A multi-objective optimization method fusing random local search and boundary weight encryption

Abstract

The application discloses a multi-objective optimization method combining random local search and boundary weight encryption, which comprises the following steps: initializing an unbounded archive set; initializing a weight vector set, including generating a dense distribution weight set and a sparse distribution weight set, screening a boundary weight vector and a center weight vector to combine to form the weight vector set; determining a neighborhood index set based on the weight vector space distance; calculating a reference point and updating the unbounded archive set; combining neighborhood search and cross mutation operators to generate offspring solutions, and then selecting the optimal offspring solution by using a scalarization function; and iteratively updating the neighborhood solution until the termination condition is met. The method enhances the search capability of the boundary region of the Pareto front by boundary weight encryption, realizes the synergy of global exploration and local development by combining neighborhood search and cross mutation, retains the non-dominated solutions in the iteration process by using the unbounded archive set, improves the convergence, uniformity and boundary solution quality of the solution set, and is suitable for solving complex multi-objective optimization problems.

Description

Technical Field

[0001] This application belongs to the field of multi-objective intelligent optimization technology, specifically involving a multi-objective optimization method that integrates random local search and boundary weight encryption. Background Technology

[0002] Multi-objective combinatorial optimization problems are a class of complex optimization problems widely found in engineering practice. They are characterized by the need to simultaneously coordinate and optimize multiple conflicting objectives. The decision variables in these problems are mostly discrete values, exhibiting strong nonlinearity, numerous constraints, and high computational complexity. They are commonly found in scenarios such as task scheduling, path planning, resource allocation, conduit process optimization, and production line optimization. For example, in conduit process optimization, it is necessary to simultaneously consider multiple performance indicators such as forming accuracy, material utilization, energy consumption, and production efficiency under limited process conditions. Traditional single-objective optimization algorithms are insufficient to meet these requirements.

[0003] Existing multi-objective optimization methods mainly employ metaheuristic algorithms, including two main categories: multi-objective evolutionary algorithms and multi-objective local search algorithms. Multi-objective evolutionary algorithms achieve global search by simulating a population evolution process, enabling optimization over a large area. However, in discrete problems, they often suffer from unstable search directions, sparse boundary solutions, and slow convergence speeds. Multi-objective local search algorithms achieve gradual improvement of solutions through neighborhood search mechanisms, obtaining high-quality local solutions in a relatively short time. However, they are prone to getting trapped in local optima and struggle to maintain the overall diversity of the solution set.

[0004] Currently, existing algorithms still have the following shortcomings for complex multi-objective combinatorial optimization problems: 1. Lack of effective integration between global search and local development. Most existing algorithms operate independently with local or evolutionary searches, lacking mechanisms for synergy and complementarity, which limits the overall performance improvement of the algorithms; 2. Low utilization of boundary solutions. Boundary solutions play a crucial role in determining the Pareto front range and guiding the search direction, but existing methods lack effective means to identify and enhance boundary solutions in discrete space, resulting in uneven distribution of solution sets and low optimization efficiency; 3. Insufficient algorithm stability and scalability. When dealing with large-scale, strongly constrained discrete multi-objective problems, traditional algorithms struggle to balance computational efficiency and solution set quality, and are prone to premature convergence or search blind spots.

[0005] Therefore, existing technologies struggle to simultaneously achieve efficient global search, accurate local exploitation, and systematic enhancement of boundary solutions in complex multi-objective combinatorial optimization problems. Summary of the Invention

[0006] To address the technical problems of insufficient fusion of global and local searches, low utilization of boundary solutions, and poor algorithm stability in existing multi-objective optimization methods for complex combinatorial optimization problems, this application provides a multi-objective optimization method that integrates random local search and boundary weight encryption to improve the quality of boundary solutions and overall convergence performance, achieve synergy between efficient global search and accurate local exploitation, and enhance the uniformity of solution set distribution and optimization efficiency.

[0007] To achieve the above technical objectives, this application specifically employs the following technical solution: In one aspect of this application, a multi-objective optimization method integrating random local search and boundary weight encryption is provided, comprising the following steps: S1. Initialize the unbounded archive set used to store non-dominated solutions; S2. Initialize the weight vector set, wherein the initialization of the weight vector set includes: generating a densely distributed weight set and a sparsely distributed weight set, selecting boundary weight vectors from the densely distributed weight set, selecting center weight vectors from the sparsely distributed weight set, and combining the boundary weight vectors and the center weight vectors to form the weight vector set; S3. Calculate the reference point based on the minimum value of the initial population on each objective function, and update the unbounded archive set based on the dominance relationship of the solution; S4. Randomly select two parent solutions corresponding to the indices in the neighborhood index set corresponding to the current weight vector, and perform crossover and mutation operations on the two parent solutions to obtain candidate child solutions; randomly select a new solution in the neighborhood of the candidate child solution, and when the new solution dominates the candidate child solution, store the new solution in the unbounded archive set; use a scalarization function to select the optimal solution from the solutions corresponding to the neighborhood index set as the child solution; S5. Repeat steps S3 to S4, and update the neighborhood solution of each weight vector using the scalarization function until the preset termination condition is met.

[0008] In one implementation, in step S2, the densely distributed weight set and the sparsely distributed weight set are generated based on the Das and Dennis methods, wherein: The number of weight vectors in the densely distributed weight set is determined by a first formula, which is: ; The number of weight vectors in the sparsely distributed weight set is determined by a second formula, which is: ; in, Divide the coordinate axes into numerical parameters. The number of target dimensions.

[0009] In one implementation, in step S2, the boundary weight vector satisfies the following condition: at least one component in the weight vector is less than a preset critical value.

[0010] In one implementation, in step S2, the combination of the boundary weight vector and the center weight vector satisfies the following constraint: The total number of boundary weight vectors and center weight vectors is equal to the preset population size; The proportion of the boundary weight vector in the total weight vector is equal to the preset estimated encryption ratio.

[0011] In one implementation, in step S3, each component of the reference point is calculated by a third formula, which is:

[0012] in, Indicates the reference point is at the 1st. The weight of each goal Indicates the initial population. Representing the solution In the Function values ​​on the objective function.

[0013] In one implementation, step S3, updating the unbounded archive set based on the dominance relationship of the solutions, includes: When a new solution is dominated by any solution in the unbounded archive set, the new solution is discarded; When the new solution is not dominated by any solution in the unbounded archive set, the new solution is added to the unbounded archive set, and all solutions in the unbounded archive set dominated by the new solution are removed.

[0014] In one implementation, in step S4, the scalarization function is the Tchebycheff scalarization function, with the expression:

[0015] in, Indicates the number of nodes in the neighborhood index set. One solution. This represents the current weight vector. This indicates that the current weight vector is at the th position. The weight of each goal Indicates a reference point. Solution In the The function value on the objective function, The total number of objective functions.

[0016] In one implementation, in step S4, the crossover operation is: generating a random number, and based on the comparison result of the random number and the preset crossover probability, determining that the value of the intermediate solution at each gene locus is taken from the first parent solution or the second parent solution.

[0017] In one implementation, in step S4, the mutation operation is to independently flip each binary bit of the intermediate solution based on a preset mutation probability.

[0018] The beneficial effects of this application are as follows: First, the weight vector set is initialized by boundary weight encryption, and the weight of the boundary region is enhanced on the unit hyperplane, so that the search process is more focused on the boundary region of the Pareto front, effectively improving the quality and distribution performance of the boundary solution, and thus improving the convergence and uniformity of the overall solution set.

[0019] Second, the neighborhood search mechanism is integrated with the crossover and mutation operator. In the global search phase, the crossover and mutation operator is used to expand the exploration range. In the local search phase, neighborhood sampling and dynamic updating of the archive set are used to achieve refined development, balancing the convergence speed and diversity preservation of the solution set, and reducing the risk of getting trapped in local optima.

[0020] Third, an unbounded archive set is used to store all non-dominated solutions generated during the iteration process. A strict dominance relationship screening mechanism is used to continuously maintain a high-quality solution set, avoid the loss of boundary solutions and uniformly distributed solutions, and provide a more complete and diverse Pareto front approximation for the output of the final solution set. Attached Figure Description

[0021] Figure 1 This is a schematic diagram of the random local search multi-objective optimization algorithm in an embodiment of the present invention; Figure 2 This is a flowchart illustrating the offspring generation method combining neighborhood search and crossover mutation operators in an embodiment of the present invention. Detailed Implementation

[0022] The technical solution of this application will be clearly and completely described below with reference to specific embodiments. However, those skilled in the art will understand that the embodiments described below are only some embodiments of this application, not all embodiments, and are only used to illustrate this application, and should not be regarded as limiting the scope of this application. Based on the embodiments in this application, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of this application.

[0023] Existing multi-objective optimization algorithms, when solving complex combinatorial optimization problems, typically execute global evolutionary search and local neighborhood search independently, lacking a synergistic and complementary mechanism. Furthermore, boundary solutions in the discrete space are difficult to effectively identify and enhance, leading to uneven solution set distribution and limited convergence efficiency. This application, based on a decomposition strategy framework, introduces an initialization method that encrypts boundary weights. By increasing the weight vector density of the boundary region on the unit hyperplane, it strengthens the search tendency for Pareto front boundary solutions. Simultaneously, it integrates random neighborhood search with crossover and mutation operators. In the global exploration phase, crossover and mutation are used to expand the sampling range; in the local development phase, neighborhood sampling and dynamic updates of the archive set achieve refined optimization, synergistically improving convergence and diversity within a unified process. An unbounded archive set is used to continuously screen and save all non-dominated solutions generated during iteration, preventing the loss of high-quality solutions. Through the cooperation of these mechanisms, a multi-objective optimization scheme that balances global search capability with precise local development is formed.

[0024] In one implementation, a multi-objective optimization method that integrates random local search and boundary weight encryption includes the following steps: S1. Initialize the unbounded archive set used to store non-dominated solutions.

[0025] After the algorithm parameters are configured, a separate storage area is allocated in memory as an unbounded archive set. Initially, the unbounded archive set is set to empty, waiting to receive non-dominated solutions generated after the initial population evaluation.

[0026] S2. Initialize the weight vector set, which includes: generating a densely distributed weight set and a sparsely distributed weight set; selecting boundary weight vectors from the densely distributed weight set; selecting center weight vectors from the sparsely distributed weight set; and combining the boundary weight vectors and the center weight vectors to form the weight vector set.

[0027] Specifically, it includes the following steps: S21. Determine the division quantity parameters and encryption ratio parameters.

[0028] Before initializing the weight vector set, the specific values ​​of two key parameters must first be determined. The first key parameter is the number of coordinate axis divisions. , Defined as a positive integer greater than zero, it controls the density of the weight vector distribution on the unit hyperplane. The second key parameter is the boundary critical value parameter. , Defined as a positive real number, it serves as the threshold for determining whether a weight vector belongs to a boundary region. The determination condition is set as follows: if at least one component in a target dimension of a weight vector has a value less than [value missing], then [the threshold is missing]. If so, the weight vector is identified as the boundary weight vector.

[0029] Division quantity parameters With boundary critical value parameters The value of is not arbitrarily specified, but must simultaneously satisfy the following two constraints. The first constraint is: the sum of the total number of boundary weight vectors and the total number of center weight vectors generated in the final generation is equal to the preset population size parameter. pop The second constraint is that the proportion of the total number of boundary weight vectors in the final total set of weight vectors is equal to the preset estimated encryption ratio parameter. .

[0030] Based on the two constraints mentioned above, the following system of mathematical equations can be established to solve for the parameters. With parameters :

[0031]

[0032] in, Indicates the number of target dimensions. and These are the parameters for the number of divisions. The relevant auxiliary calculation functions are defined as follows: ; .

[0033] S22. Generate a densely distributed weight set and filter the boundary weight vectors.

[0034] Based on the division quantity parameter determined in step S21 The uniform weight generation method proposed by Das and Dennis is used to generate a set of weight vectors that are densely distributed on the unit hyperplane. The specific generation process is based on the following weight vector expression:

[0035] in, Indicates the number of target dimensions. The partitioning parameters are used to control the density of the weight distribution. It is a non-negative integer. The generated weight vector satisfies .when When larger values ​​are taken, the weights are more densely distributed on the unit hyperplane; when When the size is smaller, the distribution is more sparse.

[0036] The total number of weight vectors contained in the generated densely distributed weight set is determined by the following formula:

[0037] This densely distributed weight set covers the entire unit hyperplane in the target space, and because the partitioning parameter uses 2... The weight vector has a high distribution density.

[0038] After obtaining the densely distributed weight set, boundary judgment filtering is performed on each weight vector in the set. The filtering rule is based on the boundary critical value parameter γ determined in step S21: traversing the M components of the weight vector, if any component value is less than... If the weight vector is selected, it is marked as a boundary weight vector and extracted into the boundary weight vector set. This filtering step aims to separate the portion of the weights located at the edge of the target space from the densely distributed weights, thereby enhancing the search tendency for Pareto front boundary solutions in subsequent search processes.

[0039] S23. Generate a sparsely distributed weight set and select the central weight vector. Combine the two weight vectors to form the final weight vector set.

[0040] Also based on the division quantity parameter determined in step S21 We employ the uniform weight generation method proposed by Das and Dennis to generate a set of weight vectors that are sparsely distributed on the unit hyperplane. The specific generation process is based on the following weight vector expression: ; In the above expression, the numerator of each target dimension For non-negative integers, the sum of all numerators equals The total number of weight vectors contained in the generated sparse weight set is determined by the following formula: .

[0041] After obtaining the sparsely distributed weight set, each weight vector in the set is filtered to extract the central weight vector located in the central region of the target space. Specifically, the filtering method is as follows: weight vectors in the sparsely distributed weight set that do not meet the boundary condition are retained, i.e., all weight components are greater than or equal to the boundary critical value parameter. The weight vectors are then incorporated into the central weight vector set. The central weight vector set is used to maintain the uniform distribution of the solution set within the target space, avoiding insufficient search density in the central region due to the densification of boundary weights.

[0042] Finally, the set of boundary weight vectors selected in step S22 and the set of center weight vectors selected in step S23 are merged to form a complete set of weight vectors. The total number of weight vectors in this set is equal to the sum of the number of boundary weight vectors and the number of center weight vectors, and this total number is equal to the population size parameter in step S21. pop Maintain consistency. The proportion of boundary weight vectors in the total set should be consistent with the estimated encryption ratio parameter. Maintain consistency.

[0043] S3. Calculate the reference point based on the minimum value of the initial population on each objective function, and update the unbounded archive set based on the dominance relationship of the solution.

[0044] The reference point is defined as a dynamic baseline vector within the objective space during multi-objective optimization, used to provide a unified reference origin for subsequent scalar function evaluation. Each component of the reference point vector corresponds to the minimum observed value achieved by a specific objective function in the current known solution set.

[0045] Specifically, let the initial population be... The number of objective functions is Regarding the first A target function is used to iterate through the initial population. All individuals Calculate the value of each body in the first... The function value of the objective function Extract the minimum value from all calculated function values; this minimum value is used as the reference point vector at the th . Components in each target dimension The calculation process is expressed by the third formula as follows:

[0046] in, Indicates the reference point is at the 1st. The weight of each goal Indicates the initial population. Represents an individual In the Each value can be taken from the objective function. Iterate through all... For each objective function, repeat the minimum value extraction operation described above, and finally obtain a... 3D reference point vector , denoted as: .

[0047] Reference point vector The establishment of this reference point provides a data-driven origin for the multi-objective optimization process. The reference point clarifies the algorithm's search direction in the objective space, enabling the decomposition-based search process to explore the boundary regions of the objective space more effectively while maintaining overall search uniformity, thus significantly enhancing the diversity of the final solution set. In subsequent iterations, as new solutions are generated, the components of the reference point vector are dynamically updated based on the newly discovered minimum objective function value to maintain the reference point's accurate description of the current search frontier.

[0048] Domination relations of solutions for unbounded archive sets Update and maintenance are in progress. Unbounded Archives Used to store all non-dominated solutions generated during algorithm execution, without being constrained by a preset capacity limit. Update operations use newly generated solutions. As input, based on the solution With archives The Pareto dominance relationship between existing solutions determines which solution to accept or reject.

[0049] The definition of dominance follows a general principle in the field of multi-objective optimization: for two solutions and If the solution Non-inferior solution on all objective functions And solve It is strictly superior to the solution on at least one of the objective functions. Then it is called a solution. Dominant Solution .

[0050] The specific operation process for updating an unbounded archive is as follows: First, the new interpretation With Unbounded Archives Each existing solution in the archive is compared for dominance relationships. If an archive set is found... There exists an existing solution that dominates a new solution. Then determine the new solution. The quality is inferior to the archive. The solutions already retained. In this case, a new solution... Not allowed to enter the archive set; the archive set is discarded immediately. The content remains unchanged. This filtering mechanism effectively prevents the introduction of inferior solutions and maintains the non-dominated property of the solution set within the archive set.

[0051] Second, if the above comparison confirms the new solution... Unarchived set If any existing solution dominates the archive, a reverse domination check is then performed. Reverse traversal of the archive set. Check all existing solutions and new solutions. Whether to dominate an existing solution. For each new solution... If an existing solution is dominant, remove that solution from the archive set. Removed from the middle. This redundant removal operation eliminates the redundancy caused by the new solution. The addition of archives The possibility of a controlled solution appearing in the archive set ensures that the archive set is secure. Each individual solution in the solution maintains a non-dominant relationship with the others.

[0052] Third, after completing the above-mentioned dominance judgment and redundant solution elimination steps, the new solution will be... Add to Unbounded Archive In the middle, the update of this archive set was completed.

[0053] Through the above update mechanism, the Unbounded Archive During algorithm iteration, non-dominated solutions generated in each generation of evolution are continuously accumulated. Since the unbounded archive is not limited by population size, it can fully preserve high-quality solutions that might be lost due to population iterations, especially boundary solutions located in the Pareto front edge region. Finally, at algorithm termination, the solutions are retrieved from the unbounded archive. The output contains all stored non-dominated solutions, which serve as the Pareto front approximate solution set for the multi-objective optimization problem.

[0054] S4. Randomly select two parent solutions corresponding to the indices in the neighborhood index set corresponding to the current weight vector, and perform crossover and mutation operations on the two parent solutions to obtain candidate child solutions; randomly select a new solution in the neighborhood of the candidate child solution, and when the new solution dominates the candidate child solution, store the new solution in the unbounded archive set; use a scalarization function to select the optimal solution from the solutions corresponding to the neighborhood index set as the child solution.

[0055] First, regarding the current weight vector to be processed... Obtain the weight vector Corresponding neighborhood index set Neighborhood index set Defined as in the vector space with the weight vector The closest Euclidean distance or angular distance A set of indices for weight vectors, where This is the preset neighborhood size parameter. Neighborhood index set. The index stored in the middle points to the population. The corresponding solution individuals.

[0056] From neighborhood index set Two distinct index values ​​are randomly selected from the data, and denoted as follows: and Based on the index and From population Extract the two corresponding solution vectors, and use them as the first parent solution. Solution with the second parent The mathematical expression for the random selection process is as follows:

[0057] in, Indicates the first in the population The solution vector for each individual.

[0058] Obtain the first parent solution Solution with the second parent Then, a crossover operation is performed on the two parent solutions to generate an intermediate solution. The crossover operation is based on a preset crossover probability parameter. Proceed. For each gene locus in the solution vector. Generate a random number that follows a uniform distribution within the interval (0,1). Based on random numbers With cross probability The comparison results determine the child solution. In the The source of values ​​at each gene locus. Offspring analysis. In the Values ​​at each gene locus The mathematical equation represents: .

[0059] Subsequently, the solution for the child generation Mutation operations are applied to introduce genetic perturbations. These operations are based on preset mutation probability parameters. Execute. With probability. Independently flip child solutions Each binary bit is used to enhance global exploration capabilities and avoid getting trapped in local optima. Its mathematical equation is expressed as:

[0060] In the formula, This represents the candidate child solutions obtained after the mutation operation. In the The final value at each gene locus. The introduction of mutation operations enhances the algorithm's ability to explore the global decision space and reduces the risk of getting trapped in local optima during the search process.

[0061] Obtain candidate sub-solutions Then, the local fine-grained search phase begins. The purpose of the local fine-grained search is to utilize candidate sub-solutions. The neighborhood information is used to further explore potential high-quality solutions, and non-dominated solution individuals in the evolution process are accumulated through the dynamic update mechanism of the unbounded archive set.

[0062] First, define candidate sub-solutions. The neighborhood. The neighborhood can be defined using a distance metric based on the decision space, such as constructing the neighborhood range by limiting a Hamming distance threshold in the case of binary encoding. In candidate sub-solutions... Perform a random selection operation within the neighborhood of the given solution to obtain a new neighborhood solution. .

[0063] Subsequently, the new solution for the neighborhood will be presented. With candidate sub-solutions Perform a dominance comparison. The dominance comparison follows the general Pareto dominance criterion in multi-objective optimization: if a new solution is found in the neighborhood... It is no worse than the candidate sub-solution on all objective functions. And a new solution in the neighborhood It is significantly superior to candidate sub-solutions on at least one objective function. Then determine the new solution in the neighborhood. Dominant candidate sub-solution .

[0064] If the above dominance relationship holds, an unbounded archive set is triggered. The dynamic update operation will add new solutions to the neighborhood. Store in an unbounded archive The specific update process follows the archive set update rules described in step S3: check for new solutions in the neighborhood. Is it included in the Unbounded Archive? If a solution has already been found, remove it if it is not already dominated. All new solutions in the neighboring region The dominant existing solution will eventually be replaced by a new solution in the neighborhood. Add to archive In the meantime, candidate child solutions... These solutions are retained and considered as candidates for scalarization function evaluation and population update in subsequent steps. This approach improves the overall quality of the solution set while maintaining population diversity, avoiding a narrowing of the search scope due to over-focusing on a single high-quality solution.

[0065] After completing the local fine-grained search, the scalarization function is used to start from the current weight vector. Corresponding neighborhood index set A solution is selected from the associated solution set as a child solution to update the population. In this implementation, the scalarization function used is the Chebyshev scalarization function.

[0066] For neighborhood index set Each index in Obtain the corresponding solution vector Calculate the solution vector. In the weight vector With reference point Chebyshev scalar value The calculation formula is as follows:

[0067] in, This represents the total number of objective functions. Represents the current weight vector In the Components in each target dimension Solution In the The function value on the objective function, Indicates reference point In the The components in each objective dimension. The geometric meaning of the Chebyshev scalarization function lies in measuring the solution. Relative to the reference point in each target dimension The weighted maximum offset; the smaller the value, the better the solution. The closer the weight vector w_i is to the Pareto front, the better.

[0068] Traversing the neighborhood index set All indexes Calculate each corresponding solution separately. Chebyshev scalar value The minimum value is selected from all calculated scalar values, and the solution corresponding to this minimum value is determined relative to the current weight vector. With reference point optimal solution The mathematical expression of the optimal selection process is as follows:

[0069] in, This represents the selected optimal offspring solution, used to replace the corresponding weight vector in the current population. The old solution is used to complete a subproblem update based on decomposition.

[0070] S5. Repeat steps S3 to S4, and update the neighborhood solution of each weight vector using the scalarization function until the preset termination condition is met.

[0071] The iterative process is as follows: In each round, step S3 is followed to update the components of the reference point and maintain the unbounded archive set; then step S4 is followed to perform parent selection, crossover mutation, local search, and scalar optimization on each weight vector. The reference point components are redefined based on the minimum value of each objective function in the current population; the unbounded archive set continuously accumulates non-dominated solutions. For each weight vector, the Chebyshev scalar value of each solution in its neighborhood index set at the current reference point is calculated, and the solution corresponding to the minimum value is selected as the updated solution for that weight vector. The termination condition can be set as the maximum number of iterations, the maximum number of function evaluations, or a convergence stagnation index. After the condition is met, all non-dominated solutions in the unbounded archive set are output as the optimization result.

[0072] Example like Figure 1 As shown, this embodiment provides a multi-objective optimization method that integrates random local search and boundary weight encryption, including the following steps: Step 1: Initialize the unbounded archive set. The unbounded archive set is used to store the non-dominated solutions generated in each generation, thereby enhancing the diversity of the solution set. Step 2: Initialize the weight vector set and use the boundary weight encryption method to improve the quality of the boundary solutions, thereby improving the overall quality of the solution set; Step 3: Calculate the reference point and update the unbounded archive set; Step 4: Combine neighborhood search and crossover mutation operators to generate offspring, select neighborhood solutions and crossover mutation solutions near each weight vector, and use a scalarization function to select the optimal solution as the offspring solution; Step 5: Repeat step 3, and use the scalarization function to update the neighborhood solution of each weight vector.

[0073] Step 2 specifically includes: Step 2.1: Determine the division quantity parameters and encryption ratio parameters. parameter The number of divisions representing the target coordinate axes, a positive integer greater than 0. (Parameter) Used to define boundary weights, the criterion for which any component in the weight vector is less than a preset critical value. These two parameters play a crucial role in the subsequent generation and combination of weight sets.

[0074] In order to determine and The values ​​of must satisfy the following two constraints: First, the total number of boundary weights and center weights must be equal to the population size. pop Second, the proportion of the total number of boundary weight vectors in the final total set of weight vectors is equal to the preset estimated encryption ratio parameter. Based on these two constraints, they can be represented by the following mathematical equations:

[0075]

[0076] In the formula and These represent the methods for calculating the number of weights: , .

[0077] Step 2.2: Generate boundary-dense weight sets using the Das and Dennis methods respectively; Step 2.3: Use the Das and Dennis methods to generate weight sets with a central sparse distribution, and combine the two weight sets.

[0078] By constructing two sets of weights, one densely distributed on the unit hyperplane and the other sparsely distributed, flexible weight configuration is achieved. The weights are generated based on the uniform weight generation method proposed by Das and Dennis. Its expression is:

[0079] in, Indicates the number of target dimensions. The partitioning parameters are used to control the density of the weight distribution. It is a non-negative integer. The generated weight vector satisfies .when When larger values ​​are taken, the weights are more densely distributed on the unit hyperplane; when When the size is smaller, the distribution is more sparse.

[0080] Step 2.3, the generation of boundary weights, aims to extract weights located in the boundary region from the densely distributed weight set, thereby enhancing the algorithm's ability to search for boundary solutions. Specifically, firstly, the uniform weight generation method proposed by Das and Dennis is used to generate a densely distributed weight set based on a given number of partitions. The number of weights is expressed by the formula:

[0081] in, Divide the coordinate axes into numbers. The target number is denoted as . The generated weight set is uniformly distributed within the target space, covering the entire solution space.

[0082] Next, the generated weight set is filtered one by one to extract the weights that meet the boundary weight determination criteria. The specific determination rule is: if any component in the weight vector is less than the parameter... If the weight is not specified, it is considered a boundary weight. The purpose of this selection process is to highlight the weight distribution density in the boundary region, thereby enhancing the ability to explore boundary solutions during the optimization process.

[0083] The generation of center weights aims to extract weights located in the central region of the target space from a sparsely distributed weight set, ensuring the global uniformity and diversity of the solution set. Specifically, the weight generation method proposed by Das and Dennis is first used to generate a sparsely distributed weight set based on a given number of partitions. The number of weights is determined by the following formula:

[0084] in, Divide the coordinate axes into numbers. The target number is denoted as . The generated weight set is uniformly distributed with a low density within the target space.

[0085] Step 3 calculates the reference point and introduces an unbounded archive set to update the archive set. The unbounded archive set stores non-dominated solutions generated in each generation, regardless of whether these solutions are retained in the current population. By recording and preserving excellent solutions that emerge throughout the evolutionary history, the unbounded archive set effectively prevents the erroneous deletion of critical solutions such as boundary solutions and solutions with good distribution uniformity. Simultaneously, the unbounded archive set provides a global perspective for the final solution set selection, ensuring that all potential Pareto front solutions are fully considered and retained.

[0086] The core purpose of updating the Unbounded Archives is to update the archives. A Dynamic maintenance is performed to ensure that the archive always contains the current non-dominated solution set, thus providing global storage support for high-quality solutions to multi-objective optimization problems. The algorithm flow is shown in Table 1. First, a uniformly distributed reference point is generated using boundary weight encryption technology, and this is used to initialize the archive set. A This establishes a high-quality initial solution set for subsequent optimization processes. During the optimization iteration process, the newly generated solutions... q Perform strict dominant judgments: if q quilt A If any solution dominates, it indicates that its quality is insufficient and should be discarded directly to avoid introducing inferior solutions; if q If not under control, further investigation is required. A Remove all solutions from the set. q The dominant solution effectively eliminates redundancy. After completing the above screening, q Add to archive A This update is now complete.

[0087] Table 1 Unbounded Archive Set Update Algorithm (UAA)

[0088] This method ensures the archive set remains optimal through a filtering mechanism that updates the unbounded archive set. Dominance judgment effectively filters out inferior solutions, preventing solution set quality degradation; while the mechanism of actively removing dominated solutions significantly improves the distribution of the solution set. The application of boundary weight encryption technology guarantees the uniformity and representativeness of the generated reference points, providing a solid foundation for algorithm convergence. Ultimately, this method achieves a uniformly distributed and comprehensive Pareto front while maintaining high convergence speed, providing stable and reliable non-dominated solution set support for multi-objective optimization problems.

[0089] In step 4, the neighbor search and crossover mutation operators are combined to generate offspring. Neighborhood solutions and crossover mutation solutions near each weight vector are selected to improve the quality of generated solutions in multi-objective optimization problems. Furthermore, the optimal solution is selected as the offspring solution using a scalarization function to ensure the diversity and convergence of the population.

[0090] The flowchart of this method is as follows: Figure 2 As shown in Table 2, by introducing a dynamic archive set and closely combining global search and local search, a good balance is achieved between the convergence and diversity of solutions, providing an efficient solution generation strategy for multi-objective optimization problems. The algorithm flow is shown in Table 2. First, in the global search phase, based on the current weight vector... The neighborhood structure, randomly select two parent solutions and Intermediate child solutions are generated by simulating crossover operations. q Subsequently q By performing polynomial or Gaussian mutations, controllable perturbations are introduced to enhance the algorithm's exploration range in the decision space, effectively avoiding premature convergence and local optima traps. Subsequently, in the local fine-search phase, the algorithm... q Candidate solutions are generated by random sampling in the neighborhood. And a dynamic archive set update mechanism is used for quality assessment: if Dominate q ,Right now Non-inferior on all objective functions q And if it is better in at least one objective, then it will Store in an unbounded archive A This enables the dynamic accumulation of high-quality solutions; simultaneously, the original solutions... q They still participate in subsequent evolutionary operations as offspring, taking into account both the convergence and diversity of the solution set.

[0091] This strategy achieves extensive sampling through crossover mutation, improves local accuracy through neighborhood search, and combines a dynamic archiving mechanism to continuously collect and maintain non-dominated solutions, thereby enhancing the algorithm's ability to approximate the Pareto front and its distribution uniformity.

[0092] Table 2. Solution generation strategy combining crossover mutation and neighborhood search

[0093] Although the embodiments of this application have been described above in conjunction with the accompanying drawings, this application is not limited to the specific embodiments and application fields described above. The specific embodiments described above are merely illustrative and instructive, not restrictive. Those skilled in the art can make many other forms based on the guidance of this specification and without departing from the scope of protection of the claims of this application, and these are all within the scope of protection of this application.

Claims

1. A multi-objective optimization method integrating random local search and boundary weight encryption, characterized in that, Includes the following steps: S1. Initialize the unbounded archive set used to store non-dominated solutions; S2. Initialize the weight vector set, wherein the initialization of the weight vector set includes: generating a densely distributed weight set and a sparsely distributed weight set, selecting boundary weight vectors from the densely distributed weight set, selecting center weight vectors from the sparsely distributed weight set, and combining the boundary weight vectors and the center weight vectors to form the weight vector set; S3. Calculate the reference point based on the minimum value of the initial population on each objective function, and update the unbounded archive set based on the dominance relationship of the solution; S4. Randomly select two parent solutions corresponding to the indices in the neighborhood index set corresponding to the current weight vector, and perform crossover and mutation operations on the two parent solutions to obtain candidate child solutions; randomly select a new solution in the neighborhood of the candidate child solution, and when the new solution dominates the candidate child solution, store the new solution in the unbounded archive set; use a scalarization function to select the optimal solution from the solutions corresponding to the neighborhood index set as the child solution; S5. Repeat steps S3 to S4, and update the neighborhood solution of each weight vector using the scalarization function until the preset termination condition is met.

2. The multi-objective optimization method according to claim 1, characterized in that, In step S2, the dense and sparse weight sets are generated based on the Das and Dennis methods, where: The number of weight vectors in the densely distributed weight set is determined by a first formula, which is: ； The number of weight vectors in the sparsely distributed weight set is determined by a second formula, which is: ； in, Divide the coordinate axes into numerical parameters. The number of target dimensions.

3. The multi-objective optimization method according to claim 1, characterized in that, In step S2, the boundary weight vector satisfies the following condition: at least one component in the weight vector is less than a preset critical value.

4. The multi-objective optimization method according to claim 1, characterized in that, In step S2, the combination of the boundary weight vector and the center weight vector satisfies the following constraints: The total number of boundary weight vectors and center weight vectors is equal to the preset population size; The proportion of the boundary weight vector in the total weight vector is equal to the preset estimated encryption ratio.

5. The multi-objective optimization method according to claim 1, characterized in that, In step S3, each component of the reference point is calculated by a third formula, which is: in, Indicates the reference point is at the 1st. The weight of each goal Indicates the initial population. Solution In the Function values ​​on the objective function.

6. The multi-objective optimization method according to claim 1, characterized in that, In step S3, updating the unbounded archive set based on the dominance relation of the solution includes: When a new solution is dominated by any solution in the unbounded archive set, the new solution is discarded; When the new solution is not dominated by any solution in the unbounded archive set, the new solution is added to the unbounded archive set, and all solutions in the unbounded archive set dominated by the new solution are removed.

7. The multi-objective optimization method according to claim 1, characterized in that, In step S4, the scalarization function is the Tchebycheff scalarization function, and its expression is: in, Indicates the number of nodes in the neighborhood index set. One solution. This represents the current weight vector. This indicates that the current weight vector is at the th position. The weight of each goal Indicates a reference point. Solution In the The function value on the objective function, The total number of objective functions.

8. The multi-objective optimization method according to claim 1, characterized in that, In step S4, the crossover operation is as follows: a random number is generated, and based on the comparison result between the random number and the preset crossover probability, the value of the intermediate solution at each gene locus is determined to be taken from the first parent solution or the second parent solution.

9. The multi-objective optimization method according to claim 1, characterized in that, In step S4, the mutation operation is to independently flip each binary bit of the intermediate solution based on a preset mutation probability.

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