Gradient information driven dimensionality perturbation mutation method for high-dimensional expensive optimization problems
Through the gradient information-driven dimensional perturbation variation method, combined with gradient descent and dimensional perturbation operations, the problem of high-dimensional expensive optimization problems is solved by using self-coding and local radial basis function proxy models, and an efficient and accurate optimization design is achieved.
Patent Information
- Application Number
- CN202311482150.6
- Authority / Receiving Office
- CN · China
- Patent Type
- Patents(China)
- Current Assignee / Owner
- Filing Date
- 2023-11-08
- Publication Date
- 2025-02-14
- Estimated Expiration
- 2043-11-08
AI Technical Summary
The existing technology is difficult to effectively solve the problem of high-dimensional expensive optimization, especially in the optimization design process of complex mechanical products, it is difficult for existing algorithms to quickly find the global optimal solution in high-dimensional space.
A gradient information-driven dimensional perturbation variation method is proposed. Combined with gradient descent variance and dimensional perturbation variation operations, efficient search is carried out through self-coded dimensionality reduction and local radial basis function proxy models, which improves optimization efficiency and accuracy.
This method effectively integrates the rapid convergence ability of gradient optimization and the global exploration ability of dimensional perturbation, significantly improves the search efficiency and optimization accuracy of high-dimensional space, and is suitable for design optimization of expensive optimization problems.
Smart Images
Figure CN117556691B_ABST
Abstract
Description
Technical Field
[0001] The present invention relates to the field of computer technology, and more specifically, to a gradient information driven dimensionality perturbation mutation algorithm for high-dimensional expensive optimization problems. Background Art
[0002] Expensive optimization problems are often reflected in the optimization design process of advanced high-end equipment such as complex mechanical products such as the optimization design of wing airfoil composite sandwich structures and the optimization design of the tooth structure of high-performance planetary roller screws. To optimize the design of these complex structures, it is usually necessary to obtain the target impact of the optimization problem through computationally time-consuming (expensive) finite element simulations, such as the flexibility and strain energy simulation of multi-fold sandwich structures or rod sandwich structures, and the fatigue simulation of the rotating shaft of the full-rotation propeller. Under the existing computing conditions (such as a workstation with a processor frequency of 3.5GHz and 64GB of memory), each simulation model analysis takes several minutes to several hours of computing time. Furthermore, the structural optimization design of complex equipment inevitably involves a large number of design parameters, which makes the design space of the original expensive optimization problem huge, which is called a high-dimensional expensive optimization problem.
[0003] In recent years, intelligent optimization algorithms assisted by surrogate models can replace time-consuming actual simulation model evaluations with a large number of computationally inexpensive surrogate model prediction points, thereby significantly reducing the number of time-consuming simulations that the algorithm needs to call during the optimization process, allowing the algorithm to provide design solutions that meet actual work accuracy and stringent requirements within the allowable actual engineering design cycle. However, the effects of such methods are highly sensitive to dimensionality and the prediction accuracy of surrogate models for high-dimensional problems is also significantly reduced, resulting in the inability to effectively solve high-dimensional expensive optimization problems.
[0004] In fact, gradient optimization algorithms are widely used in various large-scale (huge design parameters) optimization problems due to their fast convergence ability, and their convergence effect does not show significant differences with the increase of dimensions. However, such algorithms are not effective for the complex multi-peak characteristics presented by practical problems, and are prone to fall into the local optimal area close to the initial solution. In contrast, dimensional perturbation mutation operations have strong exploration capabilities for high-dimensional design spaces, which can effectively help the algorithm quickly locate global high-potential areas in high-dimensional space, but lack efficient exploration capabilities for local areas.
[0005] In summary, how to effectively integrate the local fast convergence ability of gradient optimization and the global strong exploration ability of dimensional perturbation variation to design an intelligent optimization algorithm assisted by a proxy model with efficient optimization capability for high-dimensional expensive optimization problems is the research focus of this invention, and it also provides a feasible algorithm optimization architecture for the complex structure optimization design of current high-end equipment. Summary of the invention
[0006] In view of the above limitations of the prior art or the need for improved technology, the present invention proposes a gradient information driven dimensional perturbation mutation method for high-dimensional expensive optimization problems. Based on the characteristics of existing engineering optimization problems involving time-consuming and expensive simulations and high-dimensional design variables, an efficient and high-precision gradient information driven dimensional perturbation mutation method for high-dimensional expensive optimization problems is studied and designed. The method effectively combines the fast convergence of gradient descent and the search capability of global exploration of dimensional perturbation, completes the efficient local search of high-dimensional design space through self-encoding dimensionality reduction and reconstruction, improves the optimization efficiency and optimization accuracy, and is suitable for solving high-dimensional expensive optimization problems in the field of engineering optimization.
[0007] To achieve the above object, according to one aspect of the present invention, a gradient information driven dimensional perturbation mutation method for high-dimensional expensive optimization problems is provided, the method comprising the following steps:
[0008] (1) Determine the design space and optimization objective evaluation function of the optimization problem according to the actual engineering problem to be optimized, sample the initial population sample points within the upper and lower bounds of the design space, and evaluate them through the optimization objective function to form an initial population. Design the convergence conditions according to the actual engineering design cycle and optimization accuracy requirements, and initialize the key iteration parameters in the algorithm iteration process;
[0009] (2) Based on the global database constructed so far through optimization, a global radial basis function proxy model is constructed based on the sample similarity screening criterion;
[0010] (3) Considering the characteristics of gradient descent-based mutation operations and dimensional perturbation mutation operations, a gradient information-driven dimensional perturbation mutation operation is constructed that combines the advantages of both, and then a candidate offspring pool is generated through this mutation operation;
[0011] (4) Construct a two-layer offspring selection strategy to determine the true offspring individuals for true expensive evaluation, use the true expensive objective function to evaluate the offspring individuals and update the current population, global database and key iteration parameter information;
[0012] (5) Using the self-encoding algorithm to encode and decode the original space, an efficient local search strategy based on the local radial basis function surrogate model is designed in the encoded low-dimensional design space;
[0013] (6) Update the current population, global database and key iteration parameter information to determine whether the algorithm has reached the convergence condition. If so, output the optimized solution obtained by the algorithm. Otherwise, go to step (2) until the algorithm reaches the convergence condition.
[0014] Furthermore, the global radial basis function proxy model is constructed based on the sample similarity screening criterion, specifically comprising the following steps:
[0015] (21) calculating the Euclidean distance between each evaluated sample and the remaining samples in the global database;
[0016] (22) When there are two samples whose Euclidean distance is less than the minimum distance threshold, the two samples are similar, and one of the two samples needs to be randomly deleted from the global database. Then, all the samples in the filtered global database are used to construct the global radial basis function proxy model, where the calculation formula of the minimum distance threshold is:
[0017]
[0018] In the formula, d represents the number of design variables in the design space; ub is the upper bound of each dimension in the design space; lb is the lower bound of each dimension in the design space;
[0019] (23) After determining the global proxy model modeling sample points, a global radial basis function proxy model is constructed. The mathematical expression of the global radial basis function proxy model is:
[0020]
[0021] In the formula, λ i is the i-th individual x i The corresponding weight parameter, φ(||xx i ||) represents the i-th individual x i Basis function of ||xx i || represents x and x i The Euclidean distance between them; p(x) represents a linear polynomial function used for correction.
[0022] Furthermore, the calculation formula for constructing the gradient information-driven dimensional perturbation mutation operation that combines the advantages of both is as follows:
[0023]
[0024] In the formula, v i,m Gx represents the m-th dimension component of the generated i-th variant individual; i,m is the mutated individual obtained by gradient descent mutation on the current individual; Z m means the mean is 0 and the standard deviation is 0.2 (ub m -lb m )'s normally distributed random number; ub m With lb m are the upper and lower bounds of the design space in the mth dimension respectively; rand is a uniformly distributed random number in the interval [0,1].
[0025] Furthermore, the calculation formula of the gradient descent mutation operation is as follows:
[0026]
[0027] In the formula, x i is the current individual; SS is the gradient descent step length, and its value is SS = 0.05 (ub m -lb m ); is the gradient vector of the current individual calculated according to the finite difference algorithm.
[0028] Furthermore, for each current individual, the candidate offspring pool includes all mutated individuals generated by the dimensional perturbation mutation operation driven by gradient information for the current individual.
[0029] Furthermore, the specific steps of the two-layer offspring selection strategy are as follows:
[0030] (41) For each candidate offspring pool generated by each individual in the current population, calculate the minimum Euclidean distance between each candidate offspring individual in the candidate offspring pool and other candidate offspring;
[0031] (42) Taking the minimum Euclidean distance of candidate offspring individuals and the radial basis function prediction value as two independent objectives, all non-dominated candidate offspring individuals are obtained according to the multi-objective fast non-dominated sorting method to form a temporary offspring set for the current individual;
[0032] (43) merging the temporary offspring sets corresponding to each individual in the current population, and calculating the minimum Euclidean distance between each temporary offspring in the merged individual set and each evaluated individual;
[0033] (44) Taking the minimum Euclidean distance of temporary offspring individuals and the radial basis function prediction value as two independent objectives, all non-dominated temporary offspring individuals are obtained according to the multi-objective fast non-dominated sorting method to form a set of offspring individuals that are truly used for expensive target evaluation.
[0034] Furthermore, the self-encoding algorithm is used to encode and decode the original space, and the specific steps are as follows:
[0035] (51) The encoding operation of the autoencoding algorithm constructs a latent space by capturing the main information of the original input data as much as possible. The specific expression is as follows:
[0036]
[0037] In the formula, σ(·) is the activation function, a j The bias term for calculating the jth node information in the encoding stage, w ij is the weight coefficient of the i-th dimension data when calculating the j-th node data information in the encoding stage, d is the dimension of the original input data, x iis the information of the i-th dimension of the original input data, h j is the j-th dimension information corresponding to the j-th node in the encoded hidden space;
[0038] (52) The reconstruction error is the sum of the errors between the original input data and its corresponding decoded output data. The decoding operation of the autoencoding algorithm ensures the least information loss during the entire encoding and decoding process by minimizing the reconstruction error as much as possible. The specific expression is as follows:
[0039]
[0040] In the formula, a′ i The bias term for calculating the i-th node information in the decoding stage, w′ ji is the weight coefficient of the j-th dimension data when calculating the i-th dimension data information corresponding to the i-th node in the decoding stage, m is the dimension of the hidden space data after the above encoding, and h is j is the j-th dimension information corresponding to the j-th node in the above encoded hidden space, x′ i It is the information of the reconstructed i-th dimension data in the decoding stage.
[0041] Furthermore, the calculation method of the reconstruction error in the decoding operation of the autoencoding algorithm is as follows:
[0042]
[0043] In the formula, x is all the original input data, and x′ is all the reconstructed data obtained after encoding and decoding by the autoencoder.
[0044] Furthermore, the specific steps of designing an efficient local search strategy based on a local radial basis function proxy model in the encoded low-dimensional design space are as follows:
[0045] (53) The top c individuals are selected from all evaluated individuals according to the target value sorting from small to large to form the first local proxy model modeling sample set, where: n is the number of design variables;
[0046] (54) The top n individuals are selected from all evaluated individuals according to the Euclidean distance from the current optimal individual from small to large, forming the second local proxy model modeling sample set, where: n is the number of design variables;
[0047] (55) Encode the two modeling sample sets respectively according to the encoding operation of the self-encoding algorithm to form two low-dimensional local design spaces;
[0048] (56) Local radial basis function surrogate models are constructed in two low-dimensional design spaces;
[0049] (57) The differential evolution algorithm is used to solve the optimal individuals of the corresponding local radial basis function surrogate model in two low-dimensional design spaces;
[0050] (58) According to the decoding operation of the autoencoding algorithm, the optimal individuals of the proxy models in the two low-dimensional spaces are reconstructed to obtain their corresponding individuals in the original space.
[0051] Furthermore, the modeling and local optimization regions of the two local radial basis function proxy models are composed of local regions with the maximum value of all modeling sample points in each dimension as the upper bound and the minimum value as the lower bound. The specific mathematical expressions are as follows:
[0052]
[0053]
[0054] In the formula, ubl and lbl are the upper and lower bounds of the local area respectively. is the value of the nlth individual in the dth dimension.
[0055] Furthermore, the specific steps of updating the current population, global database and key iteration parameter information are as follows:
[0056] (61) All evaluated individuals in the current iteration are merged with the individuals in the current population, and the top N better individuals are selected from small to large according to the target value to form the updated current population;
[0057] (62) adding all evaluated individuals in the current iteration to the current global database to form an updated global database;
[0058] (63) Update the current number of iterations and the current optimal individual to prepare for the next generation.
[0059] In summary, compared with the prior art, the above technical solution conceived by the present invention has the following advantages over the prior art:
[0060] Beneficial effects:
[0061] 1. Based on the sample similarity screening criterion, the modeling sample points are selected from the global database constructed so far from the optimization to construct a global radial basis function proxy model, which effectively avoids the irreversibility of the radial basis function correlation coefficient matrix caused by similar individuals and improves the construction accuracy of the proxy model.
[0062] 2. Based on the gradient descent mutation operation, a gradient information-driven dimensional perturbation mutation operation is constructed, which effectively integrates the fast convergence of gradient descent and the global search capability of dimensional perturbation, improves the overall potential and diversity of candidate offspring individuals, significantly reduces the pre-screening pressure of the proxy model, and improves the optimization efficiency of the algorithm.
[0063] 3. A two-layer offspring selection strategy that comprehensively considers the potential and distribution characteristics of offspring individuals is adopted to screen the real offspring individuals for real evaluation, avoiding the probability of the algorithm falling into local optimality. At the same time, it effectively reduces the number of calls for real expensive simulation evaluation.
[0064] 4. The self-encoding algorithm is used to encode and decode the original space, and an efficient local search strategy based on the local radial basis function proxy model is designed in the encoded low-dimensional design space. This effectively alleviates the difficulty of exploring the huge design space faced by local search in the original high-dimensional space, improves the optimization efficiency of local search, and accelerates the local exploration capability of the algorithm.
[0065] 5. The present invention can accurately predict and search high-dimensional design spaces, improve the accuracy of the optimization solution, and can be widely used in the design optimization of expensive problems, which is conducive to promotion and application. BRIEF DESCRIPTION OF THE DRAWINGS
[0066] Figure 1 A flow chart of the gradient information driven dimensional perturbation mutation method for high-dimensional expensive optimization problems provided by the present invention;
[0067] Figure 2 yes Figure 1 Schematic diagram of the overall framework of the autoencoder algorithm involved in the gradient information driven dimensional perturbation mutation method for high-dimensional expensive optimization problems;
[0068] Figure 3 It is a schematic diagram of the design space for the optimal design of a rod core sandwich structure involving a gradient information driven dimensional perturbation variation method for high-dimensional expensive optimization problems provided by an embodiment of the present invention. DETAILED DESCRIPTION
[0069] In order to make the purpose, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below can be combined with each other as long as they do not conflict with each other.
[0070] See also Figure 1 and Figure 2The present invention provides a gradient information driven dimensional perturbation mutation method for high-dimensional expensive optimization problems, which is applicable to engineering optimization problems. Specifically, the method mainly includes the following steps:
[0071] Step 1: Determine the design space and optimization target evaluation function of the optimization problem according to the actual engineering problem to be optimized, sample within the upper and lower bounds of the design space to obtain the initial population sample points, and evaluate through the optimization target function to form the initial population. Design the convergence conditions according to the actual engineering design cycle and optimization accuracy requirements, and initialize the key iteration parameters in the algorithm iteration process.
[0072] The initial population is obtained by Latin hypercube sampling, and the specific steps are as follows:
[0073] (11) Each dimension of the design space is evenly divided into N intervals, where N is the initial population size;
[0074] (12) For each dimension, randomly select an interval and generate a uniformly distributed random number in the selected interval according to a uniform distribution;
[0075] (13) The random numbers of all dimensions are combined to form a vector, which is the individual of the current population;
[0076] (14) Repeat steps (12)-(13) N-1 times until N population individuals are obtained.
[0077] Step 2: For the global database constructed so far through optimization, a global radial basis function proxy model is constructed based on sample similarity screening criteria.
[0078] The global radial basis function proxy model is constructed based on the sample similarity screening criterion, and specifically comprises the following steps:
[0079] (21) calculating the Euclidean distance between each evaluated sample and the remaining samples in the global database;
[0080] (22) When there are two samples whose Euclidean distance is less than the minimum distance threshold, the two samples are similar, and one of the two samples needs to be randomly deleted from the global database. Then, all the samples in the filtered global database are used to construct the global radial basis function proxy model, where the calculation formula of the minimum distance threshold is:
[0081]
[0082] Where d represents the number of design variables in the design space; ub is the upper bound of each dimension in the design space; lb is the lower bound of each dimension in the design space.
[0083] (23) After determining the global proxy model modeling sample points, a global radial basis function proxy model is constructed. The mathematical expression of the global radial basis function proxy model is:
[0084]
[0085] In the formula, λ i is the i-th individual x i The corresponding weight parameter, φ(||xx i ||) represents the i-th individual x i Basis function of ||xx i || represents x and x i The Euclidean distance between them; p(x) represents a linear polynomial function used for correction.
[0086] Step three, considering the characteristics of gradient descent-based mutation operations and dimensional perturbation mutation operations, construct a gradient information-driven dimensional perturbation mutation operation that combines the advantages of both, and then generate a candidate offspring pool through this mutation operation.
[0087] The calculation formula of the gradient descent mutation operation is as follows:
[0088]
[0089] In the formula, Gx i is the current individual x i The mutant individuals are generated by gradient descent; SS is the gradient descent step length, and its value is SS = 0.05 (ub m -lb m ); is the gradient vector of the current individual calculated according to the finite difference algorithm.
[0090] After the gradient descent mutation operation, the calculation formula of the gradient information-driven dimensional perturbation mutation operation is as follows:
[0091]
[0092] In the formula, v i,m Gx represents the m-th dimension component of the generated i-th variant individual; i,m is the mutated individual obtained by gradient descent mutation on the current individual; Z m means the mean is 0 and the standard deviation is 0.2 (ub m -lb m )'s normally distributed random number; ub m With lb m are the upper and lower bounds of the design space in the mth dimension respectively; rand is a uniformly distributed random number in the interval [0,1].
[0093] Specifically, for each current individual, the candidate offspring pool includes all mutated individuals generated by the dimensional perturbation mutation operation driven by gradient information for the current individual.
[0094] Step 4: Construct a two-layer offspring selection strategy to determine the true offspring individuals for real expensive evaluation, use the real expensive objective function to evaluate the offspring individuals and update the current population, global database and key iteration parameter information.
[0095] Specifically, the specific steps of the two-layer offspring selection strategy are as follows:
[0096] (41) For each candidate offspring pool generated by each individual in the current population, calculate the minimum Euclidean distance between each candidate offspring individual in the candidate offspring pool and other candidate offspring;
[0097] (42) Taking the minimum Euclidean distance of candidate offspring individuals and the radial basis function prediction value as two independent objectives, all non-dominated candidate offspring individuals are obtained according to the multi-objective fast non-dominated sorting method to form a temporary offspring set for the current individual;
[0098] (43) merging the temporary offspring sets corresponding to each individual in the current population, and calculating the minimum Euclidean distance between each temporary offspring in the merged individual set and each evaluated individual;
[0099] (44) Taking the minimum Euclidean distance of temporary offspring individuals and the radial basis function prediction value as two independent objectives, all non-dominated temporary offspring individuals are obtained according to the multi-objective fast non-dominated sorting method to form a set of offspring individuals that are truly used for expensive target evaluation.
[0100] Step five, use the self-encoding algorithm to encode and decode the original space, and design an efficient local search strategy based on the local radial basis function proxy model in the encoded low-dimensional design space.
[0101] The specific steps of the encoding and decoding operations of the self-encoding algorithm are as follows:
[0102] (51) The encoding operation of the autoencoding algorithm constructs a latent space by capturing the main information of the original input data as much as possible. The specific expression is as follows:
[0103]
[0104] In the formula, σ(·) is the activation function, a j The bias term for calculating the jth node information in the encoding stage, w ij is the weight coefficient of the i-th dimension data when calculating the j-th node data information in the encoding stage, d is the dimension of the original input data, x iis the information of the i-th dimension of the original input data, h j is the j-th dimension information corresponding to the j-th node in the encoded hidden space.
[0105] (52) The reconstruction error is the sum of the errors between the original input data and its corresponding decoded output data. The decoding operation of the autoencoding algorithm ensures the least information loss during the entire encoding and decoding process by minimizing the reconstruction error as much as possible. The specific expression is as follows:
[0106]
[0107] In the formula, a′ i The bias term for calculating the i-th node information in the decoding stage, w′ ji is the weight coefficient of the j-th dimension data when calculating the data information of the i-th node (i-th dimension) in the decoding stage, m is the dimension of the hidden space data after the above encoding, and h is j is the j-th dimension information corresponding to the j-th node in the above encoded hidden space, x′ i It is the information of the reconstructed i-th dimension data in the decoding stage.
[0108] Specifically, the calculation method of the reconstruction error in the decoding operation of the autoencoding algorithm is as follows:
[0109]
[0110] In the formula, x is all the original input data, and x′ is all the reconstructed data obtained after encoding and decoding by the autoencoder.
[0111] After encoding by the self-encoding algorithm, an efficient local search strategy based on a local radial basis function surrogate model is designed in the encoded low-dimensional design space. The specific steps are as follows:
[0112] (53) The top c individuals are selected from all evaluated individuals according to the target value sorting from small to large to form the first local proxy model modeling sample set, where: n is the number of design variables;
[0113] (54) The top c individuals are selected from all evaluated individuals according to the Euclidean distance from the current optimal individual from small to large, forming the second local proxy model modeling sample set, where: n is the number of design variables;
[0114] (55) Encode the two modeling sample sets respectively according to the encoding operation of the self-encoding algorithm to form two low-dimensional local design spaces;
[0115] (56) Local radial basis function surrogate models are constructed in two low-dimensional design spaces;
[0116] (57) The differential evolution algorithm is used to solve the optimal individuals of the corresponding local radial basis function surrogate model in two low-dimensional design spaces;
[0117] (58) According to the decoding operation of the autoencoding algorithm, the optimal individuals of the proxy models in the two low-dimensional spaces are reconstructed to obtain their corresponding individuals in the original space.
[0118] Specifically, the modeling and local optimization regions of the two local radial basis function proxy models are composed of local regions with the maximum value of all modeling sample points in each dimension as the upper bound and the minimum value as the lower bound. The specific mathematical expressions are as follows:
[0119]
[0120]
[0121] In the formula, ubl and lbl are the upper and lower bounds of the local area, respectively. is the value of the nlth individual in the dth dimension.
[0122] Step 6: Update the current population, global database and key iteration parameter information to determine whether the algorithm has reached the convergence condition. If it has converged, output the optimized solution obtained by the algorithm. Otherwise, go to step (2) until the algorithm reaches the convergence condition.
[0123] The specific steps for updating the current population, global database and key iteration parameter information are as follows:
[0124] (61) All evaluated individuals in the current iteration are merged with the individuals in the current population, and the top N better individuals are selected from small to large according to the target value to form the updated current population;
[0125] (62) adding all evaluated individuals in the current iteration to the current global database to form an updated global database;
[0126] (63) Update the current number of iterations and the current optimal individual to prepare for the next generation.
[0127] Example
[0128] See also Figure 3 In this embodiment, the gradient information driven dimensional perturbation mutation algorithm for high-dimensional expensive optimization problems provided in this embodiment is described by taking the optimization design of a rod core sandwich structure as an example. Figure 3 In the design, the coordinates and thickness information of the bar are used as design parameters, that is, each bar contains five design variables: The entire optimization design includes four bars and all bars have the same thickness, so this optimization problem contains 17 design variables. The entire optimization problem design domain is a rectangular parallelepiped, and the corresponding height × length × width is The discretized finite element mesh size is 22×80×80. Therefore, the height×length×width of the real design domain corresponding to the optimization design of the rod sandwich structure is 0.5×2×2, and is discretized to 20×80×80. Furthermore, the entire structure is fixed by the plane joint constraints at the bottom corners, and a distributed vertical load P=1 / 162 is applied along the height direction at 1 / 4 and 3 / 4 of the length of the upper surface of the structure. In the specific optimization design process, the optimization goal is to minimize flexibility, and the optimization constraints include a volume fraction less than 0.35. In this optimization problem, the dichotomy method is used to determine that all individuals in the optimization process satisfy the volume constraint, where the maximum thickness of the volume constraint is set to 2 and the minimum thickness is set to 0.01. In summary, the optimization problem expression for the rod sandwich structure is as follows:
[0129]
[0130] Where f and t are the Neumann boundary Γ t The body force density on the surface and the preset surface traction; is the Dirichlet frontier Γ u The prescribed displacement on H(·) is the Heaviside function; φ(x; D) is the multi-component topological description function constructed based on the rod structure; q is the penalty factor; E is the fourth-order elastic tensor of the isotropic solid material with given Young's modulus E and Poisson's ratio υ; ε is the second-order linear strain tensor; u is the displacement field; U ad is the permissible set of virtual displacement vectors v; is the upper bound of the available volume of solid material; is the thickness of the lth member; Binary Search(·) is the binary search algorithm mark; and They are respectively the maximum and minimum values of the thickness of the lth member during the execution of the binary algorithm.
[0131] The gradient information driven dimensionality perturbation mutation algorithm (SADE for short) for high-dimensional expensive optimization problems provided in this embodiment mainly includes the following steps:
[0132] Step 1: Determine the design space and optimization target evaluation function of the optimization problem according to the actual engineering problem to be optimized, sample within the upper and lower bounds of the design space to obtain the initial population sample points, and evaluate through the optimization target function to form the initial population. Design the convergence conditions according to the actual engineering design cycle and optimization accuracy requirements, and initialize the key iteration parameters in the algorithm iteration process.
[0133] The initial population is obtained by Latin hypercube sampling, and the specific steps are as follows:
[0134] (11) Each dimension of the design space is evenly divided into N intervals, where N is the initial population size;
[0135] (12) For each dimension, randomly select an interval and generate a uniformly distributed random number in the selected interval according to a uniform distribution;
[0136] (13) The random numbers of all dimensions are combined to form a vector, which is the individual of the current population;
[0137] (14) Repeat steps (12)-(13) N-1 times until N population individuals are obtained.
[0138] Step 2: For the global database constructed so far through optimization, a global radial basis function proxy model is constructed based on sample similarity screening criteria.
[0139] The global radial basis function proxy model is constructed based on the sample similarity screening criterion, and specifically comprises the following steps:
[0140] (21) calculating the Euclidean distance between each evaluated sample and the remaining samples in the global database;
[0141] (22) When there are two samples whose Euclidean distance is less than the minimum distance threshold, the two samples are similar, and one of the two samples needs to be randomly deleted from the global database. Then, all the samples in the filtered global database are used to construct the global radial basis function proxy model, where the calculation formula of the minimum distance threshold is:
[0142]
[0143] Where d represents the number of design variables in the design space; ub is the upper bound of each dimension in the design space; lb is the lower bound of each dimension in the design space.
[0144] (23) After determining the global proxy model modeling sample points, a global radial basis function proxy model is constructed. The mathematical expression of the global radial basis function proxy model is:
[0145]
[0146] In the formula, λ i is the i-th individual x i The corresponding weight parameter, φ(||xx i ||) represents the i-th individual x i Basis function of ||xx i || represents x and xi The Euclidean distance between them; p(x) represents a linear polynomial function used for correction.
[0147] Step three, considering the characteristics of gradient descent-based mutation operations and dimensional perturbation mutation operations, construct a gradient information-driven dimensional perturbation mutation operation that combines the advantages of both, and then generate a candidate offspring pool through this mutation operation.
[0148] The calculation formula of the gradient descent mutation operation is as follows:
[0149]
[0150] In the formula, Gx i is the current individual x i The mutant individuals are generated by gradient descent; SS is the gradient descent step length, and its value is SS = 0.05 (ub m -lb m ); is the gradient vector of the current individual calculated according to the finite difference algorithm.
[0151] After the gradient descent mutation operation, the calculation formula of the gradient information-driven dimensional perturbation mutation operation is as follows:
[0152]
[0153] In the formula, v i,m Gx represents the m-th dimension component of the generated i-th variant individual; i,m is the mutated individual obtained by gradient descent mutation on the current individual; Z m means the mean is 0 and the standard deviation is 0.2 (ub m -lb m )'s normally distributed random number; ub m With lb m are the upper and lower bounds of the design space in the mth dimension respectively; rand is a uniformly distributed random number in the interval [0,1].
[0154] Specifically, for each current individual, the candidate offspring pool includes all mutated individuals generated by the dimensional perturbation mutation operation driven by gradient information for the current individual.
[0155] Step 4: Construct a two-layer offspring selection strategy to determine the true offspring individuals for real expensive evaluation, use the real expensive objective function to evaluate the offspring individuals and update the current population, global database and key iteration parameter information.
[0156] Specifically, the specific steps of the two-layer offspring selection strategy are as follows:
[0157] (41) For each candidate offspring pool generated by each individual in the current population, calculate the minimum Euclidean distance between each candidate offspring individual in the candidate offspring pool and other candidate offspring;
[0158] (42) Taking the minimum Euclidean distance of candidate offspring individuals and the radial basis function prediction value as two independent objectives, all non-dominated candidate offspring individuals are obtained according to the multi-objective fast non-dominated sorting method to form a temporary offspring set for the current individual;
[0159] (43) merging the temporary offspring sets corresponding to each individual in the current population, and calculating the minimum Euclidean distance between each temporary offspring in the merged individual set and each evaluated individual;
[0160] (44) Taking the minimum Euclidean distance of temporary offspring individuals and the radial basis function prediction value as two independent objectives, all non-dominated temporary offspring individuals are obtained according to the multi-objective fast non-dominated sorting method to form a set of offspring individuals that are truly used for expensive target evaluation.
[0161] Step five, use the self-encoding algorithm to encode and decode the original space, and design an efficient local search strategy based on the local radial basis function proxy model in the encoded low-dimensional design space.
[0162] The specific steps of the encoding and decoding operations of the self-encoding algorithm are as follows:
[0163] (51) The encoding operation of the autoencoding algorithm constructs a latent space by capturing the main information of the original input data as much as possible. The specific expression is as follows:
[0164]
[0165] In the formula, σ(·) is the activation function, a j The bias term for calculating the jth node information in the encoding stage, w ij is the weight coefficient of the i-th dimension data when calculating the j-th node data information in the encoding stage, d is the dimension of the original input data, x i is the information of the i-th dimension of the original input data, h j is the j-th dimension information corresponding to the j-th node in the encoded hidden space.
[0166] (52) The reconstruction error is the sum of the errors between the original input data and its corresponding decoded output data. The decoding operation of the autoencoding algorithm ensures the least information loss during the entire encoding and decoding process by minimizing the reconstruction error as much as possible. The specific expression is as follows:
[0167]
[0168] In the formula, a i′ is the bias term for calculating the i-th node information in the decoding stage, w′ ji is the weight coefficient of the j-th dimension data when calculating the data information of the i-th node (i-th dimension) in the decoding stage, m is the dimension of the hidden space data after the above encoding, and h is j is the j-th dimension information corresponding to the j-th node in the above encoded hidden space, x i ′ is the information of the i-th dimension data after reconstruction in the decoding stage.
[0169] Specifically, the calculation method of the reconstruction error in the decoding operation of the autoencoding algorithm is as follows:
[0170]
[0171] In the formula, x is all the original input data, and x′ is all the reconstructed data obtained after encoding and decoding by the autoencoder.
[0172] After encoding by the self-encoding algorithm, an efficient local search strategy based on a local radial basis function surrogate model is designed in the encoded low-dimensional design space. The specific steps are as follows:
[0173] (53) The top c individuals are selected from all evaluated individuals according to the target value sorting from small to large to form the first local proxy model modeling sample set, where: n is the number of design variables;
[0174] (54) The top c individuals are selected from all evaluated individuals according to the Euclidean distance from the current optimal individual from small to large, forming the second local proxy model modeling sample set, where: n is the number of design variables;
[0175] (55) Encode the two modeling sample sets respectively according to the encoding operation of the self-encoding algorithm to form two low-dimensional local design spaces;
[0176] (56) Local radial basis function surrogate models are constructed in two low-dimensional design spaces;
[0177] (57) The differential evolution algorithm is used to solve the optimal individuals of the corresponding local radial basis function surrogate model in two low-dimensional design spaces;
[0178] (58) According to the decoding operation of the autoencoding algorithm, the optimal individuals of the proxy models in the two low-dimensional spaces are reconstructed to obtain their corresponding individuals in the original space.
[0179] Specifically, the modeling and local optimization regions of the two local radial basis function proxy models are composed of local regions with the maximum value of all modeling sample points in each dimension as the upper bound and the minimum value as the lower bound. The specific mathematical expressions are as follows:
[0180]
[0181]
[0182] In the formula, ubl and lbl are the upper and lower bounds of the local area respectively. is the value of the nlth individual in the dth dimension.
[0183] Step 6: Update the current population, global database and key iteration parameter information to determine whether the algorithm has reached the convergence condition. If it has converged, output the optimized solution obtained by the algorithm. Otherwise, go to step (2) until the algorithm reaches the convergence condition.
[0184] The specific steps for updating the current population, global database and key iteration parameter information are as follows:
[0185] (61) All evaluated individuals in the current iteration are merged with the individuals in the current population, and the top N better individuals are selected from small to large according to the target value to form the updated current population;
[0186] (62) adding all evaluated individuals in the current iteration to the current global database to form an updated global database;
[0187] (63) Update the current number of iterations and the current optimal individual to prepare for the next generation.
[0188] In order to further illustrate this embodiment, the method of this embodiment is compared with the MMA algorithm, and the maximum number of calls of the real simulation model of this embodiment is set to 1000. The experimental results are shown in Table 1. Within the same time cost limit, the method of this embodiment obtains the optimal solution, which reflects the effectiveness of this embodiment in improving the optimization accuracy. It can be considered that the method of this embodiment has good performance in high-dimensional problems with expensive calculations.
[0189] Table 1 Comparison of optimization results of different algorithms
[0190] algorithm This algorithm (SADE) MMA Target value 1.2606 1.2795
[0191] The present invention provides a gradient information driven dimensional perturbation mutation algorithm for high-dimensional expensive optimization problems. The method effectively combines the advantages of gradient descent mutation and dimensional perturbation mutation operations, improves the search efficiency for high-dimensional space with the assistance of the autoencoder algorithm, improves the optimization accuracy and efficiency of the algorithm, reduces the time cost, and has strong applicability to high-dimensional problems.
[0192] It will be easily understood by those skilled in the art that the above are only preferred embodiments of the present invention and are not intended to limit the present invention. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the present invention.
Claims
1. A gradient information driven dimensional perturbation mutation method for high-dimensional expensive optimization problems, characterized in that: The optimization design of a rod-core sandwich structure is used to illustrate the gradient information-driven dimensional perturbation mutation algorithm for high-dimensional expensive optimization problems. The rod coordinates and thickness information are used as design parameters, that is, each rod contains five design variables: D l =( , , , , ) T , the entire optimization design includes four bars and all bars have the same thickness, so this optimization problem contains 17 design variables; In the specific optimization design process, the optimization goal is to minimize flexibility, and the optimization constraints include that the volume fraction is less than 0.35; by effectively integrating the local fast convergence ability of gradient optimization and the global strong exploration ability of dimensional perturbation variation, an intelligent optimization algorithm assisted by a proxy model with efficient optimization ability for high-dimensional expensive optimization problems is designed; the method includes: (1) The design space and target evaluation function of the optimization problem are determined based on the actual engineering problem to be optimized in the optimization design of the bar-core sandwich structure. In the specific optimization design process, the optimization goal is to minimize flexibility. Sampling is performed within the upper and lower bounds of the design space with the bar coordinates and thickness information as design parameters to obtain the initial population sample points, and the initial population is formed by evaluating the minimum flexibility target function. The convergence conditions are designed based on the actual engineering design cycle and optimization accuracy requirements of the optimization design of the bar-core sandwich structure, and the key iteration parameters in the algorithm iteration process are initialized. (2) Based on the global database constructed so far through optimization, a global radial basis function proxy model is constructed based on the sample similarity screening criterion; (3) Considering the characteristics of gradient descent-based mutation operations and dimensional perturbation mutation operations, a gradient information-driven dimensional perturbation mutation operation is constructed that combines the advantages of both, and then a candidate offspring pool is generated through this mutation operation; (4) Construct a two-layer offspring selection strategy to determine the true offspring individuals for true expensive evaluation, use the true expensive objective function to evaluate the offspring individuals and update the current population, global database and key iteration parameter information; (5) The original space is encoded and decoded using a self-encoding algorithm, and an efficient local search strategy based on a local radial basis function surrogate model is designed in the encoded low-dimensional design space; (6) Update the current population, global database and key iteration parameter information to determine whether the algorithm has reached the convergence condition. If so, output the optimized solution obtained by the algorithm. Otherwise, go to step (2) until the algorithm reaches the convergence condition. The specific steps of designing an efficient local search strategy based on a local radial basis function surrogate model in the encoded low-dimensional design space are as follows: (53) Sort by target value from small to large and select the top candidates from all evaluated individuals. Individuals form the first local proxy model modeling sample set, where , is the number of design variables; (54) Sort the top individuals from all evaluated individuals according to their Euclidean distance to the current best individual from small to large. Individuals form the second local proxy model modeling sample set, where , is the number of design variables; (55) Encode the two modeling sample sets respectively according to the encoding operation of the self-encoding algorithm to form two low-dimensional local design spaces; (56) Local radial basis function surrogate models are constructed in two low-dimensional design spaces respectively; (57) The differential evolution algorithm is used to solve the optimal individuals of the corresponding local radial basis function surrogate model in two low-dimensional design spaces; (58) According to the decoding operation of the autoencoding algorithm, the optimal individuals of the proxy models in the two low-dimensional spaces are reconstructed to obtain their corresponding individuals in the original space.
2. The method according to claim 1, characterized in that The global radial basis function proxy model is constructed based on the sample similarity screening criterion, and specifically includes the following steps: (21) calculating the Euclidean distance between each evaluated sample and the remaining samples in the global database; (22) When there are two samples whose Euclidean distance is less than the minimum distance threshold, the two samples are similar, and one of the two samples needs to be randomly deleted from the global database. Then, all the samples in the filtered global database are used to construct the global radial basis function proxy model, where the minimum distance threshold is calculated as: In the formula, represents the number of design variables in the design space; is the upper bound of each dimension in the design space; is the lower bound of each dimension in the design space; (23) After determining the global proxy model modeling sample points, a global radial basis function proxy model is constructed. The mathematical expression of the global radial basis function proxy model is: In the formula, For the Individual The corresponding weight parameter, Indicates Individual The basis functions of express and The Euclidean distance between represents a linear polynomial function used for correction.
3. The method according to claim 1, characterized in that The calculation formula for constructing the gradient information-driven dimensional perturbation mutation operation that combines the advantages of both is as follows: In the formula, Indicates the generated The variant individual Dimensional components; is the mutated individual obtained by performing gradient descent mutation on the current individual; means the mean is 0 and the standard deviation is Normally distributed random numbers; and The design space is Upper and lower bounds on dimensions; is a uniformly distributed random number in the interval [0,1].
4. The method according to claim 3, characterized in that The calculation formula of the gradient descent mutation operation is as follows: In the formula, For the current individual; is the gradient descent step size, and its value is ; is the gradient vector of the current individual calculated according to the finite difference algorithm.
5. The method according to claim 1, characterized in that The specific steps of the two-layer offspring selection strategy are as follows: (41) For each candidate offspring pool generated by each individual in the current population, calculate the minimum Euclidean distance between each candidate offspring individual in the candidate offspring pool and other candidate offspring; (42) Taking the minimum Euclidean distance of candidate offspring individuals and the radial basis function prediction value as two independent objectives, all non-dominated candidate offspring individuals are obtained according to the multi-objective fast non-dominated sorting method to form a temporary offspring set for the current individual; (43) Merge the temporary offspring sets corresponding to each individual in the current population, and calculate the minimum Euclidean distance between each temporary offspring in the merged individual set and each evaluated individual; (44) Taking the minimum Euclidean distance of temporary offspring individuals and the predicted value of radial basis function as two independent objectives, all non-dominated temporary offspring individuals are obtained according to the multi-objective fast non-dominated sorting method to form a set of offspring individuals truly used for expensive target evaluation.
6. The method according to claim 1, characterized in that The self-encoding algorithm is used to encode and decode the original space, and the specific steps are as follows: (51) The encoding operation of the self-encoding algorithm constructs a latent space by capturing the main information of the original input data. The specific expression is as follows: In the formula, is the activation function, Calculate the encoding phase The bias term of node information, Calculate the encoding phase Node data information The weight coefficient of the dimensional data, is the dimension of the original input data, The original input data Dimensional data information, is the first hidden space after encoding The node corresponding to Dimensional information; (52) The reconstruction error is the sum of the errors between the original input data and its corresponding decoded output data. The decoding operation of the autoencoder algorithm ensures the least information loss during the entire encoding and decoding process by minimizing the reconstruction error. The specific expression is as follows: In the formula, Calculate the first The bias term of node information, Calculate the first Node data information The weight coefficient of the dimensional data, is the dimension of the hidden spatial data after the above encoding, is the first hidden space after the above encoding The node corresponding to Dimensional information, To achieve the reconstruction in the decoding stage dimensional data information.
7. The method according to claim 6, characterized in that The calculation method of the reconstruction error in the decoding operation of the autoencoding algorithm is as follows: In the formula, For all original input data, It is all the reconstructed data obtained after encoding and decoding from the encoder.
8. The method according to claim 1, characterized in that The specific steps for updating the current population, global database and key iteration parameter information are as follows: (61) Combine all evaluated individuals in the current iteration with the individuals in the current population, and select the top individuals according to the target value from small to large. better individuals to form the updated current population; (62) adding all evaluated individuals in the current iteration to the current global database to form an updated global database; (63) Update the current number of iterations and the current optimal individual to prepare for the next generation.