Method and device for solving an interval-constrained multi-objective optimization problem

Abstract

The application relates to a method and device for solving an interval constraint multi-objective optimization problem, which is used for optimizing a composite material laminated structure. The method comprises the following steps: updating a first population according to a dynamic interval constraint dominance relation and screening out a first sub-generation; updating a second population according to an interval Pareto dominance relation and screening out a second sub-generation; updating a third population according to the dynamic interval constraint dominance relation and the interval Pareto dominance relation and screening out a third sub-generation; merging the first sub-generation, the second sub-generation and the third sub-generation to obtain a sub-generation population in the next iteration; and obtaining a final solution set of the multi-objective optimization problem in response to reaching a maximum iteration number. The method provided by the application directly solves the interval constraint multi-objective optimization problem, completely retains valuable uncertain information, retains high-quality feasible solutions and infeasible solutions with good convergence, and a three-population collaborative evolution framework balances the feasibility, convergence and diversity of the final solution.

Description

Technical Field

[0001] This invention belongs to the field of material laminate structure and multi-objective solution technology, specifically relating to a method and apparatus for solving interval-constrained multi-objective optimization problems. Background Technology

[0002] Composite laminate structures are widely used in aerospace and other fields due to their excellent specific stiffness and specific strength. The core of their optimization design is to achieve the overall optimization of multiple indicators. However, composite laminate structures contain multiple layers of different types of materials. The strength of such structures is affected by the uncertainties of material properties and geometric dimensions. At the same time, parameters such as Young's modulus and density of each layer of the composite material fluctuate within a range due to factors such as preparation and measurement. Ignoring this uncertainty will lead to a large deviation between the optimization results and the actual situation. In addition, there are coupling constraints between multiple indicators, making it difficult to take into account both multi-objective optimization and preset constraints. This makes it impossible to fully adapt to the anisotropic characteristics of laminate structures, resulting in insufficient reliability and specificity of structural solutions.

[0003] Constrained multi-objective optimization problems (CMOPs) are ubiquitous in many real-world fields, including engineering design, resource scheduling, and path planning, and can be used to solve optimization problems in composite laminate structures. However, due to factors such as measurement errors and randomness, many real-world problems exhibit inherent uncertainty, typically manifested as parameter uncertainty. Through uncertainty propagation, these parameters affect the evaluation of objective functions and constraints, significantly increasing the difficulty and complexity of solving optimization problems. Currently, uncertain parameters are generally classified into three categories: random numbers, fuzzy numbers, and interval numbers. Among these, compared to the probability distribution of random numbers and the membership function of fuzzy numbers, interval numbers intuitively represent the possible range of parameter values ​​in the form of upper and lower bounds, which has a significant advantage when historical data is insufficient. Furthermore, random numbers and fuzzy numbers can be converted into intervals using confidence levels and cutoffs, respectively. There is increasing attention in related technologies to constrained multi-objective optimization problems with interval parameters. Among these, CMOPs with interval parameters are called interval CMOPs (ICMOPs), where both objective functions and constraints are represented in interval form. Solving ICMOPs requires optimizing multiple interval-valued objective functions while satisfying a set of interval constraints.

[0004] Interval-constrained multi-objective optimization problems involve optimizing multiple conflicting objectives within interval constraints, where the propagation of uncertainty between objectives and constraints presents significant challenges. However, existing solution techniques cannot be directly applied to interval-constrained multi-objective optimization problems, and indirect methods based on transformation models inevitably lead to the loss of uncertainty information. Summary of the Invention

[0005] To address the aforementioned technical problems, this invention provides a method and apparatus for solving interval-constrained multi-objective optimization problems.

[0006] This invention includes a method for solving a multi-objective optimization problem with interval constraints, used to optimize composite laminate structures. The method includes: obtaining the optimization objective of the laminate structure, interval constraints of the laminate structure, and uncertain material parameters; wherein the optimization objective includes minimizing the total mass and maximizing the vibration frequency of the composite laminate structure; the interval constraints include ensuring that the vibration frequency interval of the composite laminate structure meets a preset frequency constraint, and ensuring that the total mass interval of the composite laminate structure meets a preset mass constraint; the uncertain material parameters include the Young's modulus interval and density interval of each layer of the composite laminate structure; constructing a multi-objective optimization problem based on the optimization objective, the interval constraints, and the uncertain material parameters, and generating multiple initial solutions to the multi-objective optimization problem; the solutions to the multi-objective optimization problem include the thickness of each layer of the composite laminate structure; generating a first population, a second population, and a third population based on the initial solutions to the multi-objective optimization problem; and using the first population as the offspring population for the first iteration. Iteratively execute: Based on the union of the first population and the offspring population, update the first population at least according to the dynamic interval constraint dominance relationship, and select multiple solutions from the union of the first population and the offspring population as the first offspring; Based on the union of the second population and the offspring population, update the second population according to the interval Pareto dominance relationship, and select multiple solutions from the union of the second population and the offspring population as the second offspring; Based on the union of the third population and the offspring population, according to the dynamic interval constraint dominance relationship and the interval... The Pareto dominance relation updates the third population, and multiple solutions are selected from the union of the third population and the offspring population as the third offspring; the first offspring, the second offspring, and the third offspring are merged to obtain the offspring population for the next iteration; in response to reaching the maximum number of iterations, the iteration ends, and the updated first population from the last iteration is returned as the final solution set of the multi-objective optimization problem; the thickness of each layer in the composite laminate structure is determined based on the solutions in the final solution set; the dynamic interval constraint dominance relation includes: in the... In the next iteration, for any solution reconciliation If any one of conditions one, two, or three is satisfied, then the solution is... Dynamic interval constraint governing solution The calculation formulas for conditions one, two, and three above include:

[0007] in, This is a dynamic constraint threshold that decays in each iteration based on the current iteration number. ; To solve constraint violation value, To solve The constraint violation value is defined as follows: a constraint violation value less than or equal to zero indicates that the solution satisfies all of the above-mentioned lamination structure interval constraints, and a constraint violation value greater than zero indicates that the solution violates at least some of the above-mentioned lamination structure interval constraints. The constraint violation value is proportional to the degree to which the solution violates the above-mentioned lamination structure interval constraints. The symbol used to represent the Pareto dominance relation in an interval. Solution Interval Pareto Dominant Solution and indicate the solution The objective function interval value is better than the solution The objective function interval values ​​are the interval values ​​of the total mass and vibration frequency of the composite laminate structure corresponding to the solution.

[0008] Optionally, the above dynamic constraint threshold The calculation formulas include:

[0009] in, This is the initial dynamic constraint threshold. The maximum value among all the constraint violation values ​​of the above initial solutions; The maximum number of iterations mentioned above. This refers to the current iteration number. For dynamic constraint threshold The attenuation rate, .

[0010] Optionally, the formula for calculating the above constraint violation value includes:

[0011]

[0012] in, To solve constraint violation value, To solve In the The constraint violation values ​​of the interval inequality constraints are as follows: the first interval inequality constraint is to make the vibration frequency interval value of the above composite laminate structure satisfy the above frequency constraint; the second interval inequality constraint is to make the total mass interval value of the above composite laminate structure satisfy the above mass constraint. To solve In the Interval values ​​on interval inequality constraints The upper bound of the interval is... The lower bound of the interval, For interval values.

[0013] Optionally, updating the first population and selecting the first offspring, updating the second population and selecting the second offspring, and updating the third population and selecting the third offspring all include: calculating the interval crowding distance for each solution in the population, and updating the population and selecting offspring based on the interval crowding distance; the formula for calculating the interval crowding distance includes:

[0014] The multi-objective optimization problem includes multiple interval objectives. The objective space formed by these interval objectives includes multiple dominance layers, and each solution is assigned to one of these dominance layers according to its dominance relationship. To solve Left neighbor in the same dominance layer, solution To solve The right neighbor in the same dominance layer; To solve The distance between congested areas To solve and solution The degree of congestion between them To solve and solution The degree of congestion between them To solve The hypervolume in the aforementioned target space; the formula for calculating the crowding degree between the two solutions includes:

[0015] in, To solve and solution The degree of congestion between them To solve The hypervolume in the aforementioned target space; the aforementioned dominance layer includes a non-dominated front, and solutions in the aforementioned non-dominated front are not dominated by any other solutions. To solve and solution The proportion of the influence region formed in the aforementioned target space relative to the influence region formed by the entire aforementioned non-dominated frontier in the aforementioned target space. To solve and solution The non-overlapping hypervolume between solutions and solution The non-overlapping proportion of the total hypervolume.

[0016] Optionally, the formula for calculating the above-mentioned hypervolume includes:

[0017] in, To solve The hypervolume in the aforementioned target space To solve In the Interval values ​​on a given interval target The upper bound of the interval is... The lower bound of the interval, The total number of targets in the interval. For interval values.

[0018] Optionally, the formula for calculating the above-mentioned influence ratio includes:

[0019] in, To solve and solution The proportion of the influence region formed in the aforementioned target space relative to the influence region formed by the entire aforementioned non-dominated frontier in the aforementioned target space. The total number of targets in the interval. For interval value parameters, For the solution in the above non-dominated frontier, the first... The maximum upper bound of a target interval. For the solution in the above non-dominated frontier, the first... The minimum lower bound of the interval target; To solve In the Interval values ​​on a given interval target The upper bound of the interval is... This is the lower bound of the interval; To solve In the Interval values ​​on a given interval target The upper bound of the interval is... This is the lower bound of the interval.

[0020] Optionally, the formula for calculating the above non-overlapping ratio includes:

[0021]

[0022] in, To solve and solution The non-overlapping hypervolume between solutions and solution The non-overlapping proportion of the total hypervolume. To solve The hypervolume in the aforementioned target space To solve The hypervolume in the aforementioned target space To solve and solution Overlapping hypervolume between them To solve In the Interval values ​​on a given interval target To solve In the Interval values ​​on a given interval target Solution and solution In the Overlapping areas on the target intervals The total number of targets in the interval. For interval values.

[0023] Based on the same inventive concept, this invention also provides a multi-objective optimization problem solving device with interval constraints for optimizing composite laminate structures, comprising: an acquisition module for acquiring the optimization objective of the laminate structure, interval constraints of the laminate structure, and uncertain material parameters; wherein the optimization objective of the laminate structure includes minimizing the total mass and maximizing the vibration frequency of the composite laminate structure; the interval constraints of the laminate structure include satisfying a preset frequency constraint for the vibration frequency interval of the composite laminate structure and satisfying a preset mass constraint for the total mass interval of the composite laminate structure; the uncertain material parameters include the Young's modulus interval and density interval of each layer of material in the composite laminate structure; a problem construction module for constructing a multi-objective optimization problem based on the optimization objective of the laminate structure, the interval constraints of the laminate structure, and the uncertain material parameters, and generating multiple initial solutions to the multi-objective optimization problem; the solutions to the multi-objective optimization problem include the thickness of each layer of material in the composite laminate structure; a population generation module for generating a first population, a second population, and a third population based on the initial solutions to the multi-objective optimization problem; and a population update module for updating the first population... As the offspring population in the first iteration, the following iterative processes are performed: Based on the union of the first population and the offspring population, the first population is updated at least according to the dynamic interval constraint dominance relation, and multiple solutions are selected from the union of the first population and the offspring population as the first offspring; Based on the union of the second population and the offspring population, the second population is updated according to the interval Pareto dominance relation, and multiple solutions are selected from the union of the second population and the offspring population as the second offspring; Based on the union of the third population and the offspring population, the solutions are updated according to the dynamic interval constraint dominance relation and the interval Pareto dominance relation. The third population is updated using the dominance relation, and multiple solutions are selected from the union of the third population and the offspring population as the third offspring; the first offspring, the second offspring, and the third offspring are merged to obtain the offspring population for the next iteration; the output module is used to end the iteration in response to reaching the maximum number of iterations and return the updated first population as the final solution set of the multi-objective optimization problem; the output module is also used to determine the thickness of each layer of the composite laminate structure based on the solutions in the final solution set; the dynamic interval constraint dominance relation includes: in the... In the next iteration, for any solution reconciliation If any one of conditions one, two, or three is satisfied, then the solution is... Dynamic interval constraint governing solution The calculation formulas for conditions one, two, and three above include:

[0024] in, This is a dynamic constraint threshold that decays in each iteration based on the current iteration number. ; To solve constraint violation value, To solve The constraint violation value is defined as follows: a constraint violation value less than or equal to zero indicates that the solution satisfies all of the above-mentioned lamination structure interval constraints, and a constraint violation value greater than zero indicates that the solution violates at least some of the above-mentioned lamination structure interval constraints. The constraint violation value is proportional to the degree to which the solution violates the above-mentioned lamination structure interval constraints. The symbol used to represent the Pareto dominance relation in an interval. Solution Interval Pareto Dominant Solution and indicate the solution The objective function interval value is better than the solution The objective function interval values ​​are the interval values ​​of the total mass and vibration frequency of the composite laminate structure corresponding to the solution.

[0025] Based on the same inventive concept, the present invention also provides an electronic device, comprising: a processor; a memory for storing executable instructions; wherein the processor is configured to read the executable instructions from the memory and execute the executable instructions to implement any of the methods described above.

[0026] Based on the same inventive concept, the present invention also provides a computer-readable storage medium having a computer program stored thereon, wherein when the computer program is executed by a processor, the processor performs any of the above-described methods.

[0027] The beneficial effects of this invention are as follows: The method provided by this invention directly solves the interval-constrained multi-objective optimization problem in the optimization of composite laminate structures, fully preserving valuable uncertain information. It fully considers the interval uncertainties of Young's modulus and density of each layer, avoiding optimization deviations caused by ignoring parameter fluctuations and improving the reliability of the optimization results. Simultaneously, this invention provides a dynamic interval constraint dominance relationship, which incorporates dynamically shrinking interval constraint boundaries. Under interval uncertainty, it can simultaneously preserve high-quality feasible solutions and infeasible solutions exhibiting good convergence within the dynamic interval constraint boundaries, thus providing more valuable candidate solutions during the evolution process. This invention also provides a three-group co-evolutionary framework, maximizing the preservation of excellent feasible solutions with good diversity and low uncertainty, achieving a good balance between feasibility, convergence, and diversity in the final solution. This improves the solution performance for interval-constrained multi-objective optimization problems, collaboratively achieving the dual objectives of minimizing the total mass of the laminate structure and maximizing the vibration frequency, thereby improving the design accuracy and engineering applicability of the laminate structure. Attached Figure Description

[0028] The accompanying drawings, which are incorporated in and form part of this specification, illustrate embodiments consistent with the invention and, together with the description, serve to explain the principles of the invention.

[0029] To more clearly illustrate the technical solutions in the embodiments of the present invention or the prior art, the drawings used in the description of the embodiments or the prior art will be briefly introduced below. Obviously, for those skilled in the art, other drawings can be obtained based on these drawings without creative effort.

[0030] Figure 1 A flowchart illustrating a method for solving an interval-constrained multi-objective optimization problem provided in an embodiment of the present invention; Figure 2 This is a schematic diagram of a composite material laminate structure provided in an embodiment of the present invention; Figure 3 This is a schematic diagram illustrating the distribution of non-dominated solutions of interval constraint dominance relations and dominance relations in related technologies, provided in embodiments of the present invention. Figure 4 A flowchart illustrating another method for solving interval-constrained multi-objective optimization problems provided in an embodiment of the present invention; Figure 5 A schematic diagram of an interval for an inequality constraint provided in an embodiment of the present invention; Figure 6 This is a schematic diagram of an interval for an equality constraint provided in an embodiment of the present invention; Figure 7 A schematic diagram of a target space provided in an embodiment of the present invention; Figure 8 This is another schematic diagram of the target space provided in an embodiment of the present invention; Figure 9 This is another schematic diagram of the target space provided in an embodiment of the present invention; Figure 10 A bar chart of comparative experimental results provided in an embodiment of the present invention; Figure 11 A bar chart of ablation experiment results provided in an embodiment of the present invention; Figure 12 This is a schematic diagram of a device for solving interval-constrained multi-objective optimization problems provided in an embodiment of the present invention; Figure 13 This is a schematic diagram of the hardware structure of an electronic device provided in an embodiment of the present invention.

[0031] Among them, 10 is the acquisition module; 20 is the problem construction module; 30 is the population generation module; 40 is the population update module; 50 is the output module; 1101 is the processor; 1102 is the memory; 1103 is the transceiver; and 1104 is the bus. Detailed Implementation

[0032] To better understand the above-mentioned objectives, features, and advantages of the embodiments of the present invention, the solutions of the embodiments of the present invention will be further described below. It should be noted that, unless otherwise specified, the embodiments of the present invention and the features thereof can be combined with each other.

[0033] Many specific details are set forth in the following description in order to provide a full understanding of the embodiments of the present invention, but the embodiments of the present invention may also be implemented in other ways different from those described herein; obviously, the embodiments in the specification are only some embodiments of the embodiments of the present invention, and not all embodiments.

[0034] Compared to deterministic constraints, interval constraints significantly impact the feasibility assessment of solutions and the size of the feasible region, greatly increasing the difficulty of searching the optimal solution space. In traditional CMOPs, deterministic constraint violations can intuitively determine the feasibility of a solution. However, in ICMOPs, interval objective functions and interval constraints make the spatial relationship between solutions and the feasible region more complex. Some interval solutions simultaneously cross both the feasible and infeasible regions, thus existing in an intermediate state between strictly feasible and strictly infeasible. This further increases the difficulty of assessing the individual feasibility of interval solutions, calculating constraint violations, and determining constraint dominance relationships.

[0035] Related technologies also include multi-objective optimization problems with interval parameters (IMOPs). IMOPs add interval objectives with interval parameters to CMOPs, but their constraints remain deterministic. The key challenge of IMOPs lies in effectively tracking the interval Pareto front (IPF), considering only the uncertainty propagation in the interval objective function. Compared to IMOPs, ICMOPs are doubly affected by both interval-valued objectives and interval constraints. The feasible region induced by the interval constraints further influences the feasibility and continuity of the true Pareto front. This dual uncertainty propagation further escalates the core challenge of ICMOPs to rapidly tracking the interval-constrained Pareto front (ICPF), requiring optimization algorithms to achieve efficient equilibrium solutions under interval uncertainty, considering feasibility, convergence, and diversity.

[0036] Meanwhile, most methods for solving IMOPs and ICMOPs in related technologies are indirect methods, that is, transforming the multi-objective optimization problem involving intervals into deterministic multi-objective optimization problems (i.e., CMOPs) through different mathematical transformation models before solving them. Although indirect methods can effectively reduce the optimization difficulty, they inevitably lead to the loss of valuable uncertain information. In addition, the initial interval multi-objective optimization problems obtained by different models will produce significantly different deterministic optimization problems, which will lead to a large deviation between the obtained optimal solution and the true optimal solution.

[0037] To address the aforementioned technical problems, embodiments of the present invention provide a method for solving interval-constrained multi-objective optimization problems and for optimizing composite laminate structures. Unlike the indirect methods described above, the method provided by the embodiments of the present invention directly quantifies the impact of uncertainty propagation, enabling the direct acquisition of optimal solutions for complex interval-constrained multi-objective optimization problems without any transformation, thus fully preserving valuable uncertainty information. Figure 1 As shown, the method provided in this embodiment of the invention includes: S1. Obtain the optimization objective of the laminated structure, the interval constraints of the laminated structure, and the uncertain material parameters.

[0038] The optimization objectives of the laminated structure include minimizing the total mass and maximizing the vibration frequency of the composite laminated structure; the range constraints of the laminated structure include ensuring that the vibration frequency range of the composite laminated structure meets the preset frequency constraints, and ensuring that the total mass range of the composite laminated structure meets the preset mass constraints; the uncertain material parameters include the Young's modulus range and density range of each layer of material in the composite laminated structure.

[0039] The following embodiments of the present invention will specifically illustrate each parameter, which will not be elaborated here.

[0040] S2. Based on the optimization objective of the laminated structure, the interval constraints of the laminated structure, and the uncertain material parameters, construct a multi-objective optimization problem and generate initial solutions for multiple multi-objective optimization problems. The solutions to the multi-objective optimization problems include the thickness of each layer of material in the composite laminated structure.

[0041] S3. Generate the first, second, and third populations based on the initial solution of the multi-objective optimization problem.

[0042] Specifically, the initial solutions are numerous solutions generated within the feasible range of the solution, without considering the multi-objective optimization problem of the layered structure optimization objective and the layered structure interval constraints, for subsequent optimization. These initial solutions are randomly and evenly divided into the first group, the second group, and the third group mentioned above. In some embodiments, all initial solutions can also be copied, that is, the first group, the second group, and the third group are populations including the same initial solutions. It is understood that the first group, the second group, and the third group are mainly distinguished by the following different iterative update methods, so the difference in population composition between the first group, the second group, and the third group is not limited here.

[0043] The first population is used as the offspring population in the first iteration, and S4 and S5 are executed iteratively. S4 and S5 specifically include: S4, based on the union of the first population and the offspring population, the first population is updated at least according to the dynamic interval constraint dominance relationship, and multiple solutions are selected from the union of the first population and the offspring population as the first offspring; based on the union of the second population and the offspring population, the second population is updated according to the interval Pareto dominance relationship, and multiple solutions are selected from the union of the second population and the offspring population as the second offspring; based on the union of the third population and the offspring population, the third population is updated according to the dynamic interval constraint dominance relationship and the interval Pareto dominance relationship, and multiple solutions are selected from the union of the third population and the offspring population as the third offspring. S5. Merge the first, second, and third offspring generations to obtain the offspring population for the next iteration.

[0044] Specifically, the interval constraint dominance relationship prioritizes whether the solution violates the interval constraints (in this embodiment, the interval constraints refer to the aforementioned laminated structure interval constraints, which will not be elaborated further). The dynamic interval constraint dominance relationship, based on the interval constraint dominance relationship, dynamically relaxes the constraint boundaries during the iterative update process to retain high-quality feasible solutions and valuable infeasible solutions close to the feasible region. The first group of solutions uses a feasibility-priority strategy for iterative updates. The interval Pareto dominance relationship prioritizes the contribution of the solution to the interval objective. That is, if the interval objective is to minimize a certain objective function, the interval Pareto dominance relationship will determine the solution that makes the interval value of the objective function smaller (in this embodiment, the objective function and its interval value are the interval values ​​of the total mass and vibration frequency of the composite laminated structure, which will not be elaborated further) as the better solution. Based on this, the second group of solutions uses a convergence-priority strategy for iterative updates. The third group of solutions provides an enhanced diversity iterative update strategy by simultaneously considering feasibility and convergence. The above-mentioned S4 and S5 incorporate three weakly cooperative populations, enabling the interval environment selection strategies of prioritizing feasibility, prioritizing convergence, and enhancing diversity to conduct complementary searches in the solution space, thereby achieving a good balance between feasibility, convergence, and diversity in the final solution.

[0045] In response to reaching the maximum number of iterations, the iteration ends and execution is performed.

[0046] S6: Return the first population as updated in the last iteration as the final solution set for the multi-objective optimization problem.

[0047] S7. Determine the thickness of each layer of material in the composite laminate structure based on the solutions in the final solution set.

[0048] Among them, the aforementioned dynamic interval constraint dominance relationship includes: in the first... In the next iteration, for any solution reconciliation If any one of conditions one, two, or three is satisfied, then the solution is... Dynamic interval constraint governing solution The calculation formulas for conditions one, two, and three include:

[0049] in, This is a dynamic constraint threshold that decays in each iteration based on the current iteration number. ; To solve constraint violation value, To solve The constraint violation value is defined as follows: a constraint violation value less than or equal to zero indicates that the solution satisfies all lamination structure interval constraints, and a constraint violation value greater than zero indicates that the solution violates at least some lamination structure interval constraints. The constraint violation value is proportional to the degree to which the solution violates the lamination structure interval constraints. The symbol used to represent the Pareto dominance relation in an interval. Solution Interval Pareto Dominant Solution and indicate the solution The objective function interval value is better than the solution The objective function interval values ​​are the interval values ​​of the total mass and vibration frequency of the composite laminate structure corresponding to the solution.

[0050] Specifically, the aforementioned dynamic constraint threshold This effectively replaces the zero boundary, meaning it replaces the logic in related technologies that determines whether a solution violates an interval constraint based on the relationship between the constraint violation value and zero. Thus, for two solutions located at the constraint boundary, for example, solution... The constraint violation value is less than zero, solution The constraint violation value is greater than zero, but the solution reconciliation The constraint violation values ​​all approach zero infinitely. In related technologies, the solution is directly determined. Dominant Solution However, through the above-mentioned dynamic constraint threshold of the present invention ,untie reconciliation Condition one can be used to enter a second-order decision based on Pareto dominance; if the solution... The contribution to the objective within the interval is greater than the solution. Then the solution will not be determined directly. Dominant Solution To avoid solutions that still have value Excluded, the dynamic interval constraint dominance relation improves the convergence of the non-dominated set and mitigates the risk of diversity loss by increasing the survival probability of certain infeasible solutions with superior interval objective values ​​and smaller constraint violation values ​​within the dynamic interval constraint boundary. Meanwhile, the dynamic constraint threshold... As the number of iterations decreases, the constraint boundary is continuously tightened, thereby ensuring the feasibility of the final solution.

[0051] This invention provides a Dynamic Interval Constraint Dominance Relationship (DICPD), which, during the iterative update process, uses a dynamic constraint threshold... Dynamically relax the boundary constraints to preserve high-quality feasible solutions as well as valuable infeasible solutions near the feasible region.

[0052] The method provided in this invention directly solves the interval-constrained multi-objective optimization problem in composite laminate structure optimization, fully preserving valuable uncertainty information. It comprehensively considers the interval uncertainties of Young's modulus and density of each layer, avoiding optimization deviations caused by ignoring parameter fluctuations and improving the reliability of the optimization results. Simultaneously, this invention provides a dynamic interval constraint dominance relationship, incorporating dynamically shrinking interval constraint boundaries. Under interval uncertainty, this relationship simultaneously preserves high-quality feasible solutions and infeasible solutions exhibiting good convergence within the dynamic interval constraint boundaries, thus providing more valuable candidate solutions during the evolution process. Furthermore, this invention provides a three-group co-evolutionary framework, maximizing the preservation of excellent feasible solutions with good diversity and low uncertainty, achieving a good balance between feasibility, convergence, and diversity in the final solution. This improves the solution performance for interval-constrained multi-objective optimization problems, collaboratively achieving the dual objectives of minimizing the total mass of the laminate structure and maximizing the vibration frequency, thereby enhancing the design accuracy and engineering applicability of the laminate structure.

[0053] In some embodiments, the aforementioned dynamic constraint threshold The calculation formulas include:

[0054] in, This is the initial dynamic constraint threshold. The maximum value among all constraint violation values ​​of the initial solutions; The maximum number of iterations, This represents the current iteration number; For dynamic constraint threshold The attenuation rate, Preferred .

[0055] During the iterative evolution process, the dynamic interval constraint boundary gradually shrinks towards the feasible region boundary. The dynamic interval constraint dominance relationship provided by the embodiments of the present invention ensures that valuable infeasible solutions gradually converge from the infeasible region to the Pareto front, while valuable feasible solutions gradually approach the Pareto front from the interior of the feasible region, thereby simultaneously improving the feasibility and convergence of the non-dominated set.

[0056] To facilitate understanding of the following other embodiments by those skilled in the art, the present invention provides several related definitions.

[0057] First, an interval number is the set of all points between a specified lower bound and an upper bound. It directly characterizes the possible range of values ​​for an uncertain parameter using the corresponding upper and lower bounds. This can be expressed as follows:

[0058] in, and Representing intervals The lower and upper bounds. express The midpoint, express The radius, and .when Time, interval It degenerates into a real number.

[0059] The result of interval operations is the smallest closed interval containing the set of all exact results obtained by applying the corresponding scalar operation to all possible input values. Given two interval numbers... and The interval operation rules are defined as follows:

[0060] Without loss of generality, the interval-constrained multi-objective optimization problem (ICMOP) in related technologies can be expressed in the following form:

[0061]

[0062] in, Solution The interval objective function vector is given by Different interval targets Composition. Solution For a by A vector consisting of deterministic decision variables. Solution The Each decision variable is bounded by a corresponding lower bound. and the Upper Realm limit. It is a collection Interval value parameters The vector. Interval equality constraints and interval inequality constraints are respectively determined by... and It indicates that the number of constraints is respectively and . , , and All are represented in interval form.

[0063] Based on the above definition, this invention provides a more specific embodiment applied to a range-constrained multi-objective optimization problem of a four-layer composite laminate structure, and performs finite element simulation using ANSYS software, as shown below. Figure 2 As shown, the first and last layers of the composite material are composed of type I material, and the middle layer is composed of type II material. The orientations of the composite material layers are 0°, 45°, -45°, and 90°, and the thicknesses to be solved are... (cm). The structure is a 3m × 4m rectangular plate composed of the aforementioned layers, discretized into smaller plate elements. All nodes along the 3-meter long side are defined as fixed boundaries, forming a cantilevered composite laminate structure. The main optimization objective is to minimize the total mass of the entire plate while maximizing the vibration frequency. The optimization objective and interval constraints of the laminate structure are as follows:

[0064]

[0065] in, The goal is to optimize the laminated structure. For the laminar structure interval constraint. This represents the total mass of the composite laminate structure, which must satisfy a mass constraint greater than or equal to 0. The first-order vibration frequency of the composite laminate structure must be less than or equal to a threshold value. Frequency constraints; uncertain material parameters include: for type I materials, the uncertainty of Young's modulus is... The uncertainty of density is For type II materials, the uncertainty of Young's modulus is: The uncertainty of density is , , , , All values ​​are interval values. The calculation of the objective function value depends on the finite element model. Specific uncertain material parameters are shown in Table 1: Table 1:

[0066] By solving the above modeling formula using the method provided in this embodiment of the invention, the optimal plate thickness that minimizes the total mass of the entire plate while maximizing the vibration frequency can be obtained. Those skilled in the art can substitute the above parameters into formulas in other embodiments based on the disclosure of this invention, which will not be elaborated upon here.

[0067] Specifically, based on the above basic definition, the aforementioned interval Pareto dominance (IPD) includes: given two solutions and Interval Pareto dominance It is valid if and only if the following two conditions are met simultaneously:

[0068] Among them, the interval dominance probability The calculation method is as follows:

[0069] The above Interval Constraint Dominance Relation (ICDP) includes: given two solutions and A state is said to be in a state where any of the following conditions are met. Interval constraint domination , recorded as :

[0070] in, To solve constraint violation value, To solve The constraint violation value.

[0071] Interval Pareto Optimal Set (ICPS): If a solution A solution that is not dominated by any other Pareto solution in an interval is called an interval Pareto optimal solution. The set of all interval Pareto optimal solutions is called the interval Pareto optimal set, denoted as . :

[0072] Interval Pareto Front (ICPF): The interval Pareto Front is the solution set. The mapping in the target space composed of interval targets is called :

[0073] Figure 3 The illustration shows the distribution of non-dominated solutions obtained from a population of 16 individuals using the Dynamic Interval Domination Relationship (DICPD), the Interval Pareto Domination Relationship (IPD), and the Interval Domination Relationship (ICDP) provided by this embodiment of the invention. Specifically, a non-dominated solution is a solution that is not dominated by any other solution; conversely, a solution dominated by any other solution is a dominant solution.

[0074] like Figure 3As shown in (a), ICDP selects solutions with better interval constraint violation values ​​within the feasible region, resulting in only 3 non-dominated solutions. Although the interval values ​​of these non-dominated solutions strictly satisfy the feasibility constraints, they exhibit insufficient diversity and convergence, which hinders the algorithm from exploring infeasible regions and prevents the population from escaping local optima. Figure 3 As shown in (b), IPD selects solutions with better interval objective values ​​to prioritize convergence, resulting in five non-dominated solutions within the infeasible region. Although these non-dominated interval solutions exhibit good convergence and diversity, the lack of consideration for constraints often leads them to be far from the feasible region, thus failing to provide effective information for problem solving. Neither of the two interval dominance relationships mentioned above adequately preserves and utilizes valuable infeasible solutions near the feasible region that possess superior interval objective functions and lower interval constraint violations when determining non-dominated interval solutions.

[0075] Figure 3 (c) to Figure 3 (d) shows the DICPD provided in the embodiments of the present invention in different... The distribution of non-dominated solutions obtained under the given value. These figures illustrate that the non-dominated set obtained by DICPD simultaneously preserves non-dominated solutions within the feasible region and potentially valuable infeasible solutions, effectively balancing feasibility, convergence, and diversity. Furthermore, with... As the constraint is gradually reduced, DICPD eliminates infeasible solutions with large constraint violations from the non-dominated solution set, thus allowing the retained infeasible solutions to gradually converge from the infeasible region to ICPF. At that time, the nondominated solution obtained by DICPD is consistent with the result of ICDP.

[0076] Based on the above analysis, compared with IPD which prioritizes convergence and ICDP which prioritizes feasibility, the DICPD provided by the embodiments of the present invention can better balance the feasibility, diversity and convergence of the non-dominated solution set, and can make full use of information from valuable feasible and infeasible solutions, thereby enhancing the optimization potential of the algorithm when searching for ICPF.

[0077] Figure 4 The overall framework of the method proposed in the embodiments of the present invention is shown, and detailed pseudocode is shown in Table 2. In the initialization phase, three scales are first randomly initialized. Initial solution population: First population second group and the third group Offspring population Initialize to Then, the objective function interval values ​​and ICV (interval constraint violation) are calculated for the three populations. Accordingly, the maximum ICV value among all solutions is used as the initial dynamic constraint threshold. The minimum lower bound of the objective function for all intervals in the three populations is set as the initial ideal point. In each iteration of the main loop, the dynamic constraint threshold is updated first. and ideal point Subsequently, interval environment selection strategies were designed for the three populations, prioritizing feasibility, convergence, and enhanced diversity.

[0078] By respectively in three groups and The corresponding interval environment selection strategy is executed on the union of the three populations to obtain the updated three populations. , and and its corresponding mating pool , and Each mating pool is generated independently by randomly applying the GA and DE operators. Each offspring, obtaining the first offspring. Second generation With the third generation These offspring are merged to form the next generation. When the iteration counter Reaching the maximum number of iterations When, return As the final solution set.

[0079]

[0080] In the calculation of constraint violation values ​​in CMOPs, solutions can be directly classified into strictly feasible and strictly infeasible solutions based on the precise distance between the solution and the constraint boundary. However, in ICMOPs, the inherent uncertainty of the interval causes some solutions to simultaneously cross both feasible and infeasible regions. These solutions are called partially feasible solutions. Therefore, effectively distinguishing these three different types of solutions is crucial in the calculation of interval constraint violation values ​​in ICMOPs. In view of this, in some embodiments, the formula for calculating constraint violation values ​​in this invention includes:

[0081]

[0082] in, To solve constraint violation value, To solve In the The constraint violation values ​​of the interval inequality constraints are as follows: the first interval inequality constraint is to ensure that the vibration frequency interval value of the composite laminate structure meets the frequency constraint, and the second interval inequality constraint is to ensure that the total mass interval value of the composite laminate structure meets the mass constraint. To solve In the Interval values ​​on interval inequality constraints The upper bound of the interval is... The lower bound of the interval, For interval values.

[0083] The above embodiments of the present invention propose a novel ICV calculation method. For interval inequality constraints, it can intuitively distinguish strictly feasible solutions, strictly infeasible solutions, and partially feasible solutions based on the calculated ICV values.

[0084] Specifically, for For example, Figure 5 As shown, when Time (i.e., Case 1), interval If the solution falls entirely within the feasible region, it indicates that the solution is correct. In the It is strictly feasible under the constraints of the inequalities, therefore it will Set to 0. When (i.e., Case 2) Simultaneously crossing both feasible and infeasible regions indicates a solution. In the If only some of the inequality constraints are satisfied, the solution is considered locally feasible. Correspondingly... The value ranges from 0 to 1; when When (Case 3) the entire interval is in the infeasible region, it indicates that the solution is not feasible. In the It is strictly infeasible under the constraints of these inequalities. Set to a value greater than 1.

[0085] In other embodiments, when the multi-objective optimization problem also includes equality constraints, the formula for calculating the constraint violation value in this invention includes:

[0086]

[0087]

[0088] in, To solve constraint violation value, To solve In the The constraint violation value of each interval inequality constraint. To solve In the The constraint violation value of an interval equality constraint. Let be the number of interval inequality constraints. The number of interval equality constraints; To solve In the Interval values ​​on interval inequality constraints The upper bound of the interval is... The lower bound of the interval, For interval values; To solve In the Interval values ​​on interval equality constraints The upper bound of the interval is... The lower bound of the interval, To relax the constraints on parameters, Preferred .

[0089] The above embodiments of the present invention propose a novel ICV calculation method. For interval equality constraints and interval inequality constraints, it can intuitively distinguish strictly feasible solutions, strictly infeasible solutions, and partially feasible solutions based on the calculated ICV values.

[0090] Specifically, for For example, Figure 6 As shown, in In Case 1, the solution Considered as in the Strictly feasible under the equality constraints, Set to 0. In Cases 2, 3, and 4, Simultaneously traversing both feasible and infeasible regions, the solution will be... This is called a locally feasible solution, and its corresponding... The value ranges from 0 to 1. In Case 5 and Case 6, The fact that the solution lies entirely within the infeasible region indicates that the solution is... Strictly not feasible, its The value is greater than 1.

[0091] In some embodiments, the processes of updating the first population and selecting the first offspring, updating the second population and selecting the second offspring, and updating the third population and selecting the third offspring in S4 above all include: Calculate the inter-crowding distance for each solution in the population, and update the population and select offspring based on the inter-crowding distance.

[0092] Interval crowding distance is a distributional metric proposed in interval-constrained multi-objective optimization algorithms for interval-valued objective functions (i.e., the objective value is an interval range rather than a precise numerical value). It assesses the sparsity of solutions by calculating the interval distance between adjacent solutions in the objective space (common methods include based on the interval midpoint, interval width, or probability model). Given the complex spatial relationships between individuals in multidimensional intervals, relying solely on the upper / lower bounds or midpoint of the interval to calculate the interval crowding distance cannot accurately assess the crowding degree of individuals in the objective space, thus hindering the effective maintenance of population diversity. Furthermore, reducing the uncertainty of individuals is also crucial when solving ICMOPs. Therefore, this invention proposes a method for calculating interval crowding distance, which comprehensively considers diversity and uncertainty from aspects such as the hypervolume of interval individuals (i.e., each solution), the overlap rate between interval individuals, and the proportion of the influence area of ​​interval individuals. For interval individuals, those exhibiting greater diversity and lower uncertainty will be assigned a correspondingly larger interval crowding distance.

[0093] Specifically, the formula for calculating the interval congestion distance provided in this embodiment of the invention includes:

[0094] The multi-objective optimization problem includes multiple interval objectives. The objective space formed by the interval objectives includes multiple dominance layers, and each solution is assigned to one of the dominance layers according to its dominance relationship. To solve Left neighbor in the same dominance layer, solution To solve The right neighbor in the same dominance layer; To solve The distance between congested areas To solve and solution The degree of congestion between them To solve and solution The degree of congestion between them To solve Hypervolume in the target space; Indicates when the solution The hypercube is located on the boundary of the dominance layer, and its crowding is given an ideal value of 1.

[0095] The formula for calculating the crowding degree between two solutions includes:

[0096] in, To solve and solution The degree of congestion between them To solve The hypervolume in the target space; the dominance layer includes a non-dominated front, where solutions are not dominated by any other solutions. To solve and solution The proportion of the influence region formed in the target space relative to the influence region formed by the entire non-dominated frontier in the target space (solution) and solution The largest rectangular region formed in the objective space is considered to be the influence region of these two solutions. To solve and solution The non-overlapping hypervolume between solutions and solution The non-overlapping proportion of the total hypervolume.

[0097] This invention proposes a method for calculating the congestion distance between intervals. This distance comprehensively considers the hypervolume of individuals within the interval, the overlap rate between individuals within the interval, and the proportion of the area affected by individuals within the interval in the target space, thereby improving the survival probability of individuals exhibiting high diversity and low uncertainty in the environmental selection process.

[0098] Figures 7 to 9 This intuitively illustrates the impact of regional proportions, individual overvolume, and non-overlapping proportions on the crowding level of individuals within a given interval. Figures 7 to 9 The coordinate system shown can be considered as the target space. The red boxes represent the hypervolume of each solution in the target space, and the areas within the dashed boxes represent the influence regions of the two solutions. For example... Figure 7 As shown, the solution and solution The hypervolume and non-overlap ratio remain unchanged, but from Figure 7 (a) to Figure 7 (b) then to Figure 7 (c), with the solution and solution The influence region (i.e., the proportion of influence) occupies an increasingly larger proportion of the objective space, and the crowding of each solution in the objective space also increases. For example... Figure 8 As shown, the solution and solution The areas of influence and the proportion of non-overlapping areas are fixed, but from Figure 8 (a) to Figure 8 (b) then to Figure 8 (c) As the hypervolume of each solution becomes smaller, its distribution in the target space becomes sparser. For example... Figure 9 As shown, the solution and solution The area of ​​influence and the supervolume remain unchanged, but from Figure 9 (a) to Figure 9 (b) then to Figure 9 (c), with the solution and solution The greater the proportion of non-overlapping elements, the better their diversity in the target space.

[0099] In some embodiments, the formula for calculating the above-mentioned excess volume includes:

[0100] in, To solve Hypervolume in the target space To solve In the Interval values ​​on a given interval target The upper bound of the interval is... The lower bound of the interval, The total number of targets in the interval. For interval values. By solving The solution is obtained by multiplying the interval widths over all interval targets, thus measuring the solution. Uncertainty.

[0101] In some embodiments, the formula for calculating the above-mentioned influence ratio includes:

[0102] in, To solve and solution The proportion of influence of the region formed in the target space relative to the influence of the entire non-dominated frontier in the target space. The total number of targets in the interval. For interval value parameters, For the solution in the non-dominated frontier, the th The maximum upper bound of a target interval. For the solution in the non-dominated frontier, the th The minimum lower bound of the interval target; To solve In the Interval values ​​on a given interval target The upper bound of the interval is... This is the lower bound of the interval; To solve In the Interval values ​​on a given interval target The upper bound of the interval is... This is the lower bound of the interval.

[0103] In some embodiments, the formula for calculating the non-overlapping ratio includes:

[0104]

[0105] in, To solve and solution The non-overlapping hypervolume between solutions and solution The non-overlapping proportion of the total hypervolume. To solve Hypervolume in the target space To solve Hypervolume in the target space To solve and solution Overlapping hypervolume between them To solve In the Interval values ​​on a given interval target To solve In the Interval values ​​on a given interval target Solution and solution In the Overlapping areas on the target intervals The total number of targets in the interval. For interval values.

[0106] The pseudocode in Table 3 below illustrates the process in S4 of updating the first population based on the union of the first population and the offspring population, at least according to the dynamic interval constraint dominance relationship, and selecting multiple solutions from the union of the first population and the offspring population as the first offspring. Specifically, this process includes two parts: population update and mating pool selection.

[0107] In the first part, the dynamic interval constraint dominance relationship is determined based on the above. and Non-dominated solution set in the union .if The size is smaller than the required offspring size. Then from Random copy Find a solution and add it to the solution. China as the next generation (i.e., update) Otherwise, from The lower bound minimum and upper bound maximum values ​​of the objective function for all intervals are extracted as ideal points. and worst point Subsequently, using the determined and right The interval objective function values ​​of all individuals are normalized. The interval crowding distance for each solution is calculated based on the normalized interval objective function values. Select the first section in descending order of congestion distance. A solution to form the next generation .

[0108] The mating pool in the second part During the construction process, to prioritize the feasibility of the solution, the Interval Constraint Dominance Dependency (ICDP) is applied to the updated solution. Extracting non-dominated solution sets .if The size is smaller than (recorded as) ), then from Random copy Find a solution and add it to In order to build Otherwise, based on interval function normalization and interval congestion distance calculation, from Select the best one The solution is as follows Finally, the mating pool Independently generated by randomly applying the GA and DE operators. Each offspring, obtaining the aforementioned first offspring. .

[0109]

[0110] The pseudocode in Table 4 below illustrates the process in S4 of updating the second population based on the union of the second population and the offspring population according to the interval Pareto dominance relation, and selecting multiple solutions from the union of the second population and the offspring population as the second offspring. Specifically, this process also includes two parts: population update and mating pool selection.

[0111] In updating the population At that time, firstly based on the interval Pareto dominance relationship, from and The non-dominated solution set is selected from the set. .if Then from Random copy Find a solution and add it to In order to form the next generation Otherwise, from The maximum upper bound of the objective function in all intervals is extracted as the worst-case scenario. Use the ideal point of input. and right The interval objective function values ​​of all solutions are normalized. Then, the interval congestion distance for each solution is calculated based on the normalized interval objective function values. Select the one with the largest inter-regional congestion distance Each solution is needed to form the next generation of the population. .

[0112] In constructing mating pools In order to prioritize the convergence of solutions, the interval Pareto dominance relation is also used to determine the dominance relationship between solutions. Because of the updated... All solutions are non-dominated with respect to the interval Pareto dominance relation. Therefore, based on interval objective normalization and interval congestion distance calculation, we can directly calculate from... Select the best one The solution is as follows Finally, the mating pool Independently generated by randomly applying the GA and DE operators. Each offspring, obtaining the aforementioned second offspring. .

[0113]

[0114] The pseudocode in Table 5 below illustrates the process in S4 of updating the third population based on the union of the third population and the offspring population, according to the dynamic interval constraint dominance and interval Pareto dominance, and selecting multiple solutions from the union of the third population and the offspring population as the third offspring.

[0115] Specifically, in updating the population At that time, based on the dynamic interval constraint dominance relationship from the mixed population Selecting the non-dominated solution set .Will Reset to an empty set and generate A uniformly distributed reference vector Calculate the average of the minimum angular distances among all reference vectors, and use this as the adaptive neighborhood radius. Then, calculate. Find the midpoint of the interval objective function for all solutions, and extract the maximum value from the midpoint of all these interval objective function values ​​as the worst-case point. Use the ideal point of input. and Normalize the midpoint of the objective function over the interval. Based on the normalized midpoint, calculate... Each solution and The angular distance between all reference vectors in the dataset. For each reference vector, identify the angular distance between it and the reference vector that is less than [value missing]. solution set .

[0116] if Then Add the solution with the smallest angular distance to the reference vector to In the middle. Otherwise, from Further extract feasible solution sets .if Then Add the solution with the smallest ICV value to Middle. Otherwise, will Add the solution with the smallest angular distance to the reference vector to .

[0117] In constructing mating pools At that time, the interval Pareto dominance relation is applied to the updated... Extracting non-dominated solution sets .if Then from Random copy Find a solution and add it to In order to form Otherwise, The maximum upper bound of the objective function across all intervals is determined as the worst-case scenario. Then, using the input ideal point and right The interval-valued objective vectors of all solutions are normalized. Based on the normalized interval-valued objective vectors, the interval congestion distance for each solution is calculated. Finally, from... Select the one with the largest inter-regional congestion distance A solution to form a mating pool Finally, the mating pool Independently generated by randomly applying the GA and DE operators. Each offspring, obtaining the aforementioned third offspring. .

[0118]

[0119] To comprehensively evaluate the performance of the method provided in this embodiment of the invention for solving ICMOPs, the invention conducts experiments based on the following five performance indicators: (1) Feasibility Rate (FR): The proportion of strictly feasible solutions in the final population obtained by the algorithm is used as the feasibility rate to evaluate the feasibility performance of the algorithm; (2) Inverse Generation Distance Plus (IGD+): IGD+ is a widely used indicator for evaluating the population diversity and convergence of CMOPs. Under interval conditions, the average of the best and worst case IGD+ values ​​between the reference point set and the approximate ICPS is calculated to reflect the overall optimization performance of the algorithm. The reference point set consists of 10,000 points uniformly distributed on the CPF of the initial CMOP. The best and worst case IGD+ values ​​are calculated based on the upper and lower bounds of the interval target in the approximate ICPS, respectively. The lower the IGD+ value, the better the algorithm’s performance in terms of diversity and convergence; (3) Hypervolume (HV): Similar to IGD+, under interval conditions, the average of the worst and best hypervolumes formed by the reference point set and the approximate ICPS is used as HV. The reference point is set in the normalized target space. The worst hypervolume and the best hypervolume are calculated based on the upper and lower bounds of the target value in the approximate ICPS. The larger the HV value, the better the algorithm’s performance in terms of diversity and convergence; (4) Average Inaccuracy (MI): MI is the average hypervolume of the approximate ICPS obtained by the algorithm in the target space. The smaller the value, the better the approximate Pareto optimal solution obtained by the algorithm performs in terms of inaccuracy; (5) Running Time (RT): The shorter the running time, the better the algorithm’s performance in terms of computational efficiency.

[0120] The experiment was adapted from a novel MW suite in related technologies to evaluate the performance of the algorithm in solving ICMOPs, denoted as IMW. The IMW test suite contains 11 interval-constrained biobjective optimization problems, namely IMW1–IMW3, IMW5–IMW7, and IMW9–IMW13, which are constructed by introducing perturbation coefficients into the objective function of the corresponding MW test problem.

[0121] In the experiment, the number of deterministic decision variables was set to D=15, the number of objective functions was set to M=2, the population size of the algorithm was set to N=100, the number of iterations was set to Gen=500, and the ICV was uniformly calculated using the method proposed in the embodiments of this invention.

[0122] The DICEA algorithm proposed in this invention is compared with seven state-of-the-art algorithms (INSGAⅡ, ICCMO, IMTCMO, EG-ICMOEA, NPS-CNSGAⅡ, IP-NSGAⅡ, and PF-ICMOA) on the IMW suite in terms of five metrics to evaluate its comprehensive performance in terms of feasibility, convergence, diversity, uncertainty, and computational efficiency. Figure 10 A bar chart of the indicators is shown. From Figure 10It is evident that the DICEA proposed in this embodiment ranks first in FR, HV, and RT metrics, and achieves the second-best performance in IGD+, slightly inferior to ICCMO. Compared to the seven state-of-the-art ICCMOEAs, the algorithm proposed in this embodiment demonstrates superior performance in handling ICMOPs, effectively balancing convergence, diversity, and uncertainty while ensuring feasibility and computational efficiency.

[0123] The embodiments of this invention also conducted a series of ablation experiments to verify the effectiveness of the core components in the proposed DICEA algorithm, including the DICPD relationship, interval crowding distance, and three-group co-evolution. By removing the above components, the experiments of this invention created six variants, named DICEA-V1 to V6, and compared their performance with the original DICEA algorithm on the IMW test suite. Figure 11 The average ranking results of DICEA and all its variants on the IMW test suite are shown.

[0124] To verify the effectiveness of the proposed DICPD relationship, this invention first designed a variant called DICEA-V1, replacing the DICPD relationship in DICEA with an ICDP relationship. From Figure 11 The original DICEA significantly outperforms DICEA-V1 across all four metrics of the IMW test suite, providing strong evidence for the effectiveness of the proposed DICPD relation. Compared to ICDP, which only prioritizes feasible solutions, DICPD additionally preserves potentially valuable infeasible solutions within relaxed constraint boundaries, thus effectively maintaining population diversity. Furthermore, the relaxed constraint boundaries gradually tighten during evolution, guiding the population to search for the optimal region simultaneously from both feasible and infeasible regions.

[0125] To verify the effectiveness of the proposed interval congestion distance, this invention designed two variants, named DICEA-V2 and DICEA-V3, respectively, replacing the interval congestion distance in DICEA with the interval congestion distances in EG-ICMOEA and NPS-CNSGAII. Figure 11As shown, DICEA performs comparably to DICEA-V2 and DICEA-V3 on the FR metric, significantly outperforms them on the IGD+ and HV metrics, but lags behind on the MI metric. The advantage of DICEA-V2 and DICEA-V3 in terms of uncertainty stems from their populations only covering a portion of the ICPF with lower uncertainty, leading to decreased diversity and convergence. Experiments demonstrate that, compared to the interval crowding distances in EG-ICMOEA and NPS-CNSGAII, the interval crowding distance proposed in this invention achieves a better balance between convergence, diversity, and uncertainty, improving DICEA's performance in searching the complete ICPF.

[0126] To verify the effectiveness of the proposed three-population co-evolution, this invention designed three variants, named DICEA-V4, DICEA-V5, and DICEA-V6. Specifically, variant DICEA-V4 retains only the feasibility-priority population P1, thus degenerating into a single-population algorithm. Variant DICEA-V5 maintains the co-evolution of populations P1 and P2, eliminating the diversity-enhancing population P3. Variant DICEA-V6 retains the co-evolution of populations P1 and P3, removing the convergence-priority population P2. Figure 11 As shown, DICEA significantly outperforms DICEA-V4 in all four metrics, indicating that a single-population strategy struggles to balance exploration and exploitation in the solution space. Furthermore, while DICEA performs comparably to DICEA-V5 in FR and MI metrics, it significantly outperforms DICEA-V5 in IGD+ and HV metrics. DICEA-V6 performs comparably to DICEA in FR, but significantly underperforms DICEA in the other three metrics. Experiments confirm that the three-population co-evolutionary approach proposed in this invention, while ensuring feasibility, can better balance convergence, diversity, and uncertainty, thus improving the performance of DICEA in solving ICMOPs.

[0127] from Figure 11 It is clear that DICEA achieves the best average ranking across FR, IGD+, and HV metrics. While DICEA lags behind DICEA-V2 and DICEA-V3 in uncertainty, it significantly outperforms them in convergence and diversity. In conclusion, the results of the ablation experiments provide strong evidence for the effectiveness of these three key components in enhancing DICEA's ability to handle ICMOPs.

[0128] It should be noted that the method of this embodiment can be executed by a single device, such as a computer or server. The method of this embodiment can also be applied to a distributed scenario, where multiple devices cooperate to complete the task. In such a distributed scenario, one of these devices may execute only one or more steps of the method of this embodiment, and the multiple devices will interact with each other to complete the above method.

[0129] It should be noted that some embodiments of the present invention have been described above. Furthermore, the processes depicted in the accompanying drawings do not necessarily require a specific or sequential order to achieve the desired results. In some embodiments, multitasking and parallel processing are possible or may be advantageous.

[0130] Based on the same inventive concept, and corresponding to the methods of any of the above embodiments, this invention also provides a solution device for interval-constrained multi-objective optimization problems, used to optimize composite laminate structures, such as... Figure 12 As shown, it includes: The acquisition module 10 is used to acquire the optimization objective of the laminated structure, the interval constraints of the laminated structure, and the uncertain material parameters. The optimization objective of the laminated structure includes minimizing the total mass and maximizing the vibration frequency of the composite laminated structure. The interval constraints of the laminated structure include ensuring that the vibration frequency interval of the composite laminated structure meets the preset frequency constraints and that the total mass interval of the composite laminated structure meets the preset mass constraints. The uncertain material parameters include the Young's modulus interval and density interval of each layer of material in the composite laminated structure.

[0131] Problem construction module 20 is used to construct multi-objective optimization problems based on the optimization objective of the laminated structure, the interval constraints of the laminated structure, and the uncertain material parameters, and to generate initial solutions to multiple multi-objective optimization problems; the solutions to the multi-objective optimization problems include the thickness of each layer of material in the composite laminated structure.

[0132] Population generation module 30 is used to generate a first population, a second population, and a third population based on the initial solution of a multi-objective optimization problem.

[0133] The population update module 40 is used to take the first population as the offspring population in the first iteration and iteratively execute the following: based on the union of the first population and the offspring population, update the first population at least according to the dynamic interval constraint dominance relationship, and select multiple solutions from the union of the first population and the offspring population as the first offspring; based on the union of the second population and the offspring population, update the second population according to the interval Pareto dominance relationship, and select multiple solutions from the union of the second population and the offspring population as the second offspring; based on the union of the third population and the offspring population, update the third population according to the dynamic interval constraint dominance relationship and the interval Pareto dominance relationship, and select multiple solutions from the union of the third population and the offspring population as the third offspring; merge the first offspring, the second offspring, and the third offspring to obtain the offspring population in the next iteration.

[0134] Output module 50 is used to terminate the iteration in response to reaching the maximum number of iterations and return the first population updated at the time of the last iteration as the final solution set of the multi-objective optimization problem.

[0135] Output module 50 is also used to determine the thickness of each layer of material in the composite laminate structure based on the solutions in the final solution set.

[0136] Dynamic interval constraint dominance relationships include: in the first... In the next iteration, for any solution reconciliation If any one of conditions one, two, or three is satisfied, then the solution is... Dynamic interval constraint governing solution The calculation formulas for conditions one, two, and three include:

[0137] in, This is a dynamic constraint threshold that decays in each iteration based on the current iteration number. ; To solve constraint violation value, To solve The constraint violation value is defined as follows: a constraint violation value less than or equal to zero indicates that the solution satisfies all lamination structure interval constraints, and a constraint violation value greater than zero indicates that the solution violates at least some lamination structure interval constraints. The constraint violation value is proportional to the degree to which the solution violates the lamination structure interval constraints. The symbol used to represent the Pareto dominance relation in an interval. Solution Interval Pareto Dominant Solution and indicate the solution The objective function interval value is better than the solution The objective function interval values ​​are the interval values ​​of the total mass and vibration frequency of the composite laminate structure corresponding to the solution.

[0138] The apparatus provided in this invention directly solves the interval-constrained multi-objective optimization problem in composite laminate structure optimization, fully preserving valuable uncertainty information. It comprehensively considers the interval uncertainties of Young's modulus and density of each layer, avoiding optimization deviations caused by ignoring parameter fluctuations and improving the reliability of the optimization results. Simultaneously, this invention provides a dynamic interval constraint dominance relationship, incorporating dynamically shrinking interval constraint boundaries. Under interval uncertainty, this relationship can simultaneously preserve high-quality feasible solutions and infeasible solutions exhibiting good convergence within the dynamic interval constraint boundaries, thus providing more valuable candidate solutions during the evolution process. Furthermore, this invention provides a three-group co-evolutionary framework, maximizing the preservation of excellent feasible solutions with good diversity and low uncertainty, achieving a good balance between feasibility, convergence, and diversity in the final solution. This improves the performance of solving interval-constrained multi-objective optimization problems, collaboratively achieving the dual objectives of minimizing the total mass of the laminate structure and maximizing the vibration frequency, thereby enhancing the design accuracy and engineering applicability of the laminate structure.

[0139] For ease of description, the above apparatus is described in terms of its functions, divided into various modules. Of course, in implementing this invention, the functions of each module can be implemented in one or more software and / or hardware components.

[0140] The apparatus described above is used to implement the corresponding interval-constrained multi-objective optimization problem solving method in any of the foregoing embodiments, and has the beneficial effects of the corresponding method embodiments, which will not be repeated here.

[0141] Figure 13 A schematic diagram of the structure of an electronic device provided by an embodiment of the present invention is shown.

[0142] like Figure 13 As shown, the electronic device may include a processor 1101 and a memory 1102 storing computer program instructions.

[0143] Specifically, the processor 1101 may include a central processing unit (CPU), an application specific integrated circuit (ASIC), or one or more integrated circuits that can be configured to implement the embodiments of the present invention.

[0144] Memory 1102 may include a mass storage device for information or instructions. For example, and not limitingly, memory 1102 may include a hard disk drive (HDD), a floppy disk drive, flash memory, optical disk, magneto-optical disk, magnetic tape, or a Universal Serial Bus (USB) drive, or a combination of two or more of these. Where appropriate, memory 1102 may include removable or non-removable (or fixed) media. Where appropriate, memory 1102 may be internal or external to the integrated gateway device. In a particular embodiment, memory 1102 is a non-volatile solid-state memory. In a particular embodiment, memory 1102 includes read-only memory (ROM). Where appropriate, the ROM may be a mask-programmed ROM, a programmable ROM (PROM), an erasable PROM (EPROM), an electrically erasable programmable PROM (EEPROM), an electrically alterable ROM (EAROM), or flash memory, or a combination of two or more of these.

[0145] The processor 1101 reads and executes the computer program instructions stored in the memory 1102 to perform the steps of the interval-constrained multi-objective optimization problem solving method provided in the embodiments of the present invention.

[0146] In one example, the electronic device may also include a transceiver 1103 and a bus 1104. Wherein, as... Figure 13 As shown, the processor 1101, memory 1102 and transceiver 1103 are connected via bus 1104 and communicate with each other.

[0147] Bus 1104 may include hardware, software, or both. For example, and not limitingly, the bus may include an Accelerated Graphics Port (AGP) or other graphics bus, an Extended Industry Standard Architecture (EISA) bus, a Front Side Bus (FSB), a Hyper Transport (HT) interconnect, an Industrial Standard Architecture (ISA) bus, an Infinite Bandwidth Interconnect, a Low Pin Count (LPC) bus, a memory bus, a MicroChannel Architecture (MCA) bus, a Peripheral Component Interconnect (PCI) bus, a PCI-Express (PCI-X) bus, a Serial Advanced Technology Attachment (SATA) bus, a Video Electronics Standards Association Local Bus (VLB) bus, or other suitable buses, or a combination of two or more of these. Where appropriate, bus 1104 may include one or more buses. Although specific buses are described and illustrated in the embodiments of the present invention, the present invention is contemplated by any suitable bus or interconnect.

[0148] The following are embodiments of a computer-readable storage medium provided in this invention. This computer-readable storage medium and the interval-constrained multi-objective optimization problem solving methods in the above embodiments belong to the same inventive concept. For details not described in detail in the embodiments of the computer-readable storage medium, please refer to the embodiments of the interval-constrained multi-objective optimization problem solving methods described above.

[0149] This embodiment provides a storage medium containing computer-executable instructions, which, when executed by a computer processor, are used to perform a method for solving an interval-constrained multi-objective optimization problem.

[0150] Of course, the computer-executable instructions provided in the embodiments of the present invention are not limited to the above-described method operations, but can also execute related operations in the interval-constrained multi-objective optimization problem solving method provided in any embodiment of the present invention.

[0151] Based on the above description of the implementation methods, those skilled in the art can clearly understand that the present invention can be implemented using software and necessary general-purpose hardware, and of course, it can also be implemented using hardware, but in many cases the former is a better implementation method. Based on this understanding, the technical solution of the present invention, or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product can be stored in a computer-readable storage medium, such as a computer floppy disk, read-only memory (ROM), random access memory (RAM), flash memory, hard disk, or optical disk, etc., including several instructions to cause a computer cloud platform (which can be a personal computer, server, or network cloud platform, etc.) to execute the interval-constrained multi-objective optimization problem solving method provided in the various embodiments of the present invention.

[0152] It should be noted that, in this document, relational terms such as "first" and "second" are used merely to distinguish one entity or operation from another, and do not necessarily require or imply any such actual relationship or order between these entities or operations. Furthermore, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or apparatus. Without further limitations, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or apparatus that includes the aforementioned element.

[0153] The above description is merely a specific embodiment of the present invention, enabling those skilled in the art to understand or implement the invention. Various modifications to these embodiments will be readily apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the invention. Therefore, the present invention is not to be limited to the embodiments described above, but is to be accorded the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. A method for solving interval-constrained multi-objective optimization problems, characterized in that, Used to optimize composite laminate structures, including: The optimization objectives, interval constraints, and uncertain material parameters of the laminated structure are obtained. The optimization objectives include minimizing the total mass and maximizing the vibration frequency of the composite laminated structure. The interval constraints include ensuring that the vibration frequency interval of the composite laminated structure meets a preset frequency constraint and that the total mass interval of the composite laminated structure meets a preset mass constraint. The uncertain material parameters include the Young's modulus interval and density interval of each layer of material in the composite laminated structure. Based on the laminated structure optimization objective, the laminated structure interval constraints, and the uncertain material parameters, a multi-objective optimization problem is constructed, and multiple initial solutions to the multi-objective optimization problem are generated; the solution to the multi-objective optimization problem includes the thickness of each layer of material in the composite laminated structure; The first, second, and third groups are generated based on the initial solution of the multi-objective optimization problem. The first population is used as the offspring population for the first iteration, and the following iterative steps are performed: Based on the union of the first population and the offspring population, the first population is updated at least according to the dynamic interval constraint dominance relationship, and multiple solutions are selected from the union of the first population and the offspring population as the first offspring; Based on the union of the second population and the offspring population, the second population is updated according to the interval Pareto dominance relationship, and multiple solutions are selected from the union of the second population and the offspring population as the second offspring; Based on the union of the third population and the offspring population, the third population is updated according to the dynamic interval constraint dominance relationship and the interval Pareto dominance relationship, and multiple solutions are selected from the union of the third population and the offspring population as the third offspring; The first offspring, the second offspring, and the third offspring are merged to obtain the offspring population for the next iteration; In response to reaching the maximum number of iterations, the iteration ends, and the first population updated at the time of the last iteration is returned as the final solution set of the multi-objective optimization problem; The thickness of each layer of material in the composite laminate structure is determined based on the solutions in the final solution set; The dynamic interval constraint dominance relationship includes: in the... In the next iteration, for any solution reconciliation If any one of conditions one, two, or three is satisfied, then the solution is... Dynamic interval constraint governing solution The calculation formulas for conditions one, two, and three include: in, This is a dynamic constraint threshold that decays in each iteration based on the current iteration number. ; To solve constraint violation value, To solve The constraint violation value is defined as follows: a constraint violation value less than or equal to zero indicates that the solution satisfies all the lamination structure interval constraints, and a constraint violation value greater than zero indicates that the solution violates at least some of the lamination structure interval constraints. The constraint violation value is proportional to the degree to which the solution violates the lamination structure interval constraints. The symbol used to represent the Pareto dominance relation in an interval. Solution Interval Pareto Dominant Solution and indicate the solution The objective function interval value is better than the solution The objective function interval value, wherein the objective function interval value is the interval value of the total mass of the composite laminate structure and the interval value of the vibration frequency corresponding to the solution; The formula for calculating the constraint violation value includes: in, To solve constraint violation value, To solve In the The constraint violation values ​​of the interval inequality constraints are as follows: the first interval inequality constraint is to make the vibration frequency interval value of the composite laminate structure satisfy the frequency constraint, and the second interval inequality constraint is to make the total mass interval value of the composite laminate structure satisfy the mass constraint. To solve In the Interval values ​​on interval inequality constraints The upper bound of the interval is... The lower bound of the interval, For interval values.

2. The method for solving interval-constrained multi-objective optimization problems as described in claim 1, characterized in that, The dynamic constraint threshold Calculation formula include: in, This is the initial dynamic constraint threshold. The maximum value among all the constraint violation values ​​of the initial solutions; The maximum number of iterations, The current iteration number; For dynamic constraint threshold The attenuation rate, .

3. The method for solving interval-constrained multi-objective optimization problems as described in claim 1, characterized in that, Updating the first population and selecting the first offspring, updating the second population and selecting the second offspring, and updating the third population and selecting the third offspring all include: Calculate the inter-crowding distance for each solution in the population, and update the population and select offspring based on the inter-crowding distance; The formula for calculating the congestion distance between intervals includes: The multi-objective optimization problem includes multiple interval objectives, and the objective space formed by the interval objectives includes multiple dominance layers. Each solution is assigned to one of the dominance layers according to its dominance relationship. To solve Left neighbor in the same dominance layer, solution To solve The right neighbor in the same dominance layer; To solve The distance between congested areas To solve and solution The degree of congestion between them To solve and solution The degree of congestion between them To solve The hypervolume in the target space; The formula for calculating the crowding degree between two solutions includes: in, To solve and solution The degree of congestion between them To solve The hypervolume in the target space; the dominance layer includes a non-dominated front, where solutions in the non-dominated front are not dominated by any other solutions. To solve and solution The proportion of the influence region formed in the target space relative to the influence region formed by the entire non-dominated front in the target space. To solve and solution The non-overlapping hypervolume between solutions and solution The non-overlapping proportion of the total hypervolume.

4. The method for solving interval-constrained multi-objective optimization problems as described in claim 3, characterized in that, The formula for calculating the hypervolume includes: in, To solve The hypervolume in the target space, To solve In the Interval values ​​on a given interval target The upper bound of the interval is... The lower bound of the interval, The total number of targets in the interval. For interval values.

5. The method for solving interval-constrained multi-objective optimization problems as described in claim 3, characterized in that, The formula for calculating the influence ratio includes: in, To solve and solution The proportion of the influence region formed in the target space relative to the influence region formed by the entire non-dominated front in the target space. The total number of targets in the interval. For interval value parameters, For the solution in the non-dominated front, the first... The maximum upper bound of a target interval. For the solution in the non-dominated front, the first... The minimum lower bound of the interval target; To solve In the Interval values ​​on a given interval target The upper bound of the interval is... This is the lower bound of the interval; To solve In the Interval values ​​on a given interval target The upper bound of the interval is... This is the lower bound of the interval.

6. The method for solving interval-constrained multi-objective optimization problems as described in claim 3, characterized in that, The formula for calculating the non-overlapping ratio includes: in, To solve and solution The non-overlapping hypervolume between solutions and solution The non-overlapping proportion of the total hypervolume. To solve The hypervolume in the target space, To solve The hypervolume in the target space, To solve and solution Overlapping hypervolume between them To solve In the Interval values ​​on a given interval target To solve In the Interval values ​​on a given interval target Solution and solution In the Overlapping areas on the target intervals The total number of targets in the interval. For interval values.

7. A device for solving interval-constrained multi-objective optimization problems, characterized in that, Used to optimize composite laminate structures, including: The acquisition module is used to acquire the optimization objective of the laminated structure, the interval constraints of the laminated structure, and the uncertain material parameters. The optimization objective includes minimizing the total mass and maximizing the vibration frequency of the composite laminated structure. The interval constraints include ensuring that the vibration frequency interval of the composite laminated structure meets a preset frequency constraint, and ensuring that the total mass interval of the composite laminated structure meets a preset mass constraint. The uncertain material parameters include the Young's modulus interval and density interval of each layer of material in the composite laminated structure. The problem construction module is used to construct a multi-objective optimization problem based on the laminated structure optimization objective, the laminated structure interval constraints, and the uncertain material parameters, and generate multiple initial solutions to the multi-objective optimization problem; the solution to the multi-objective optimization problem includes the thickness of each layer of material in the composite laminated structure; The population generation module is used to generate a first population, a second population, and a third population based on the initial solution of the multi-objective optimization problem. The population update module is used to take the first population as the offspring population for the first iteration and iteratively execute the following steps: based on the union of the first population and the offspring population, update the first population according to at least the dynamic interval constraint dominance relationship, and select multiple solutions from the union of the first population and the offspring population as the first offspring; based on the union of the second population and the offspring population, update the second population according to the interval Pareto dominance relationship, and select multiple solutions from the union of the second population and the offspring population as the second offspring; based on the union of the third population and the offspring population, update the third population according to the dynamic interval constraint dominance relationship and the interval Pareto dominance relationship, and select multiple solutions from the union of the third population and the offspring population as the third offspring; merge the first offspring, the second offspring, and the third offspring to obtain the offspring population for the next iteration; The output module is used to end the iteration in response to reaching the maximum number of iterations and return the first population updated at the last iteration as the final solution set of the multi-objective optimization problem; The output module is also used to determine the thickness of each layer of material in the composite laminate structure based on the solutions in the final solution set; The dynamic interval constraint dominance relationship includes: in the... In the next iteration, for any solution reconciliation If any one of conditions one, two, or three is satisfied, then the solution is... Dynamic interval constraint governing solution The calculation formulas for conditions one, two, and three include: in, This is a dynamic constraint threshold that decays in each iteration based on the current iteration number. ; To solve constraint violation value, To solve The constraint violation value is defined as follows: a constraint violation value less than or equal to zero indicates that the solution satisfies all the lamination structure interval constraints, and a constraint violation value greater than zero indicates that the solution violates at least some of the lamination structure interval constraints. The constraint violation value is proportional to the degree to which the solution violates the lamination structure interval constraints. The symbol used to represent the Pareto dominance relation in an interval. Solution Interval Pareto Dominant Solution and indicate the solution The objective function interval value is better than the solution The objective function interval value, wherein the objective function interval value is the interval value of the total mass of the composite laminate structure and the interval value of the vibration frequency corresponding to the solution; The formula for calculating the constraint violation value includes: in, To solve constraint violation value, To solve In the The constraint violation values ​​of the interval inequality constraints are as follows: the first interval inequality constraint is to make the vibration frequency interval value of the composite laminate structure satisfy the frequency constraint, and the second interval inequality constraint is to make the total mass interval value of the composite laminate structure satisfy the mass constraint. To solve In the Interval values ​​on interval inequality constraints The upper bound of the interval is... The lower bound of the interval, For interval values.

8. An electronic device, characterized in that, include: processor; A memory is provided for storing executable instructions; wherein the processor is configured to read the executable instructions from the memory and execute the executable instructions to implement the interval-constrained multi-objective optimization problem-solving method according to any one of claims 1 to 6.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that, The storage medium stores a computer program, which, when executed by a processor, causes the processor to implement the interval-constrained multi-objective optimization problem-solving method according to any one of claims 1 to 6.

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