forming a discrete point set on the boundary of the target space.

[0144] Step S2: pre-process the boundary points of the target space, and then use the space boundary convex hull algorithm to construct the convex set boundary range of the target space.

[0145] Step S21: Based on the actual scheduling requirements, outliers are eliminated and similar values ​​are deduplicated for the boundary points of the target space to obtain the optimized boundary points of the target space.

[0146] The directly searched boundary point set may contain outliers due to numerical calculation errors or redundant points due to overly dense angle divisions. This step uses quantitative criteria to clean the data, which can improve the stability and efficiency of the subsequent convex hull algorithm and obtain a more accurate feasible region boundary.

[0147] Calculate any two points P in the boundary point set a and P b The normalized Euclidean distance between them. For example, set the distance threshold d sim =0.01. If D(P a ,P b )<0.01, the two points are considered similar, one of them is retained (for example, the Pareto-dominant point is retained), and the other is deleted.

[0148] For each objective function f i The boundary point value sequence of is judged by the 3σ principle. Specifically, the mean μ of the sequence is calculated i and standard deviation σ i If a point P a The function value f i,a Does not fall within [μ i -3σ i ,μ i +3σ i ] interval, then point P a Mark as outliers and remove them.

[0149] Step S22: For the optimization boundary points of the target space, a spatial boundary convex hull algorithm is used to obtain the convex set boundary range of the target space of the multi-objective optimization problem of cross-basin water transfer.

[0150] A standard convex hull algorithm such as Quickhull is used. Its core is to iteratively add points to the convex hull and determine whether the newly added points will change the current convex hull structure.

[0151] In step S223, it is determined whether a point Ptest is outside the constructed convex structure. The specific mathematical method is as follows:

[0152] In two-dimensional space: Assume that an edge of the convex hull is defined by two vertices P1=(x1,y1) and P2=(x2,y2). For the point P to be measured test =(x t, y t ), we can calculate the z component of the vector cross product: V=(x2-x1)(y t -y1)-(y2-y1)(x t -x1). According to the convention that convex hull vertices are ordered counterclockwise, if V > 0, the point is outside the edge. If a point is outside an edge of the convex hull, then the point is outside the convex hull.

[0153] In three or higher dimensional space: An n-dimensional convex hull is composed of multiple n-1 dimensional facets. Each facet defines a hyperplane N T Pc=0. Select a point inside the convex hull (such as the center of mass of all points) as the reference point P ref , so that it satisfies N T P ref -c<0. For the point P to be measured test , calculate V=N T P test -c. If for a certain patch, the calculated V>0, then it proves that P test Outside the patch, that is, outside the entire convex hull, the convex hull structure needs to be updated.

[0154] Through this step, we finally get the convex set boundary range defined by a series of hyperplane equations or vertex sets.

[0155] Step S3: Generate an ε-constrained threshold combination within the convex set boundary range of the target space; for each ε-constrained threshold combination, transform the multi-objective optimization problem into multiple single-objective optimization sub-problems, and use the sparse optimization interior point method to solve them separately to generate a non-inferior solution set of the inter-basin water transfer scheme.

[0156] Step S31: within the convex set boundary of the target space, set a gridding fraction q, and for each gridding fraction, calculate the ε constraint threshold of the target function to form an ε constraint threshold combination.

[0157] While traditional ε-constraint methods generate a mesh within the theoretical range of the entire objective function, the present invention generates a mesh within the practically feasible convex set boundary determined in step S2. This ensures that the vast majority of generated ε-constraint threshold combinations are feasible, fundamentally avoiding a large number of inefficient computations and significantly improving the efficiency and quality of generating non-inferior solution sets.

[0158] Step S32: Convert each ε constraint threshold combination into a constraint situation, and combine it with the main objective function to convert the multi-objective optimization problem into multiple single-objective optimization sub-problems.

[0159] For each threshold combination (ε 1,j , ε 2,j ), construct the following single-objective optimization subproblem: min f3(x); the constraint condition is: f1(x)≤ε 1,j ; f2(x)≤ε 2,j ;h(x)=0;g(x)≤0;

[0160] Step S33: Use the sparse optimization interior point method to solve the above single-objective optimization sub-problems respectively to obtain a non-inferior solution set of the inter-basin water transfer scheme.

[0161] For each single-objective subproblem, the KKT system is solved by constructing a Lagrangian function and applying the Newton method. The core of the problem is to solve the linear equation system HΔx = -g. Because the Hessian matrix H in inter-basin water transfer problems is typically large and sparse, the conjugate gradient method (CG) is used for iterative solution.

[0162] In step S333, the key parameter selection principles and termination conditions of the conjugate gradient method are as follows:

[0163] Step factor α k and β k Selection principles:

[0164] α k Usually, a fixed value is not used, but it is dynamically determined in each iteration through the LineSearch method to ensure that the objective function has a sufficient decrease. For example, the Armijo criterion or the Wolfe criterion can be used to find an α that meets the conditions. k .

[0165] β k Calculate using the standard conjugate gradient method. For example, the Fletcher-Reeves formula can be used: k =(r k+ 1 T r k+1 ) / (r k T r k ), or the Polak-Ribière formula.

[0166] Iteration termination condition: set a tolerance threshold ε tol , for example, ε tol =10-6.

[0167] After each iteration, the residual vector r is calculated k+1 The L2 norm of (i.e., the Euclidean length of the vector).

[0168] When |||rk+1|||2<ε tol When , the iteration is considered to have converged, the iteration is stopped, and the current solution is output.

[0169] Optionally, you can also set a maximum number of iterations (such as k max =1000), to prevent the algorithm from falling into an infinite loop. The solutions of all single-objective optimization subproblems are summarized and the Pareto inferior solutions are eliminated. The final set obtained is the high-quality non-inferior solution set of the multi-objective optimization problem of cross-basin water transfer. Through the above steps, the present invention first accurately characterizes the feasible domain boundary of the multi-objective problem, and then performs efficient ε-constraint solving within this boundary, and adopts a high-performance numerical algorithm based on the characteristics of the problem, thereby greatly improving the solution efficiency while ensuring the coverage and accuracy of the solution set, and providing decision makers with reliable and diverse scheduling solutions.

[0170] According to another aspect of the present application, searching for boundary points of the target space in step S13 specifically includes:

[0171] Perform an initial coarse-grained scan to generate an initial set of boundary points;

[0172] Based on the initial boundary point set, determine the angle interval to be encrypted;

[0173] The angle interval to be encrypted is iteratively encrypted until the preset termination condition is met to generate the boundary points of the final target space.

[0174] Preferably, determining the angle interval to be encrypted specifically includes:

[0175] For each boundary point generated in the coarse-grained scan or iterative encryption, the corresponding primal-dual variable vector is extracted from the optimization solver used to solve the boundary point to form a primal-dual variable sequence;

[0176] Based on the original dual variable sequence, calculate the dual variable gradient between adjacent angles;

[0177] The dual variable gradient is compared with a preset gradient threshold to determine one or more angle intervals to be encrypted.

[0178] Preferably, before calculating the dual variable gradient, the method further includes:

[0179] The original dual variable sequence is input into the Kalman filter as observations;

[0180] The Kalman filter is used to smooth the original dual variable sequence to generate a smoothed dual variable sequence;

[0181] The calculation of the dual variable gradient is based on the smoothed dual variable sequence.

[0182] Specifically, this embodiment provides a method for implementing the boundary point search process in step S1. This method aims to address the issues of uneven sampling point distribution and inaccurate capture of the key knee point region that arise when using a fixed angle step size for boundary point search. This method, with limited computing resources, allows for intelligently acquiring the boundary point set with the highest information density and the most valuable value for decision-making.

[0183] Searching for boundary points of the target space corresponding to other objective functions can also be:

[0184] Step S1a: perform a coarse-grained scan to obtain an initial boundary point set and an initial dual variable sequence.

[0185] The dual variable, also known as the Lagrange multiplier, is a variable associated with the constraints in a constrained optimization problem. Its value reflects the rate of change in the optimal value of the primary objective function when the constraint changes by one unit, characterizing the marginal substitution relationship between different objectives.

[0186] First, a large, preset angle step size, such as Δθ, is used. coarse =5, perform a global, sparse scan of the n-dimensional target space determined in step S11. i , solve the weighted and single-objective optimization problem minw(θ i ) T f(x).

[0187] This step yields:

[0188] Initial boundary point set P initial ={P1,P2,…,P M}, where each point P i Corresponding to a solved boundary point coordinate. This point set will serve as the basis for subsequent iterative encryption.

[0189] Initial dual variable sequence Λ raw ={λ1,λ2,…λ i ,…,λ M}, where each vector λ i is extracted from the solver and is related to the boundary point P i The corresponding, noisy primal dual variables. This sequence will serve as the input observations of the Kalman filter.

[0190] Step S1b: Apply Kalman filtering to smooth the initial dual variable sequence to generate a smoothed dual variable sequence.

[0191] Directly use the primal-dual variable λ obtained from the solver i It may contain significant noise due to the convergence problem of numerical calculation. If the judgment is made directly based on its gradient, it is easy to misjudge the encrypted area. This step introduces the Kalman filter, which regards the dual variable changing along the angle θ as the time series signal of the dynamic system. Through filtering and smoothing, random noise is effectively suppressed and a smooth signal reflecting the trend of the real marginal replacement rate is extracted. Specifically,

[0192] Construct a state space model: take the angle θ as the time dimension. Define the state of the system as the real, noise-free dual variable, and define the observation value as the initial dual variable sequence Λ obtained in step S1a. raw .

[0193] Perform filtering and smoothing: Λ raw As input, a standard Kalman smoothing algorithm (such as the RTS smoother) is applied to the entire sequence.

[0194] The smoothed dual variable sequence Λ*={λ*1,λ*2,…,λ* M This sequence is a better estimate of the true dual variable and will be used for subsequent gradient calculations and encryption judgments.

[0195] Step S1c: Calculate the gradient of the smoothed dual variable and determine the angle interval to be encrypted.

[0196] Using the smoothed dual variable sequence Λ*, the normalized gradient between adjacent angles is calculated. For each interval [θ i ,θ i+1 ], calculate its gradient norm G i =||θ i+1 -θ i λ * i+1 -λ* i ∣∣2.

[0197] Set a gradient threshold λ thresh (For example, it can be set to 1.5 times the average of all calculated gradient values ​​Gi). i >λ thresh , then the angle interval [θ i ,θ i+1 ] is determined as the decision-sensitive knee point area.

[0198] Get the list of angle intervals to be encrypted L refine This list will guide the next iterative encryption operation.

[0199] Step S1d: iteratively encrypt the angle interval to be encrypted until a termination condition is met.

[0200] Traverse the list of angle intervals to be encrypted Lrefine, and for each interval [θ i ,θ i+1 ], take the midpoint angle θ new =(θ i +θ i+1 ) / 2 to search for a new boundary point and obtain a new boundary point P new and the new primal-dual variable λ new .

[0201] P new and λ new Insert into the existing set of boundary points and dual variable sequence in angular order.

[0202] With the updated data as input, steps S1b and S1c are repeatedly executed, that is, Kalman filter smoothing is re-performed, gradients are recalculated, and a new list of intervals to be encrypted is determined.

[0203] The iterative process continues until, in a complete cycle, the list of angle intervals to be encrypted outputted by step S1c is L refine is empty, or the preset maximum number of iterations has been reached.

[0204] Finally, after the above adaptive iterative search process, the final boundary point set obtained is the final output of step S1, which is used for preprocessing and convex hull construction in the subsequent step S2.

[0205] Through the above embodiments, the method of the present invention can intelligently concentrate computing resources on the key areas of the non-inferior frontier. Compared with the fixed step size method, it improves the efficiency and accuracy of boundary point search and lays the foundation for generating high-quality non-inferior solution sets.

[0206] According to one aspect of the present application, the spatial boundary convex hull algorithm is used in step S22 to obtain the convex set boundary range, which can also be:

[0207] Obtaining the optimized boundary point set, and determining a corresponding uncertainty value for each optimized boundary point in the set to form a boundary point uncertainty sequence;

[0208] The optimized boundary point set and the boundary point uncertainty sequence are used as input to train a probability model for learning the functional relationship between the coordinate position in the target space and the boundary of the feasible region, wherein the uncertainty value is used to adjust the influence weight of the corresponding optimized boundary point in the model training process;

[0209] Based on the trained probability model, a probabilistic feasible domain model is generated, and the model is used to output a probability value that any query point in the target space belongs to the feasible domain.

[0210] Preferably, it further comprises:

[0211] receiving the probabilistic feasible domain model as input and setting a confidence threshold;

[0212] According to the confidence threshold, a confidence feasible domain is defined from the probabilistic feasible domain model, where the confidence feasible domain is composed of all points whose feasible probability is not lower than the confidence threshold; wherein the confidence feasible domain is passed to step S3 as the spatial range for generating the ε constraint threshold combination.

[0213] Preferably, determining a corresponding uncertainty value for each optimized boundary point specifically includes:

[0214] When solving each boundary point in step S1, the final convergence information of the optimization solver solving the boundary point is recorded;

[0215] Extracting one or more indicators based on the final convergence information, including solving the residual norm or the satisfaction of the KKT condition;

[0216] Quantify the indicators into comprehensive uncertainty values.

[0217] Preferably, the probability model is a Gaussian process model;

[0218] The optimized boundary point set is used as the training input of the Gaussian process model, the scalar representing the boundary position is used as the training output, and the boundary point uncertainty sequence is used to define the observation noise variance in the Gaussian process model.

[0219] For example, this embodiment provides a detailed description of the process for constructing the convex set boundary in step S2. This aims to address the problem that standard convex hull algorithms are sensitive to numerical errors and may produce overly optimistic or distorted boundaries due to single outliers. By introducing a probabilistic model, this embodiment constructs a more robust and realistic soft boundary, thereby improving the effectiveness of subsequent steps.

[0220] Specifically, the convex set boundary range of the target space constructed in step S22 can also be:

[0221] Step S2a: quantify the uncertainty of the boundary points and generate boundary point uncertainty data.

[0222] The boundary point uncertainty data is a numerical value or vector corresponding to the optimized boundary points outputted in step S21, and is used to describe the confidence level or possible error range of the position coordinates of each boundary point.

[0223] Because the boundary points obtained in step S1 are the result of numerical optimization and are not absolutely accurate, adding uncertainty information to them is the basis for subsequent probabilistic modeling. Points with higher uncertainty should have a smaller weight or influence when constructing the final boundary, making the final boundary less sensitive to noise.

[0224] The sources of boundary point uncertainty data can be:

[0225] Extract the residual norm or the degree of satisfaction of the KKT condition at convergence from the optimization solver used to solve the boundary point. Larger residuals indicate lower solution quality and higher uncertainty.

[0226] If step S1 uses an adaptive search based on a Kalman filter, the state covariance matrix output by the Kalman filter can be used directly. This matrix gives the estimated variance of the dual variable corresponding to each boundary point, and this variance is positively correlated with the uncertainty of the boundary point position.

[0227] The uncertainty sequence of boundary points is obtained and used together with the optimized boundary point set as the input for the next step of Gaussian process fitting.

[0228] Step S2b: Fit the Gaussian process model to generate a probabilistic feasible domain model.

[0229] In this scenario, a Gaussian process is used to learn and fit the shape of the non-inferior frontier. Its unique feature is that for any point in space, the GP not only predicts the expected function value (i.e., whether the point is on the frontier), but also provides the variance of the prediction (i.e., the confidence level of the prediction).

[0230] The optimized boundary point set output from step S21 is used as the input of the training data X, and the boundary position is expressed as a scalar (for example, in two-dimensional space, for a given f1 value, the corresponding boundary f2 value) as the output of the training data (y). The boundary point uncertainty sequence obtained from step S2a is used as the noise variance of the training data y.

[0231] Using this data, we train a Gaussian process regression model to obtain a trained probabilistic feasible region model. This model, by inputting the coordinates of any point in the target space, outputs a probability value P that the point belongs to the feasible region. This can replace the convex set boundary generated by the convex hull algorithm.

[0232] Step S2c: Determine a trusted feasible region based on the probabilistic feasible region model.

[0233] Set a confidence threshold, such as P thresh =0.95. Traverse the target space, all feasible probabilities P predicted by the probabilistic feasible domain model (feasible) ≥P threshThe area formed by the points is the trusted feasible region finally determined in this step.

[0234] The resulting confidence feasible region serves as the effective range for generating the ε-constrained threshold combination in step S3. This ensures that the ε-constrained threshold is generated within a boundary that is robust to noise and closer to real physical limits, improving the reliability of the entire method.

[0235] According to one aspect of the present application, the gridding fraction q is set in step S31 to form an ε constraint threshold combination, which may also be:

[0236] Generate and solve the initial, sparse ε-constrained threshold combinations to obtain the initial non-inferior solution set;

[0237] In each iteration, based on the currently generated non-inferior solution set, the optimal ε constraint threshold combination to be solved next is determined within the feasible domain by maximizing the preset information gain index;

[0238] Solve the ε constraint threshold combination to be solved, obtain a new non-inferior solution, and update the new non-inferior solution to the non-inferior solution set;

[0239] The aforementioned steps of determining, solving, and updating are repeated until a preset computing resource budget or convergence condition is met.

[0240] Preferably, the information gain indicator is the expected hypervolume improvement value;

[0241] This value is used to quantitatively evaluate the expected increase in the hypervolume of the current set of non-inferior solutions that can be achieved by solving any candidate ε-constraint threshold combination in the feasible domain, where the hypervolume is a comprehensive measure of the convergence and diversity of the current set of non-inferior solutions.

[0242] Preferably, determining the ε constraint threshold combination to be solved specifically includes:

[0243] Based on all currently solved ε-constraint threshold combinations and their corresponding optimal solutions to the primary objective function, a proxy model is constructed and trained. The proxy model is used to approximate the mapping relationship between the ε-constraint threshold combinations and the optimal solutions to the primary objective function.

[0244] Using the surrogate model, predicting a set of candidate ε-constraint threshold combinations within a feasible domain, and estimating the expected hypervolume improvement value of each candidate combination;

[0245] The candidate ε-constrained threshold combination with the maximum expected supervolume improvement value is selected as the optimal ε-constrained threshold combination to be solved next.

[0246] Specifically, this embodiment provides an alternative to the ε-constrained threshold combination generation process in step S3. This approach addresses the uneven allocation of computing resources and low information efficiency inherent in traditional gridding methods when exploring the feasible region. By incorporating active learning, this embodiment prioritizes computing resources to regions most likely to improve the quality of the entire set of non-inferior solutions, achieving an efficient and intelligent solution process.

[0247] In this embodiment, step S3 generates and solves the ε constraint threshold combination, which specifically includes the following sub-steps:

[0248] Step S3a: perform initial sampling and build a proxy model.

[0249] A surrogate model is a computationally inexpensive approximation of a computationally expensive real function. In this scenario, it is used to approximate the complex mapping between a combination of ε-constrained thresholds and the optimal solution to the primary objective function.

[0250] Within the feasible region determined in step S2 (eg, the aforementioned trusted feasible region), a very small number of sparse initial ε-constraint threshold combination sets are generated.

[0251] Perform step S33 on each combination in the initial set to obtain an initial non-inferior solution set.

[0252] A surrogate model is constructed using the solved ε-constraint threshold combination as input and the corresponding primary objective function solution as output. Preferably, this surrogate model can be a Gaussian process model, providing both the predicted value (mean) and the predicted uncertainty (variance). This yields an initial set of non-inferior solutions and a trained surrogate model. These two together guide the selection of sampling points in the next step.

[0253] Step S3b: Calculate the expected super volume improvement value based on the current non-inferior solution set.

[0254] The Hypervolume Index (HVI) is a comprehensive metric for measuring the performance of a non-inferior solution set. Its value is equal to the volume of the multidimensional space enclosed by the points in the solution set and the preset worst-case reference point. A higher HVI indicates a higher overall solution quality (including convergence and diversity). The Expected Hypervolume Improvement (EHVI) is an information gain metric. Using a surrogate model, each unsampled candidate point in the feasible region is evaluated to predict the expected increase in the total hypervolume of the current solution set if the point is actually solved.

[0255] The calculation process is as follows: Calculate the hypervolume of the current initial non-inferior solution set. Using the surrogate model trained in step S3a, traverse all candidate ε-constrained threshold points in the feasible region and calculate the EHVI value of each point. Obtain an EHVI distribution map for all candidate points in the feasible region.

[0256] Step S3c: determine and solve the next optimal sampling point, and iteratively update it.

[0257] From the EHVI distribution map output in step S3b, the candidate ε-constrained threshold combination with the largest EHVI value is selected as the next optimal sampling point.

[0258] The optimal sampling point is solved in step S33 to obtain a new non-inferior solution. This new solution is incorporated into the current set of non-inferior solutions, and the surrogate model in step S3a is updated with this new (sampling point-solution) data pair.

[0259] Repeat steps S3b and S3c, that is, based on the updated solution set and the surrogate model, recalculate the EHVI distribution, find the next optimal sampling point, and solve and update again.

[0260] The iterative process continues until the total number of solutions reaches a preset computing resource budget (for example, solving 1000 points), or the maximum value of EHVI falls below a very small threshold, indicating that it is difficult to find points that can significantly improve the solution set.

[0261] Through the above embodiment, constraint combinations are no longer generated blindly. Instead, each step is carefully considered, so that computing resources are used in the most critical places. As a result, at the same computing cost, a non-inferior solution set with quality far exceeding that of the uniform grid method is obtained.

Claims

1. A method for generating multi-objective non-inferior solution sets for inter-basin water transfer based on convex set feasible region, characterized by: The following steps are involved: Step S1: Collect hydrological data of the study area, select a main objective function from the pre-constructed objective functions, and search for the boundary points of the target space corresponding to other objective functions; Step S2: pre-process the boundary points of the target space, and then use the space boundary convex hull algorithm to construct the convex set boundary range of the target space; Step S3: within the convex set boundary of the target space, set a gridding fraction q, calculate the ε constraint threshold of the objective function for each gridding fraction, and form an ε constraint threshold combination; convert each ε constraint threshold combination into a constraint condition, and combine it with the main objective function to convert the multi-objective optimization problem into multiple single-objective optimization sub-problems; The sparse optimization interior point method is used to solve the above single-objective optimization sub-problems respectively, and the non-inferior solution set of the inter-basin water transfer scheme is obtained; Step S1 include: Step S11: Collect hydrological data for the study area, build a multi-objective optimization model for inter-basin water transfer, including at least two objective functions, and determine one of them as the main objective function; Step S12: construct an n-1 dimensional plane set covering all angles in the space corresponding to the n objective functions other than the main objective function; Step S13: respectively solve the extreme values ​​of the n-1 dimensional plane set covering the full angle, and find the corresponding boundary point of the target space for each extreme value; Among them, the sparse optimization interior point method is used to solve the above single-objective optimization sub-problems respectively, and the non-inferior solution set of the inter-basin water transfer scheme is obtained, including: For the single-objective optimization sub-problem of inter-basin water transfer, a Lagrangian function is constructed to combine the objective function and constraints; For the above Lagrangian function, construct the Hessian matrix and sparse it to obtain a sparse matrix; The conjugate gradient method is used to iteratively solve the sparse matrix and obtain the solution of the single-objective optimization subproblem, which is the non-inferior solution set of the multi-objective optimization problem of inter-basin water transfer.

2. The method for generating a multi-objective non-inferior solution set for inter-basin water transfer based on a convex set feasible region according to claim 1 is characterized in that: Step S12 includes: Step S121: Determine the target space dimension n according to the number of objective functions other than the main objective function, construct a general formula for an n-1-dimensional plane set covering all angles, and set a normal vector weight combination; Step S122: Divide the normal vector angle intervals of the plane set according to different objective functions to obtain the normal vector angle intervals corresponding to each objective function. For resource-related, economic-related, and ecological-related objective functions, the division accuracy is gradually reduced. Step S123: In each angle interval of the normal vector corresponding to each type of objective function, uniformly select an angle and combine them to form an n-1 dimensional plane set covering all angles.

3. The method for generating a multi-objective non-inferior solution set for inter-basin water transfer based on a convex set feasible region according to claim 1 is characterized in that: Step S2 includes: Step S21: Based on the actual scheduling requirements, the boundary points of the target space are subjected to outlier removal and similarity deduplication to obtain the optimized boundary points of the target space; Step S22: For the optimization boundary points of the target space, a spatial boundary convex hull algorithm is used to obtain the convex set boundary range of the target space of the multi-objective optimization problem of cross-basin water transfer.

4. The method for generating a multi-objective non-inferior solution set for inter-basin water transfer based on a convex set feasible region according to claim 1 is characterized in that: The n-1 dimensional plane set covering all angles is: Plane equation y(x)=w1f1+w2f2+…+w n f n ; Normal vector weight combination \(w = (\sin\theta_1\sin\theta_2\cdots\sin\theta\) n-2 \(\cos\theta\) n-1 , \(\sin\theta_1\sin\theta_2\cdots\sin\theta\) n-2 \(\sin\theta\) n-1 , \(\sin\theta_1\sin\theta_2\cdots\cos\theta\) n-2 , \(\cdots\), \(\sin\theta_1\cos\theta_2\), \(\cos\theta_1)\), \(\theta_1\), \(\theta_2\), \(\cdots\), \(\theta\) n-2 \(\in[0,\pi]\), \(\theta\) n-1 \(\in[0,2\pi]\); where f1, f 2…， f n is the objective function; w1, w 2…， w n is the plane normal vector; w is the normal vector weight combination; θ1, θ2, ..., θ n-1 is the normal vector angle, by changing θ 1， θ 2， …, θ n-1 The value of can realize the free rotation of the plane in all directions in space.

5. The method for generating a multi-objective non-inferior solution set for inter-basin water transfer based on a convex set feasible region according to claim 3 is characterized in that: Step S22 is specifically as follows: Step S221: Select an initial reference point from the optimized boundary points, and for each point, calculate its direction and angle relative to the reference point to determine its relative position in space; Step S222: sort each optimized boundary point in ascending order according to its relative position in space; Step S223: Select the first n points after sorting, construct an initial convex hull, form a minimum n-1 dimensional convex structure, process the remaining points one by one, and determine whether they are outside the convex structure. If so, update the convex hull structure to obtain the convex set boundary range of the target space corresponding to the multi-objective optimization problem of cross-basin water transfer.

6. The method for generating a multi-objective non-inferior solution set for inter-basin water transfer based on a convex set feasible region according to claim 5 is characterized in that: The ascending sorting method is as follows: In two-dimensional space, all points are arranged in a counterclockwise order from smallest to largest angle. In three-dimensional or higher-dimensional space, the azimuth and elevation of each point relative to a reference point are calculated, the spatial position of the point is converted into angle information, and all points are sorted by azimuth. If multiple points have the same azimuth, they are further sorted by elevation to form an ordered point set.

7. The method for generating a multi-objective non-inferior solution set for inter-basin water transfer based on a convex set feasible region according to claim 1 is characterized in that: For each extreme value, find the corresponding boundary point of the target space, including: Perform an initial coarse-grained scan to generate an initial set of boundary points; Based on the initial boundary point set, determine the angle interval to be encrypted; The angle interval to be encrypted is iteratively encrypted until the preset termination condition is met to generate the boundary points of the final target space.

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