Coevolution method for solving expensive constrained multi-modal problem

Abstract

The invention discloses a coevolution method for solving an expensive constrained multi-modal problem, which belongs to the technical field of multi-modal optimization, and comprises the following steps: S1, inputting a target function f, constraint functions {g1,..., gm} and the maximum number TEmax of real evaluation particles, entering an initialization stage, and updating the number FE of real evaluation times; s2, when FElt; tEmax, entering an iteration period; s3, respectively constructing / updating global agent models of the target and each constraint by using samples in the database; s4, designing a two-stage auxiliary problem construction mechanism and a main-auxiliary problem coevolution strategy; s5, executing a multi-index guided two-stage agent model management strategy; s6, proposing a two-stage elite solution local search strategy; s7, updating the database, the feasible solution set, the infeasible solution set and the number of real evaluation times, wherein Cycle = Cycle + 1; and S8, stopping iteration, and outputting a feasible solution set and an optimal solution set. By adopting the method, the main-auxiliary problem can quickly track the optimal solution in multiple modes of the main problem through bidirectional information migration.

Description

Technical Field

[0001] The present invention relates to the technical field of multimodal optimization, and in particular to a collaborative evolutionary method for solving expensive constrained multimodal problems. Background Art

[0002] In practical applications, many optimization problems are not only subject to various constraints, but the computational cost of their objective functions and constraints is also very high. For example, problems such as computational electromagnetics and automotive suspension component structural design are often referred to as expensive constraint optimization problems (ECOPs). In addition, in some cases, such problems also exhibit multimodal characteristics, that is, there are multiple global or local optimal solutions within the feasible region. Typical cases include mechanical design and electric vehicle concept design. Decision makers usually seek multiple optimal solutions in this situation to meet different requirements. Such problems are called expensive constraint multimodal optimization problems (ECMMOPs), and their form is as follows:

[0003]

[0004] Where: X=(x1,…,x D ) is the D-dimensional decision vector, S is the value space of X, R D is a D-dimensional search space, f is an objective function with multiple optimal solutions, g i (x) is the i-th inequality constraint, and m is the number of constraints. In addition, the constraint violation (CV) is often used to evaluate the feasibility of candidate solutions as follows:

[0005]

[0006] Obviously, X is feasible if and only if CV(X)=0.

[0007] In multimodal optimization problems, decision makers hope to find multiple global or local optimal solutions so that they can flexibly choose and meet the different needs of different occasions. At present, a large number of evolutionary algorithms (EAs) have been proposed to deal with multimodal optimization problems. However, there are relatively few studies on constrained multimodal optimization problems. Deb and Saha transformed the constrained multimodal optimization problem into a bi-objective problem to capture all global and local optima in the weak Pareto set. Rebello et al. drew the optimal confidence region of the population based on likelihood testing and used the constrained sliding particle swarm optimization algorithm to solve the constrained multimodal optimization problem to find multiple local and global optimal solutions. However, when dealing with ECMMOPs, these methods still have the disadvantage of high computational cost because all generated individuals need to be truly evaluated.

[0008] Surrogate-assisted evolutionary algorithms (SAEAs) are a common approach for reducing high evaluation costs. Their main idea is to use computationally inexpensive surrogate models (such as Kriging, radial basis functions, support vector machines, and artificial neural networks) to approximate expensive objective or constraint functions, thereby reducing the costly evaluation of the true function during the evolutionary process. However, traditional SAEAs typically only find a single optimal solution in a single run because they lack specialized mechanisms for maintaining multiple optimal solutions. To address multimodal problems, recent research has incorporated strategies such as search space partitioning, clustering, and multi-population coevolution into SAEAs. Representative approaches include the dual-surrogate-assisted two-swarm particle swarm optimization algorithm proposed by Ji et al., the heterogeneous ensemble surrogate-assisted interval particle swarm optimization algorithm, the multi-surrogate-assisted multi-task particle swarm optimization algorithm, the RBFN-based decomposition-based differential evolution algorithm proposed by Gao et al., and the multi-layer perceptron-based surrogate-assisted evolutionary algorithm proposed by Ji and Tan. These methods can simultaneously locate multiple optimal solutions to a problem. However, due to the lack of effective constraint handling strategies, these methods are unable to handle ECMMOPs.

[0009] Existing techniques for handling constraint problems can be roughly divided into five categories. The first category is penalty function methods; the second category is feasibility rules, specifically: when two solutions are both feasible, the solution with the better fitness value wins; when one solution is feasible and the other is not, the feasible solution wins; when both solutions are not feasible, the solution with the least constraint violation wins; the third category is multi-objective optimization methods, in which all constraints are converted into objectives; the fourth category is ε-constraint methods, in which one objective function is selected as the primary objective and the other objectives are treated as constraints with a threshold ε; and the fifth category is random ranking methods, which use a probability parameter to balance the objective function value and the constraint violation degree to rank the solutions. All of these methods can be directly applied to handling expensive constraints in ECMMOPs. However, they are too time-consuming due to the need to realistically evaluate the values ​​of all individual constraint functions.

[0010] Although SAEAs have been widely applied to ECOPs and expensive multimodal optimization problems, they still face numerous challenges when dealing with ECMMOPs, which are computationally expensive, constrained, and multimodal. The complexity of ECMMOPs lies in the fact that algorithms must not only identify and maintain multiple global and local optimal solutions while maintaining computational constraints, but also strictly satisfy the constraints, making simple extensions of existing algorithms ineffective. Therefore, how to ensure solution diversity while also considering solution feasibility and effectively balancing global exploration and local exploitation becomes a key issue in solving ECMMOPs.

[0011] To date, only Ji et al. have systematically studied ECMMOPs. Their core concept is to construct a collaborative framework for surrogate models with mutually guided objective and constraint functions. This approach trains surrogate models for the objective and constraint functions at the global and local levels, respectively, and combines partial evaluation strategies with a hybrid particle swarm update mechanism to fully utilize limited real-world evaluations while avoiding resource waste in non-critical areas. However, this approach has limitations in rapidly locating the feasible region, which is often the primary step in constrained optimization. Summary of the Invention

[0012] The purpose of the present invention is to provide a collaborative evolutionary method for solving expensive constrained multimodal problems, so as to solve the problems mentioned in the background technology.

[0013] To achieve the above objectives, the present invention provides a collaborative evolutionary method for solving expensive constrained multimodal problems, comprising the following steps:

[0014] S1, input objective function f, constraint function {g1,…,g m} and the maximum number of true evaluation particles TE max , enter the initialization phase and update the real evaluation times FE;

[0015] S2, when FE <TE max , enter the iteration cycle;

[0016] S3, using samples in the database to build / update the global surrogate model of the target and each constraint;

[0017] S4. Design a two-stage auxiliary problem construction mechanism and a main-auxiliary problem co-evolution strategy;

[0018] S5. Execute a two-stage agent model management strategy guided by multiple indicators, specifically: select a set of samples from the main problem and conduct real evaluation, conduct real evaluation on the selected samples, and update the database, feasible solution set, and infeasible solution set;

[0019] S6. A two-stage elite solution local search strategy is proposed;

[0020] S7, update the database, feasible solution set, infeasible solution set and the number of true evaluations, Cycle = Cycle + 1;

[0021] S8. Terminate the iteration and output the feasible solution set and the optimal solution set. Specifically, after the entire iteration is completed, output the feasible solution set FS, and filter the optimal solution set OS from FS.

[0022] Preferably, the S1 includes:

[0023] S11. Generate an initial population of size NP by using the Latin hypercube sampling method, conduct a true evaluation on it, and store the target value and constraint value in the database;

[0024] S12. Calculate the true objective function value and true constraint function value of the population Pop, and store (Pop, PF, PG) in the database;

[0025] S13. storing the feasible solutions and infeasible solutions in the database in a feasible solution set and an infeasible solution set respectively;

[0026] S14. Set a flag bit according to whether there is a feasible solution in the database, the flag bit is represented by flag, and update the number of real evaluations, represented by FE, FE=NP, the iteration cycle is represented by Cycle, and Cycle=1.

[0027] Preferably, the setting of the flag according to the database is specifically: if there is a feasible solution in the database, setting flag=1; otherwise, setting flag=0.

[0028] Preferably, the S4 includes:

[0029] S41. If flag=0, construct / update auxiliary problem 1, and the constraints of the main problem are constructed as auxiliary problem 1;

[0030] S42, use NCDE to perform T-generation evolution on the main problem and auxiliary problem 1 respectively;

[0031] S43. Replace the individual with the largest constraint violation in the main problem with the individual with the smallest objective value in auxiliary problem 1;

[0032] S44. If flag=1, construct / update auxiliary problem 2, and construct auxiliary problem 2 into a two-objective optimization problem;

[0033] S45, evolve the main problem and auxiliary problem 2 for T generations based on NCDE and NSGA-II respectively;

[0034] S46. Transfer the feasible solutions in the main problem to the auxiliary problem 2, and transfer the non-dominated solutions in the auxiliary problem 2 to the main problem;

[0035] S47. Prune the population sizes of the main problem and auxiliary problem 2 to ensure that the population sizes are NP.

[0036] Preferably, the auxiliary question 1 in S41 is expressed by the following formula:

[0037]

[0038] Where, CV i (X)=max(0,g i (X)) is the constraint violation degree of the i-th constraint, is the weight of the i-th constraint.

[0039] Preferably, the auxiliary question 2 in S44 is expressed by the following formula:

[0040]

[0041] Where, To assist the first goal of Problem 2, To assist Problem 2, the second objective is the diversity index of individual X, NP is the population size, and Div(X) represents the average distance between individual X and all other individuals in the population.

[0042] Preferably, the S5 includes:

[0043] S51, when flag=0, sampling standard I is used to select samples; when flag=1, sampling standard II is used to select samples;

[0044] S52, conduct a true evaluation of the representative solution set;

[0045] S53. Update the database, feasible solution set, and infeasible solution set.

[0046] Preferably, the sampling criteria I and II in S51 are respectively:

[0047] Sampling Criterion I: The individuals in the master problem perform non-dominated sorting on the master problem according to the two objectives provided by the following formula, and select Pareto non-dominated solutions as the representative solution set:

[0048] maxPoF(X);

[0049] maxDiv(X);

[0050] Among them, PoF(X) is the feasibility probability and Div(X) is the diversity index.

[0051] Sampling Criterion II: The individuals in the master problem perform non-dominated sorting on the master problem according to the two objectives provided by the following formula, and select Pareto non-dominated solutions as the representative solution set:

[0052] maxCEI(X);

[0053] maxDiv(X);

[0054] Among them, CEI(X) is the constrained expected improvement and Div(X) is the diversity index.

[0055] Preferably, the S6 includes:

[0056] S61. When flag = 0, select M individuals with the smallest constraint violation from IFS; perform APC on these selected individuals; select the central individual from each cluster and obtain the solution XP;

[0057] S62, when flag = 1, perform affinity propagation clustering on the feasible solution set; select the best individual from each cluster and obtain the solution set, denoted by XP;

[0058] S63, executing a loop for each solution in XP, with the loop variable i ranging from 1 to |XP|, where |XP| is the number of solutions in the solution set XP;

[0059] S64 vs XP i Conduct a realistic assessment.

[0060] Preferably, the loop operation in S63 is:

[0061] When i=1, execute the following loop:

[0062] Build XP i Local RBFN;

[0063] XPi Adopting sequential quadratic programming SQP;

[0064] Determine whether i is |XP|. If i is not equal to |XP|, then i=i+1 and continue to execute the loop. If i=|XP|, stop the loop.

[0065] Therefore, the present invention adopts the above-mentioned collaborative evolution method for solving expensive constrained multimodal problems, which has the following beneficial effects:

[0066] (1) A two-stage main-auxiliary problem collaborative optimization framework was constructed. In the first stage, constraint-related auxiliary problems were constructed to help the main problem quickly locate the feasible region. In the second stage, dual-objective auxiliary problems were constructed to help the main problem find optimal solutions with good diversity. During the evolutionary process, the main-auxiliary problem can quickly track the optimal solutions in multiple modes of the main problem through bidirectional information migration.

[0067] (2) A two-stage agent model management strategy guided by multiple indicators is designed. When there is no feasible solution, the filling samples are screened based on the feasibility probability index and the diversity index in order to obtain high-feasibility samples on multiple modes. When there is a feasible solution, the filling samples are screened based on the constrained expected gain index and the diversity index in order to obtain high-quality samples on multiple modes. This strategy weighs the feasibility, pros and cons, and diversity of the solution.

[0068] (3) A two-stage elite solution local search strategy is proposed; based on the existence of feasible solutions in the current population, two different local search schemes are implemented, in order to discover feasible solutions when no feasible solutions exist, and to quickly converge to multiple optimal solutions when feasible solutions exist.

[0069] The technical solution of the present invention is further described in detail below through the accompanying drawings and embodiments. BRIEF DESCRIPTION OF THE DRAWINGS

[0070] Figure 1 This is a flow chart of a collaborative evolutionary method for solving expensive constrained multimodal problems according to an embodiment of the present invention;

[0071] Figure 2 Schematic diagram of the information migration process of the main question and auxiliary question according to an embodiment of the present invention;

[0072] Figure 3 The running time of the proposed method and OCSPSO on G and M problems;

[0073] Figure 4 Comparison of the optimized rotor airfoil geometry and pressure coefficient obtained by the optimization algorithm. DETAILED DESCRIPTION

[0074] In order to make the purpose, technical solutions and advantages of the embodiments of the present invention clearer, the technical solutions in the embodiments of the present invention will be clearly and completely described below in conjunction with the drawings in the embodiments of the present invention. Obviously, the described embodiments are part of the embodiments of the present invention, not all of the embodiments. The components of the embodiments of the present invention generally described and shown in the drawings herein can be arranged and designed in various different configurations. Therefore, the following detailed description of the embodiments of the present invention provided in the drawings is not intended to limit the scope of the claimed invention, but merely represents selected embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0075] It should be noted that similar reference numerals and letters denote similar items in the following drawings, and therefore, once an item is defined in one drawing, it does not need to be further defined or explained in subsequent drawings.

[0076] Example

[0077] like Figure 1 As shown, the present invention provides a collaborative evolutionary method for solving expensive constrained multimodal problems, comprising the following steps:

[0078] S1, input objective function f, constraint function {g1,…,g m}, the maximum number of true evaluation particles TE max , then enter the initialization phase and update the true evaluation times FE.

[0079] S11. Generate an initial population Pop of size NP using the Latin hypercube sampling method, perform a true evaluation on it, and store the target value and constraint value in the database DB. NP is the population size, and the main population (Pop A) = the auxiliary population (Pop A) = Pop.

[0080] S12. Calculate the true objective function value PF and the true constraint function value PG of the population Pop, and store (Pop, PF, PG) in DB.

[0081] S13. Store the feasible solutions and infeasible solutions in the DB in the feasible solution set FS and the infeasible solution set IFS respectively.

[0082] S14. Set flag according to DB; update the number of real evaluations FE, FE=NP, iteration cycle Cycle=1; set flag according to whether there is a feasible solution in DB: if there is a feasible solution in DB, set flag=1; otherwise, set flag=0.

[0083] S2, when FE <TE max, enter the iteration cycle. Execute steps S3 to S6 cyclically until the termination condition is met.

[0084] S3. Use the samples in the DB to build / update the global surrogate model of the target and each constraint. The global surrogate model uses the Kriging model, which is directly used by the model provided by the SURROGATES software package.

[0085] S4. Design a two-stage auxiliary problem construction mechanism and a main-auxiliary problem co-evolution strategy (CCTMA) to simultaneously locate multiple optimal solutions after quickly finding a feasible solution.

[0086] Two-stage auxiliary question construction mechanism:

[0087] Constructing constraint-related auxiliary problems helps the main problem quickly locate feasible regions, aiming to quickly find feasible solutions in the first phase. When solving ECMMOPs, it is desirable that the constructed auxiliary problems first focus on helping the main problem find feasible solutions, and then shift the focus to maintaining solution diversity. Therefore, in the two-stage main-auxiliary problem construction mechanism, the type of auxiliary problem constructed is adaptively determined based on the state of a flag.

[0088] Based on the flag value, the auxiliary problem is constructed or updated. Before evolution begins, if flag = 1, auxiliary problem 2 is constructed directly; otherwise, auxiliary problem 1 is constructed first. Once the flag value changes from 0 to 1, the auxiliary problem is updated to auxiliary problem 2. The main problem is the original optimization problem, with population A corresponding to the main problem and population B corresponding to the auxiliary problem. When the problem changes from main and auxiliary problem 1 to main and auxiliary problem 2, population B needs to be reinitialized.

[0089] In the first stage, it is stated that there is no feasible solution in the database, and the design of the auxiliary problem should focus on finding a feasible solution.

[0090] In the second stage, since feasible solutions are already available in the DB, the design of auxiliary problems focuses on helping the main problem locate the optimal solution in multiple modes.

[0091] This phased approach allows for more efficient use of computing resources, ensuring that feasible solutions are found while gradually approaching multiple optimal solutions.

[0092] Coevolutionary strategies for primary and secondary problems (including coevolution I and coevolution II):

[0093] Collaboratively optimize the primary and auxiliary problems and exchange information. Pop A and Pop B are optimized simultaneously and exchange information between them, evolving continuously for T generations.

[0094] Constructing a dual-objective auxiliary problem helps the main problem find a diverse optimization solution, that is, simultaneously locating multiple high-quality optimization solutions by enhancing the diversity of solutions.

[0095] The collaborative solution of the proposed primary and secondary problems, using NCDE and NSGA-II, aims to quickly locate multiple optimal solutions to the problem. Given that auxiliary problem 1 is inherently multimodal, both the primary and secondary problems are solved using NCDE to balance convergence and diversity in the population; auxiliary problem 2 is solved using NSGA-II.

[0096] Coevolution I: Both the main problem and auxiliary problem 1 evolve for T generations with the help of agent models, followed by information transfer. The specific transfer strategy is to select individuals with the lowest objective value (i.e., the lowest constraint violation) in Pop B for migration, as these individuals are more likely to be feasible solutions. Subsequently, the same number of individuals with high constraint violations in Pop A are replaced. This exchange increases the probability of Pop A finding a feasible solution.

[0097] Coevolution II: Similar to the first stage, the main problem and auxiliary problem 2 also evolve for T generations with the help of the surrogate model, and then information transfer is carried out. This time, information transfer is bidirectional. The specific transfer strategy is as follows: Pop B performs non-dominated sorting and selects the Pareto optimal solution set as migration individuals to enhance the diversity of Pop A; Pop A uses the feasible solutions predicted by the surrogate model as migration individuals, prompting Pop B to find diverse solutions within the feasible region. Then, individuals from Pop A and Pop B migrate to each other. After this exchange, the size of both populations will exceed NP, so pruning is required. For Pop A, individuals are sorted according to the feasibility rule, and lower-ranked individuals are removed until the population size returns to NP. For Pop B, all individuals are ranked according to non-dominated sorting and crowding distance, and individuals exceeding the NP size are removed.

[0098] In this embodiment, step S4 includes:

[0099] S41. If flag = 0, construct / update auxiliary problem 1. At this time, there is no feasible solution in the database, and the design of the auxiliary problem should focus on finding a feasible solution. Therefore, the constraints of the main problem are constructed as auxiliary problem 1, as shown in the following formula:

[0100]

[0101] Where, CV i (X)=max(0,g i (X)) is the constraint violation degree of the i-th constraint; is the weight of the i-th constraint. This formula is calculated based on the constraint proxy model built for the main problem.

[0102] Obviously, f 辅1 (X) focuses on minimizing constraint violations in the original problem and can achieve multiple minima around zero, demonstrating that auxiliary problem 1 is multimodal in nature. Once a solution is found that predicts a zero target value, it is passed to the main problem for the actual constraint violation evaluation. When a feasible solution is found, the first phase of the search stops and enters the second phase.

[0103] S42. Use NCDE to perform T-generation evolution on the main problem (Pop A) and the auxiliary problem (Pop B). The auxiliary problem is auxiliary problem 1.

[0104] S43. Replace the individual with the largest constraint violation (CV) in Pop A with the individual with the smallest target value in Pop B.

[0105] S44. If flag = 1, construct / update auxiliary problem 2. Since feasible solutions are already available in the DB, the design of the auxiliary problem focuses on helping the main problem locate the optimal solution in multiple modes. Therefore, auxiliary problem 2 is constructed as a two-objective optimization problem, as shown in the following formula:

[0106]

[0107] Where, is the first objective of auxiliary problem 2, i.e. the predicted value of the objective function of the main problem; To assist Problem 2, the second objective is the diversity index for individual X, where NP is the population size. Div(X) represents the average distance between individual X and all other individuals in the population. A higher Div(X) value indicates a greater distance between individuals X and the population, indicating greater diversity. Conversely, a lower Div(X) value indicates a denser population of individuals in the area surrounding X. This diversity index, Div(X), is directly calculated based on the real-time spatial distances between individuals in the population, eliminating the need for a separate proxy model.

[0108] S45. Evolve the main problem (Pop A) and the auxiliary problem (Pop B) for T generations based on NCDE and NSGA-II, respectively. The auxiliary problem is now Auxiliary Problem 2.

[0109] S46. Migrate the feasible solutions in Pop A to Pop B, and migrate the non-dominated solutions in Pop B to Pop A. Pop A is used to evolve the main problem, and Pop B is used to evolve the auxiliary problem, such as Figure 2 shown.

[0110] S47. Prune the population size of Pop A and Pop B to ensure that the size is NP.

[0111] The present invention couples NCDE and NSGA-II to construct a master-slave co-evolution framework, aiming to quickly locate multiple feasible optimal solutions within a limited computational budget.

[0112] Neighborhood-based Crowding Differential Evolution (NCDE) introduces an adaptive neighborhood mechanism into the classic differential evolution "mutation-crossover-selection" process to achieve multimodal search. The algorithm dynamically partitions the population into n local ecological niches based on Euclidean distance, with each individual competing only with its nearest neighbors. This allows for the simultaneous preservation of multiple dominant peaks within a single run. Experiments have shown that NCDE can simultaneously discover and maintain multiple global and local optima, almost independently of population size, making it particularly well-suited for the second phase of the present invention, which involves rapidly tracking multimodal solutions within the feasible domain.

[0113] NSGA-II, a fast multi-objective evolutionary algorithm with elite preservation, is used to solve the dual-objective auxiliary problem. Its core steps include:

[0114] 1. Fast non-dominated sorting: Divide the merged population into Pareto frontiers F1, F2, ...F n and assign a rank to the individual;

[0115] 2. Crowding distance calculation: estimate the individual crowding degree based on the target projection interval within the same front;

[0116] 3. Double-Tier Tournament Selection: Parents are selected based on the criteria of "rank first, high congestion first" and offspring are generated through multi-point crossover and polynomial mutation;

[0117] 4. Elite merger: After the parent and child generations merge, the above process is repeated, and only the top *N* individuals are retained to enter the next generation.

[0118] With the help of the above mechanism, NSGA-II can obtain a uniformly covered Pareto optimal set in one evolution, achieve a balance between convergence and diversity with very few parameters, and provide high-quality and highly complementary non-dominated solutions to the main problem.

[0119] S5. Execute a two-stage surrogate model management strategy guided by multiple indicators. A set of samples is screened from Pop A and evaluated truthfully. The screened samples are then evaluated truthfully and the DB, FS, and IFS are updated. This step is used to select a small number of representative individuals and conduct truthful evaluations on them, thereby populating new samples for the surrogate model update.

[0120] The role of proxy model management is to select representative samples (RepSet) from the current population for real-world evaluation. These samples are used to update the database and the corresponding proxy models to improve the accuracy of the proxy models in key areas. Sample selection is divided into two cases, which are distinguished by flags.

[0121] In this embodiment, step S5 includes:

[0122] S51. When flag = 0, sampling standard I is adopted, focusing on finding samples with high feasibility; when flag = 1, sampling standard II is adopted, giving priority to high-quality and diverse samples.

[0123] Sampling Criterion I: Since there are no feasible solutions in the database, the criterion balances feasibility and diversity when selecting samples. Specifically, individuals in Pop A perform non-dominated sorting on Pop A according to the two objectives provided by the following formula and select Pareto non-dominated solutions as the representative solution set RepSet.

[0124] maxPoF(X);

[0125] maxDiv(X);

[0126] Among them, PoE(X) is the feasibility probability; Div(X) is the diversity index.

[0127] Sampling Criterion II: When feasible solutions are already available in the database, this criterion balances performance, feasibility, and diversity when selecting samples. Specifically, individuals in Pop A perform a non-dominated sort on Pop A according to the two objectives provided by the following formula and select Pareto non-dominated solutions as the representative solution set RepSet.

[0128] maxCEI(X);

[0129] maxDiv(X);

[0130] Among them, CEI(X) is the constrained expected improvement, which takes into account the expected improvement and feasibility of individuals; Div(X) is the diversity index.

[0131] S52. Perform a true evaluation on the representative solution set RepSe.

[0132] S53. Update DB, FS, and IFS.

[0133] To enhance the algorithm's local exploitation capabilities when dealing with expensive constrained optimization problems, a two-stage elitist local search strategy (TEL) is proposed. This strategy aims to deeply mine promising solutions and improve the algorithm's overall performance. Specifically, a local search is performed on the FS or IFS based on flags to find feasible solutions or improve the accuracy of multiple optimal solutions. This local search employs Sequential Quadratic Programming (SQP) to locally search for solutions at a specified location.

[0134] The core ideas of the two-stage elite solution local search strategy include: (1) selecting solutions from different sets for clustering and local search based on whether there are feasible solutions, thereby reducing unnecessary development of invalid solutions; (2) conducting local searches based on different optimization objectives in order to meet different requirements at different stages, regardless of whether there are feasible solutions.

[0135] Phase 1: When the FS is empty, the local search first aims to identify feasible solutions. Therefore, the top M individuals with the lowest constraint violations (CVs) are selected from the IFS and clustered using affinity propagation (APC). Local search is then performed at the cluster centers. It should be noted that APC is an automatic clustering method that avoids the need to set clustering parameters. The resulting cluster centers are then used for local search based on constraint violations, ensuring that the search is focused on candidate solutions that are close to feasible. This method avoids overexploration of invalid solutions, thereby saving computational resources.

[0136] Phase 2: Once the FS is non-empty, the goal of local search is to further optimize high-quality solutions. Therefore, the FS is automatically clustered using APC, and the best individual from each cluster is selected for local search. This approach automatically identifies the modes of each solution in the FS and simultaneously improves the performance of the best solution within each mode.

[0137] SQP is an efficient method for solving nonlinear constrained optimization problems. In the local search process of SQP, it is based on the local agent model. In stage 1, for each solution that requires local search, i , select from DB with XP i The closest K samples are used to construct the local RBFN model of all constraints. With the help of the local proxy model, based on formula (3), XP i Perform local search. The difference in stage 2 is that the objective and constraint surrogate models of the main problem are constructed simultaneously, and the feasibility rule is used to compare the individual performances; the rest remains unchanged. The scope of the SQP local search is [SL i , Su i ], i=1,2,…,Dwhere SLi and Su i are the minimum and maximum values ​​of the variables of the sample points selected when constructing the local proxy model of XP. After the local search is completed, the newly generated solution is truly evaluated, and DB, IFS, and FS are updated. It should be noted that the key parameter basis function of RBFN uses a Gaussian radial basis, and the number of implicit nodes is consistent with the number of training samples n, so as to maintain strict interpolation and avoid additional parameter adjustment. To improve numerical robustness, each local RBF contains at least 5D samples; when the number of samples is insufficient, the neighborhood points are automatically supplemented; at the same time, if the Euclidean distance between any two samples is less than 0.01, one of them is discarded to prevent matrix singularity.

[0138] In this embodiment, step S6 includes:

[0139] S61. When flag = 0, select M individuals with the smallest constraint violation from IFS; perform APC on these selected individuals; select the central individual from each cluster and obtain the solution XP;

[0140] S62, when flag=1, perform affinity propagation clustering on FS; select the best individual from each cluster and obtain the solution set XP;

[0141] S63. Loop for each solution in XP, with the loop variable i ranging from 1 to |XP|, that is, the number of solutions in the solution set XP.

[0142] When i=1, execute the following loop:

[0143] Build XP i Local RBFN;

[0144] XP i Adopting sequential quadratic programming SQP;

[0145] Determine whether i is |XP|. If i is not equal to |XP|, then i=i+1 and continue to execute the loop. If i=|XP|, stop the loop.

[0146] S64 vs XP i Conduct a realistic assessment.

[0147] S7. Update DB, FS, IFS and FE, Cycle = Cycle + 1.

[0148] S8. Terminate the iteration and output the FS and the optimal solution set OS. Specifically, after the entire iteration is completed, output the feasible solution set FS, and filter the optimal solution set OS from the FS.

[0149] In order to verify the effectiveness of this method, the maximum number of evaluations TE maxIn the experimental environment (2.6GHz IntelCore i7 CPU), it is compared with the typical expensive constrained multimodal optimization algorithm OCSPSO, and the performance is evaluated by running it independently 30 times on 13 G problems and 8 M problems.

[0150] Six performance indicators are used to evaluate the performance of the algorithm, including:

[0151] 1) Global optimal solution (GS).

[0152] 2) Feasibility Rate (FR): The ratio of the number of runs in which the algorithm finds a feasible optimal solution to the total number of runs.

[0153] 3) Success rate (SR): The ratio of the number of runs that successfully find a solution with an error less than 0.1 from the true global optimal solution to the total number of runs.

[0154] 4) First Feasible Evaluation (FFE): The actual number of evaluations required to find the first feasible solution.

[0155] 5) Number of successful evaluations (SFE): The actual number of evaluations consumed when finding the first solution whose error with the true global optimal solution is less than 0.1.

[0156] 6) Effective valley ratio (VR), the ratio of the number of valleys detected by the algorithm to the total number of valleys.

[0157]

[0158] Among them, NV is the total number of valley points of the problem; NFV i It represents the number of valley points found in the i-th run, and NR is the number of runs of the algorithm.

[0159] Indicators (GS, FR, SR) can be used to measure the quality of the algorithm in obtaining the optimal solution, indicators (FFE, SFE) can reflect the convergence speed of the algorithm, and VR indicator can be used to measure the ability of the algorithm to handle constrained MMOP.

[0160] Comparison results between this method and expensive constrained multimodal SAEAs (OCSPSO):

[0161] This method is compared with OCSPSO on 21 problems. Table 1 shows the GS mean [standard deviation], FR, and SR obtained by this method and OCSPSO on G problems, as follows:

[0162] Table 1 Comparison results between this method and OCSPSO on the G problem

[0163]

[0164]

[0165] As can be seen from Table 1, for the 13 G problems: (1) Regarding the GS index, the proposed method outperforms OCSPSO on 6 G problems and is inferior to the comparison algorithm on 4 G problems; (2) Regarding the FR index, the proposed method can find a feasible solution every time, and the running ratio of OCSPSO finding a feasible solution on the G06 problem is 87%; (3) Regarding the SR index, the proposed method outperforms the comparison algorithm OCSPSO on the G01, G04 and G07 problems respectively, but is inferior to OCSPSO on the G06, G09, G10 and G18 problems.

[0166] Table 2 shows the convergence index (FFE and SFE) of this method and OCSPSO on the G-type problem, as follows:

[0167] Table 2 Convergence comparison between this method and OCSPSO on G-type problems

[0168]

[0169] Table 2 shows the convergence index values ​​(i.e., FFE and SFE values) obtained by this method and OCSPSO on the G-type problem, as follows:

[0170] Table 2 shows that: (1) the number of true evaluations (FFE) required for the proposed method to find the first feasible solution is more than that of OCSPSO for G02, G12 and G24 problems, but less than that of OCSPSO for the other 10 problems; (2) except for G08, G12 and G24 problems, the number of true evaluations (SFE) required for the proposed method to find the first optimal solution is less than that of OCSPSO; (3) the P-value is 0.0037, indicating that the proposed method is significantly better than OCSPSO in terms of convergence performance.

[0171] Reference Figure 3 ,The running time of this method on 8 G-type problems is less than OCSPSO, especially on problem G10, the running time is significantly less than OCSPSO.

[0172] The above analysis shows that because OCSPSO uses a partial evaluation strategy, it increases the number of algorithm iterations and thus takes longer to run when dealing with optimization problems with more constraints. In summary, when dealing with G problems with a unique global optimal solution, compared to the existing OCSPSO, this method, which introduces a strategy of finding feasible solutions first, can more quickly obtain a highly competitive global optimal solution.

[0173] Table 3 shows the GS mean [standard deviation] and VR obtained by this method and OCSPSO on the M problem, as follows:

[0174] Table 3 Comparison results between this method and OCSPSO on the M problem

[0175]

[0176] In Tables 1, 2, and 3, except for the “good / average / poor” results, the P-value is adjusted using the Finner operator to show the level of significance between the present method and the comparison algorithm, with the present method serving as the control method.

[0177] As can be seen from Table 3, for the M problem: (1) For the GS index, the proposed method is superior to OCSPSO in M02, M06, and M07 problems, inferior to OCSPSO in M03 optimization problem, and there is no significant difference between the two in the remaining 4 problems; (2) For the VR index, the proposed method is superior to OCSPSO in M02 and M03 problems, and there is no significant difference between the two in the remaining 6 problems. In summary, when dealing with constrained multimodal problems with multiple global optimal solutions, the quality and quantity of the optimal solutions obtained by the proposed method are better than those of OCSPSO.

[0178] Although the P-value is 0.6033, indicating that the performance difference between this method and the OCSPSO algorithm on problems M01 to M08 is not statistically significant, Figure 3 It can be observed that the running time of our method on the six M-type problems is faster than that of OCSPSO. This indicates that when obtaining optimal solutions of similar quality, our method can find multiple optimal solutions faster.

[0179] Compressor wing optimization experiment:

[0180] This experiment tested the performance of this method on a practical problem: the aerodynamic optimization of a transonic rotor airfoil. The goal is to minimize the aerodynamic loss coefficient while satisfying constraints such as the total pressure ratio. The optimization process is based on computational fluid dynamics (CFD) simulations, which are computationally expensive. Different optimal solutions may produce similar or identical aerodynamic loss coefficient values. These characteristics make rotor airfoil optimization a representative example of ECMMOP.

[0181] In this experiment, a common transonic compressor airfoil was selected as the optimization target. The basic geometry and boundary conditions of the rotor airfoil include an inlet Mach number of 1.2, an inlet flow angle of 50°, and a pitch-chord ratio of 0.5. The optimization model is described as follows:

[0182] minη=loss;

[0183] stpre≥1.688;

[0184]

[0185] Where η is the loss coefficient of the two-dimensional wing, pre represents the total pressure ratio of the two-dimensional wing, and x i refers to the decision variables, where and They refer to the upper and lower bounds of the decision variables, respectively.

[0186] The wing geometric parameters are defined using a hierarchical approach based on elliptical topological deformation. This approach involves 22 control variables, which are divided into three levels, as shown in Table 4:

[0187] Table 4: Layered design parameters and their functions

[0188]

[0189] In this experiment, OCSPSO and MPMLS were selected as comparison algorithms. Each algorithm was independently run 30 times to calculate its statistical results. For all algorithms, the maximum number of flow field calculations (i.e., CFD calls) was 300.

[0190] Table 5 lists the results of the proposed method and three comparison methods for rotor airfoil aerodynamic calculations. It can be seen that: (1) In terms of average values, the proposed method has the lowest average value, followed by OCSPSO. (2) In terms of maximum and minimum values, the proposed method has the smallest maximum value, and the OCSPSO has the smallest minimum value. (3) Among all four methods, the proposed method has the highest ranking value, rank = 1.33.

[0191] Table 5: Comparison results of this method and the comparison algorithm on the rotor wing surface optimization example

[0192]

[0193] from Figure 4 (a) While most algorithms generate similar wing convex shapes, significant differences exist in the leading edge region. The wings generated by our method and OCSPSO exhibit smooth leading edge profiles, indicating excellent control over the local airflow acceleration near the nose. In contrast, the wing generated by MPMLS exhibits a more abrupt leading edge, which may lead to uncontrolled local airflow acceleration, exacerbating the initial flow acceleration and inducing a stronger shock wave.

[0194] from Figure 4 (b) It can be seen that the rotor wing designed according to this method can delay the onset of the shock wave farther downstream and reduce its intensity, thereby reducing the separation zone and lowering aerodynamic losses. Conversely, the wing generated by MPMLS exhibits stronger shock waves or more upstream shock locations, resulting in a larger separation zone and greater drag losses. Therefore, under certain conditions, this method can more effectively improve the aerodynamic performance of the wing.

[0195] Compared with OCSPSO, the main advantages or improvements of this method are as follows:

[0196] 1) OCSPSO does not design a strategy of “finding feasible solutions first and then finding the optimal solution”, while STCDE divides the optimization process into two stages: the first stage quickly locates the feasible region, and the second stage finds optimal solutions with good diversity.

[0197] 2) The criterion for screening samples in OCSPSO is single, while this method uses multiple indicators to guide sample screening, which can effectively balance the feasibility, quality and diversity of samples.

[0198] Therefore, the present invention adopts the above-mentioned collaborative evolutionary method for solving expensive constrained multimodal problems. The main-auxiliary problem can quickly track the optimal solution in multiple modes of the main problem through bidirectional information migration.

[0199] Finally, it should be noted that the above embodiments are only used to illustrate the technical solutions of the present invention rather than to limit the same. Although the present invention has been described in detail with reference to the preferred embodiments, those skilled in the art should understand that they can still modify or replace the technical solutions of the present invention with equivalents, and these modifications or equivalent replacements cannot cause the modified technical solutions to deviate from the spirit and scope of the technical solutions of the present invention.

Claims

1. A co-evolutionary method for solving expensive constrained multimodal problems, characterized by: The following steps are involved: S1, input objective function f, constraint function {g1,…,g m } and the maximum number of true evaluation particles TE max , enter the initialization phase and update the real evaluation times FE; S2, when FE <TE max , enter the iteration cycle; S3, using samples in the database to build / update the global surrogate model of the target and each constraint; S4. Design a two-stage auxiliary problem construction mechanism and a main-auxiliary problem co-evolution strategy; S5. Execute a two-stage agent model management strategy guided by multiple indicators, specifically: select a set of samples from the main problem and conduct real evaluation, conduct real evaluation on the selected samples, and update the database, feasible solution set, and infeasible solution set; S6. A two-stage elite solution local search strategy is proposed; S7, update the database, feasible solution set, infeasible solution set and the number of true evaluations, Cycle = Cycle + 1; S8. Terminate the iteration and output the feasible solution set and the optimal solution set. Specifically, after the entire iteration is completed, output the feasible solution set FS, and filter the optimal solution set OS from FS.

2. The collaborative evolutionary method for solving expensive constrained multimodal problems according to claim 1, characterized in that: Said S1 comprises: S11. Generate an initial population Pop of size NP by Latin hypercube sampling method, conduct real evaluation on it, and store the target value and constraint value in the database; S12. Calculate the true objective function value and true constraint function value of the population Pop, and store (Pop, PF, PG) in the database; S13. storing the feasible solutions and infeasible solutions in the database in a feasible solution set and an infeasible solution set respectively; S14. Set a flag bit according to whether there is a feasible solution in the database, the flag bit is represented by flag, and update the number of real evaluations, represented by FE, FE=NP, the iteration cycle is represented by Cycle, and Cycle=1.

3. The collaborative evolutionary method for solving expensive constrained multimodal problems according to claim 2, characterized in that: The flag bit is set according to the database as follows: if there is a feasible solution in the database, flag=1 is set; otherwise, flag=0 is set.

4. The collaborative evolutionary method for solving expensive constrained multimodal problems according to claim 3, characterized in that: The S4 includes: S41. If flag=0, construct / update auxiliary problem 1, and the constraints of the main problem are constructed as auxiliary problem 1; S42, use NCDE to perform T-generation evolution on the main problem and auxiliary problem 1 respectively; S43. Replace the individual with the largest constraint violation in the main problem with the individual with the smallest objective value in the auxiliary problem; S44. If flag=1, construct / update auxiliary problem 2, and construct auxiliary problem 2 into a two-objective optimization problem; S45, evolve the main problem and auxiliary problem 2 for T generations based on NCDE and NSGA-II respectively; S46. Transfer the feasible solutions in the main problem to the auxiliary problem 2, and transfer the non-dominated solutions in the auxiliary problem 2 to the main problem; S47. Prune the population sizes of the main problem and auxiliary problem 2 to ensure that the population sizes are NP.

5. The collaborative evolutionary method for solving expensive constrained multimodal problems according to claim 4, characterized in that: The auxiliary problem 1 in S41 is expressed as follows: Where, CV i (X)=max(0,g i (X)) is the constraint violation degree of the i-th constraint, is the weight of the i-th constraint.

6. The collaborative evolutionary method for solving expensive constrained multimodal problems according to claim 5, characterized in that: The auxiliary problem 2 in S44 is expressed as follows: Where, To assist the first goal of Problem 2, To assist Problem 2, the second objective is the diversity index of individual X, NP is the population size, and Div(X) represents the average distance between individual X and all other individuals in the population.

7. The collaborative evolutionary method for solving expensive constrained multimodal problems according to claim 6, characterized in that: The S5 includes: S51, when flag=0, sampling standard I is used to select samples; when flag=1, sampling standard II is used to select samples; S52, conduct a true evaluation of the representative solution set; S53. Update the database, feasible solution set, and infeasible solution set.

8. The collaborative evolutionary method for solving expensive constrained multimodal problems according to claim 7, characterized in that: The sampling standards I and II in S51 are respectively: Sampling Criterion I: The individuals in the master problem perform non-dominated sorting on the master problem according to the two objectives provided by the following formula, and select Pareto non-dominated solutions as the representative solution set: maxPoF(X); maxDiv(X); Among them, PoF(X) is the feasibility probability and Div(X) is the diversity index. Sampling Criterion II: The individuals in the master problem perform non-dominated sorting on the master problem according to the two objectives provided by the following formula, and select Pareto non-dominated solutions as the representative solution set: maxCEI(X); maxDiv(X); Where CEI(x) is the constrained expected improvement and Div(X) is the diversity index.

9. The collaborative evolutionary method for solving expensive constrained multimodal problems according to claim 8, characterized in that: The S6 includes: S61. When flag = 0, select M individuals with the smallest constraint violation from IFS; perform APC on these selected individuals; select the central individual from each cluster and obtain the solution XP; S62, when flag = 1, perform affinity propagation clustering on the feasible solution set; select the best individual from each cluster and obtain the solution set, denoted by XP; S63, executing a loop for each solution in XP, with the loop variable i ranging from 1 to |XP|, where |XP| is the number of solutions in the solution set XP; S64 vs XP i Conduct a realistic assessment.

10. The collaborative evolutionary method for solving expensive constrained multimodal problems according to claim 9, characterized in that: The loop operation in S63 is: When i=1, execute the following loop: Build XP i Local RBFN; XP i Adopting sequential quadratic programming SQP; Determine whether i is |XP|. If i is not equal to |XP|, then i=i+1 and continue to execute the loop. If i=|XP|, stop the loop.

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