Expensive constraint multi-objective optimization method based on integrated filling criterion
By integrating filling criteria and phased environment selection, the problems of feasibility, convergence and diversity in expensive constrained multi-objective optimization problems are solved, and Pareto optimal solution sets are obtained efficiently in complex feasible domains.
Patent Information
- Authority / Receiving Office
- CN · China
- Patent Type
- Applications(China)
- Current Assignee / Owner
- CHINA UNIV OF GEOSCIENCES (WUHAN)
- Filing Date
- 2025-10-24
- Publication Date
- 2026-04-10
AI Technical Summary
Traditional expensive constrained multi-objective optimization problems (ECMOPs) struggle to balance feasibility, convergence, diversity, and exploration within a finite number of function evaluations, especially in complex feasible domains where they are difficult to overcome obstacles and approach the Pareto optimal front.
An integrated filling criterion is adopted, which combines a Gaussian process surrogate model, angle-based calculation, PBI, IP and other criteria to screen candidate populations, and achieves multi-objective optimization with expensive constraints through phased environment selection.
Within a finite number of function evaluations, it improves the quality and efficiency of solutions in complex feasible regions, balances convergence, diversity, and feasibility, and obtains a high-quality Pareto optimal solution set.
Smart Images
Figure CN121835342A_ABST
Abstract
Description
TECHNICAL FIELD
[0001] The present application relates to the technical field of multi-objective optimization, and particularly relates to an expensive constraint multi-objective optimization method based on integrated filling criteria. BACKGROUND
[0002] In the field of industry and engineering, expensive constraint multi-objective optimization problems (ECMOPs) are very common, such as mechanical design, chemical optimization and the like. Such problems have the following significant features: on the one hand, the evaluation process needs to consume a large amount of calculation time and cost, that is, "expensiveness"; on the other hand, there are multiple conflicting optimization objectives, and there are also many constraints which are often regarded as black boxes, and it is extremely difficult to obtain gradient information. The core objective is to find a compromise solution set that satisfies the constraint conditions within a limited number of function evaluations (FEs), that is, the solution corresponding to the constraint Pareto optimal front (CPF).
[0003] However, ECMOPs face many challenges. Firstly, the feasible region is often discontinuous, the CPF can be divided into multiple segments, and part of the feasible region is narrow, so it is difficult for the algorithm to cross the infeasible obstacle and approach the CPF within a limited number of FEs; secondly, it is necessary to balance feasibility, convergence, diversity, exploration and utilization, and traditional methods are difficult to take into account these indicators.
[0004] Therefore, it is of great significance to develop an optimization algorithm that can break through the obstacle of a complex feasible region and balance the performance in multiple dimensions within a limited number of FEs for solving ECMOPs in the field of industry and engineering. SUMMARY
[0005] The purpose of the present application is to solve the problem that traditional constraint multi-objective optimization problem solving methods are difficult to balance feasibility, convergence, diversity, exploration and utilization and the like, and to provide an expensive constraint multi-objective optimization method based on integrated filling criteria.
[0006] The above purpose of the present application is realized by the following technical scheme: S1: initializing a population; S2: constructing a Gaussian process proxy model for each objective function and constraint function; in the algorithm iteration process, the Gaussian process proxy model is updated based on the search space exploration result of each generation; S3: combining the criteria based on angle calculation, the criteria based on PBI, the criteria based on IP and the criteria considering model uncertainty to select a candidate population from the generated candidate population; S4: selecting the most potential sample point from the candidate population using the comprehensive index and performing fitness evaluation, adding the new data point obtained by evaluation to the archive, and updating the Gaussian process proxy model; S5: performing stage-wise environmental selection on the candidate population to obtain a set of constrained Pareto optimal solutions in the output archive, and completing the expensive constrained multi-objective optimization.
[0007] Optionally, step S1 comprises: The Latin hypercube sampling method is adopted to generate N initial input solutions, the objective function value and the constraint function value corresponding to each solution are evaluated, and all the evaluated solutions are stored in the archive.
[0008] Optionally, step S3 comprises: Based on the angle calculation criterion, the representative solutions are selected from the generated candidate population by means of non-dominated sorting and selection of the maximum angle, that is, the candidate population is selected from the generated candidate population, and the expression is as follows:
[0009] wherein represents the number of objective functions; represents the solution on the normalized objective value of the i-th objective function; represents the candidate solution on the original objective value of the i-th objective function; represents the solution represents the candidate solution in the i-th objective space; represents the angle between the solution and the candidate solution in the i-th objective space; represents the current non-dominated solution set; represents the selected representative solution set; represents that for a solution , the minimum value in the angle between the solution and each candidate solution in the representative solution set is calculated.
[0010] Optionally, step S3 further comprises: Based on the PBI criterion, the solution that meets the requirements is selected based on the perpendicular distance of the solution to the reference vector and the PBI value, and the expression is as follows:
[0011] wherein represents the normalized vector of the solution in the objective space; represents the reference vector of the i-th sub-problem; represents the projection distance of the solution to the reference vector in the direction; represents the projection distance of the solution to the reference vector in the direction; vertical distance in the direction; represent the comprehensive fitness value of the th sub-problem. Optionally, step S3 further comprises: Based on the IP criterion, calculate the IP indicator value for each solution, and then calculate the expected improvement value of each solution, select the solution with the largest EI value, and the expression is as follows:
[0012] Wherein represents the size of the existing training set; and respectively represent the solution and the real target function value vector in the target space; represents the solution to the minimum distance of all other solutions in the existing training set; represents the normalized target vector of the solution ; represents the ideal point, which is composed of the best value of each target function in the current known solution set; represents the distance of the solution to the ideal point; represents the improvement value of the solution .
[0013] Optionally, step S5 comprises: The phased environment selection includes: an unconstrained optimization phase and a constrained optimization phase; Unconstrained optimization phase: only consider the objective function, use NSGA-III algorithm for environment selection, generate reference vector V; Select the initial population based on the minimum value of each target in the archive, generate the offspring population by simulating binary crossover and polynomial mutation, merge the parent and offspring populations, and then reselect the environment to obtain the elite population, until the maximum evolution generation of this phase is reached; Constrained optimization phase: consider all objective functions and constraint conditions, use NSGA-III-CDP algorithm with constraint dominance principle CDP for environment selection, retain the solution that satisfies the constraint and has the optimal objective function, and ensure that the search is promoted within the feasible region; Termination judgment: if the number of iterations of the algorithm reaches the preset maximum function evaluation number, output the constraint Pareto optimal solution set in the archive; otherwise, return to step S3 for iteration.
[0014] An electronic device comprises a processor, a memory, a user interface and a network interface, the memory is used to store instructions, the user interface and the network interface are used to communicate with other devices, and the processor is used to execute the instructions stored in the memory to enable the electronic device to perform an integrated filling criterion-based expensive constraint multi-objective optimization method.
[0015] A computer-readable storage medium stores instructions that, when executed, perform an integrated filling criterion-based expensive constraint multi-objective optimization method.
[0016] The technical scheme provided by the application has the beneficial effects that: An effective integrated strategy is constructed to organically combine multiple filling criteria to improve the quality and efficiency of screening candidate solutions in a complex feasible region. The limitations of traditional single filling criteria in convergence, diversity, exploration and utilization balance are solved. BRIEF DESCRIPTION OF DRAWINGS
[0017] The application will be further described below with reference to the drawings and embodiments. In the drawings: Figure 1 is a step diagram in the embodiments of the application; Figure 2 is an integrated filling criterion strategy diagram in the embodiments of the application; Figure 3 is an effect diagram in the embodiments of the application; Figure 4 is an electronic device structure schematic diagram in the embodiments of the application. DETAILED DESCRIPTION
[0018] In order to have a clearer understanding of the technical features, objectives and effects of the application, the specific embodiments of the application will be described in detail with reference to the drawings.
[0019] The embodiments of the application provide an integrated filling criterion-based expensive constraint multi-objective optimization method.
[0020] Please refer to Figure 1 , Figure 1 is a step diagram of an integrated filling criterion-based expensive constraint multi-objective optimization method in the embodiments of the application, comprising: S1: performing an initialization operation on a population; S2: constructing a Gaussian process proxy model for each target function and constraint function; in the algorithm iteration process, the Gaussian process proxy model is updated based on the search space exploration result of each generation; S3: combining the criteria based on angle calculation, the criteria based on PBI, the criteria based on IP and the criteria considering model uncertainty to screen out a candidate population from the generated candidate population; In one embodiment of the present application, by combining a plurality of different filling criteria, the advantages of each criterion are fully utilized to achieve a more efficient and comprehensive search, effectively balancing convergence, diversity and feasibility. As shown in the following figure. The integrated filling criterion includes a plurality of basic filling criteria, such as angle calculation-based criteria, PBI-based criteria, criteria considering model uncertainty, etc. The integrated filling criterion balances the advantages of each basic criterion through a dynamic weight distribution mechanism, covering the dimensions of feasibility, convergence, diversity and model uncertainty assessment. Figure 2
[0021] S4: Select the most potential sample points from the candidate population using the comprehensive index and perform fitness evaluation, add the new data points obtained by evaluation to the archive, and update the Gaussian process proxy model; In one embodiment of the present application, the most potential sample points are selected from the candidate population using the comprehensive index for expensive fitness evaluation. After adding the new data points obtained by evaluation to the archive, the algorithm updates the Gaussian process proxy model to improve the accuracy and adaptability of the model, providing more reliable support for subsequent iterations.
[0022] S5: Perform phased environmental selection on the candidate population to obtain the set of constraint Pareto optimal solutions in the output archive, and complete the expensive constraint multi-objective optimization.
[0023] The present application uses the above technical solution to address the difficulties of discontinuous and narrow feasible region in expensive multi-objective optimization problems, improves the traditional design, innovatively explores and utilizes the balance strategy, enhances the ability of the algorithm to cross the infeasible obstacles and approach the global optimal solution, and obtains a high-quality, uniformly distributed Pareto frontier solution set under a limited number of function evaluations.
[0024] Step S1 includes: A Latin hypercube sampling method is used to generate N initial input solutions, and the target function value and constraint function value corresponding to each solution are evaluated. All evaluated solutions are stored in the archive.
[0025] Step S3 includes: The angle calculation-based criterion selects representative solutions from the candidate solution set by non-dominated sorting and selecting the maximum angle, that is, filters the candidate population from the generated candidate population, and the expression is as follows:
[0026] Wherein represents the number of target functions; represents the normalized target value of the solution on the first target function; represents the candidate solution the original objective value on the th objective function; denotes the solution and the candidate solution the angle between dimensional objective space; denotes the current non-dominated solution set; denotes the selected representative solution set; denotes a solution , the minimum value of the angle between it and each candidate solution in the representative solution set is calculated.
[0027] Step S3 further comprises: Based on the PBI criterion, the solution that better meets the requirements is selected based on the perpendicular distance of the solution to the reference vector and the PBI value, and the expression is as follows:
[0028] wherein denotes the normalized vector of the solution in the objective space; denotes the reference vector of the th sub-problem; denotes the projection distance of the solution to the reference vector in the direction; denotes the perpendicular distance of the solution to the reference vector in the direction; denotes the comprehensive fitness value of the th sub-problem. Step S3 further comprises: Based on the IP criterion, the IP index value of each solution is calculated, and then the expected improvement value of each solution is calculated, and the solution with the largest EI value is selected, and the expression is as follows:
[0029] wherein denotes the size of the existing training set; and denote the real objective function value vectors of the solution and in the objective space, respectively; denotes the minimum distance of the solution to all other solutions in the existing training set; denotes the normalized objective vector of the solution ; denotes the ideal point, which is composed of the best value of each objective function in the currently known solution set; denotes the distance of the solution to the ideal point; improved value of the solution improved value of the solution
[0030] Step S5 comprises: The phased environment selection comprises: an unconstrained optimization phase and a constrained optimization phase. The unconstrained optimization phase: only considering the objective function, using the NSGA-III algorithm for environment selection, generating reference vector V. Select the initial population based on the minimum value of each target in the archive, generate the offspring population through simulated binary crossover and polynomial mutation, merge the parent and offspring populations, and then reselect the environment to obtain the elite population, until the maximum evolution generation of this phase is reached. The constrained optimization phase: considering all objective functions and constraint conditions, using the NSGA-III-CDP algorithm with constraint dominance principle CDP for environment selection, retaining solutions that meet the constraints and have optimal objective functions, and ensuring that the search is within the feasible region. Termination judgment: if the number of iterations reaches the preset maximum function evaluation number, output the constraint Pareto optimal solution set in the archive; otherwise, return to step S3 for iteration.
[0031] In one specific embodiment of the present application, the population is selected by the environment selection strategy. In phase one, the constructed optimization problem only considers the objective function, aiming to quickly converge to the UPF vicinity and cross the infeasible obstacle. At this time, the NSGA-III algorithm is used for environment selection. First, a set of reference vectors V is generated, and the minimum value corresponding to each target in the archive is obtained. From which the initial population is selected using the environment selection method of NSGA-III. In the evolution stage, the offspring population is generated through simulated binary crossover and polynomial mutation, the parent and offspring are combined to create a new population P, P is evaluated and the minimum value is updated, and then the environment selection is performed to select the elite population P, until the maximum evolution generation is reached. The strategy focuses on optimizing the objective function, guiding the search to advance near the UPF, and laying a foundation for the subsequent stage.
[0032] Phases two and three consider all objectives and constraints. The constructed surrogate model contains all objective functions and adds constraint conditions. In this phase, the NSGA-III-CDP algorithm is used for environment selection, which ensures that the search is within the feasible region and further optimizes the quality of the solution.
[0033] The CDP (Constraint Domination Principle) is integrated into the NSGA-III. The CDP is used to judge the pros and cons of solutions, and solutions that meet the constraint conditions and perform better in the objective function are preferentially selected. In the NSGA-III, there is originally an environmental selection mechanism based on non-dominated sorting and reference vectors. After the CDP is combined, the algorithm is more targeted when dealing with constraints. It will comprehensively consider the feasibility and objective function value of the solution, and the better solution will be retained to enter the next generation population.
[0034] In a specific implementation of the present application, the application and focus of each criterion are different at different stages of the optimization process. In the first stage without constraints, the main focus is on the objective function, and the criterion focuses on guiding the search to quickly approach the unconstrained Pareto frontier, such as selecting solutions with good convergence and diversity potential through non-dominated sorting and angle calculation (ND-Angle), while using the PBI-based criterion (ND-APBI) to further optimize the distribution of solutions, using the IP-based criterion (ND-IP) to consider the distance of the solution to the ideal point and the distance to the nearest neighbor. The criterion considering model uncertainty (DirHV-EI) balances exploration and utilization, laying the foundation for the search in the subsequent stages. In the second and third stages with constraints, the criterion focuses more on feasibility while considering convergence and diversity. For example, the constraint domination principle (CDP) is introduced on the basis of the previous one to form ND-Angle-CDP, ND-PBI-CDP, and ND-IP-CDP, which preferentially selects solutions that meet the feasibility requirements. For DirHV-EI, in the second and third stages, the feasibility probability (PoF) is combined to form DirHV-EI-PoF, ensuring that the feasibility of the solution is fully considered during exploration and utilization.
[0035] In a specific implementation of the present application, as shown in Figure 3 , the present experiment is based on the PlatEMO platform, and is aimed at multi-objective expensive constrained optimization problems. The LIRCMOP, DIRCMOP, and other series of test problems are selected, and the EC_MOEA_EIC algorithm of the present application is compared with the MGSAMA, PEA, and other algorithms. Each problem is independently run 20 times, and the true evaluation number is set to 400. From the experimental results, the solving accuracy and other performance of the EC_MOEA_EIC algorithm are better on multiple test problems, and the comprehensive performance exceeds the existing comparative algorithms such as MGSAMA and PEA.
[0036] The present application also discloses an electronic device. Referring to Figure 4 , Figure 4is a structural schematic diagram of an electronic device disclosed by an embodiment of the present application. The electronic device 500 can include at least one processor 501, at least one network interface 504, a user interface 503, a memory 505, and at least one communication bus 502.
[0037] The communication bus 502 is configured to realize connection and communication between the components.
[0038] The user interface 503 can include a display screen, and the optional user interface 503 can further include a standard wired interface and a wireless interface.
[0039] The network interface 504 can optionally include a standard wired interface and a wireless interface (such as a WI-FI interface).
[0040] The present application further discloses a computer readable storage medium, which stores a plurality of instructions adapted to be loaded by a processor to execute the above-mentioned expensive constraint multi-objective optimization method based on integrated filling criteria.
[0041] The above are only exemplary embodiments of the present disclosure, and cannot limit the scope of the present disclosure. Any equivalent changes and modifications made in accordance with the teachings of the present disclosure are still within the scope of the present disclosure.
[0042] The present application is intended to cover any variations, uses, or adaptive changes of the present disclosure that follow the general principles of the present disclosure and include common knowledge or conventional technical means in the technical field of the present disclosure not recorded in the present disclosure. The specification and examples are only considered as exemplary, and the scope and spirit of the present disclosure are defined by the claims.
Claims
1. A costly constrained multi-objective optimization method based on integrated filling criteria, characterized in that, The method includes the following steps: S1: Perform population initialization operations; S2: For each objective function and constraint function, construct a Gaussian process surrogate model; during the algorithm iteration process, update the Gaussian process surrogate model based on the search space exploration results in each generation; S3: Combine the criteria based on angle calculation, the criteria based on PBI, the criteria based on IP, and the criteria that take into account model uncertainty to select candidate populations from the generated candidate populations; S4: Use comprehensive indicators to select the most promising sample points from the candidate population and evaluate their fitness. Add the new data points obtained from the evaluation to the archive and update the Gaussian process surrogate model. S5: Perform phased environment selection on the candidate population to obtain the set of constrained Pareto optimal solutions in the output archive, thus completing the expensive constrained multi-objective optimization.
2. The expensive constraint multi-objective optimization method based on integrated filling criteria as described in claim 1, characterized in that, Step S1 includes: The Latin hypercube sampling method is used to generate N initial input solutions. The objective function value and constraint function value corresponding to each solution are evaluated, and all evaluated solutions are stored in the archive.
3. The expensive constraint multi-objective optimization method based on integrated filling criteria as described in claim 1, characterized in that, Step S3 includes: Based on the angle calculation criterion, representative solutions are selected from the candidate solution set through non-dominated sorting and selection of the maximum included angle. That is, candidate populations are selected from the generated candidate populations, as shown in the following expression: in Indicates the number of objective functions; Representing the solution In the Normalized target value over an objective function; Indicates candidate solutions In the The original objective value on each objective function; Representing the solution and candidate solutions exist Angle in the target space; Represents the current non-dominated solution set; This represents the selected representative solution set; This indicates that for a certain solution Calculate its relationship with each candidate solution in the representative solution set. The minimum value of the included angle.
4. The expensive constraint multi-objective optimization method based on integrated filling criteria as described in claim 1, characterized in that, Step S3 also includes: Based on the PBI criterion, a more suitable solution is selected based on the perpendicular distance from the solution to the reference vector and the PBI value. The expression is as follows: in Representing the solution A normalized vector in the target space; Indicates the first Reference vectors for each sub-problem; Representing the solution to reference vector Projected distance in the direction; Representing the solution to reference vector Vertical distance in the direction; Indicates the first The overall fitness value of each sub-problem.
5. The expensive constraint multi-objective optimization method based on integrated filling criteria as described in claim 1, characterized in that, Step S3 also includes: Based on the IP criterion, an IP index value is calculated for each solution, and then the expected improvement value for each solution is calculated. The solution with the largest EI value is selected, as shown in the following expression: in Indicates the current size of the training set; and Describe the solutions respectively and The vector of the true objective function values in the objective space; Representing the solution The minimum distance to all other solutions in the existing training set; Representing the solution The normalized target vector; The ideal point is represented by the optimal value of each objective function in the currently known solution set; Representing the solution Distance to the ideal point; Representing the solution The improvement value.
6. The expensive constrained multi-objective optimization method based on integrated filling criteria as described in claim 1, characterized in that, Step S5 includes: The phased environment selection includes: an unconstrained optimization phase and a constrained optimization phase; Unconstrained optimization stage: Considering only the objective function, the NSGA-III algorithm is used for environment selection to generate the reference vector V; The initial population is selected based on the minimum value of each objective in the archive. The offspring population is generated by simulating binary crossover and polynomial mutation. After merging the parent and offspring populations, environmental selection is performed again to obtain an elite population, until the maximum number of generations of evolution is reached at this stage. Constraint optimization phase: Considering all objective functions and constraints, the NSGA-III-CDP algorithm, which incorporates the constraint dominance principle (CDP), is used to select the environment, retaining the solution that satisfies the constraints and has the optimal objective function, and ensuring that the search proceeds within the feasible region; Termination judgment: If the number of algorithm iterations reaches the preset maximum number of function evaluations, output the constrained Pareto optimal solution set in the archive; otherwise, return to step S3 to continue iteration.
7. An electronic device, characterized in that, The device includes a processor, a memory, a user interface, and a network interface. The memory is used to store instructions, the user interface and the network interface are used to communicate with other devices, and the processor is used to execute the instructions stored in the memory to cause the electronic device to perform the method as described in any one of claims 1-6.
8. A computer-readable storage medium, characterized in that, The computer-readable storage medium stores instructions that, when executed by a computer, perform the method as described in any one of claims 1-6.