CN101887478A

CN101887478A - Sequence radial basis function agent model-based high-efficiency global optimization method

Info

Publication number: CN101887478A
Application number: CN2010102298394A
Authority: CN
Inventors: 龙腾; 刘莉; 彭磊; 李怀建; 王正平
Original assignee: Beijing Institute of Technology BIT
Current assignee: Beijing Institute of Technology BIT
Priority date: 2010-07-19
Filing date: 2010-07-19
Publication date: 2010-11-17

Abstract

The invention relates to a sequence radial basis function agent model-based high-efficiency global optimization method, and belongs to the technical field of multidisciplinary optimization in engineering design. The method comprises the following steps of: according to initial conditions given by a user, selecting sample points in a primary iteration design space, calculating a response value of a true model, constructing a radial basis function agent model, calculating the current optimal solution of the radial basis function (RBF) agent model, calculating a response value of the possible optimal solution of the current iteration in the true model, judging whether the global optimization method meets the convergence criterion, determining an important sampling space of the next iteration, increasing new sample points in the constructed important sampling space by an experimental design calculation method, saving the new sample points in a design sample point database and making k equal to k+1, and switching to the constructed radial basis function agent model for the next iteration. Through the method, the true models in the engineering design and analysis software are approximated, and the optimization design of the true models only takes several or dozens of seconds, so the period of the engineering optimization design is greatly shortened, the design cost is greatly saved and the efficiency is obviously improved.

Description

Efficient global optimization method based on sequence radial basis function agent model

Technical Field

The invention relates to an efficient global optimization method based on a sequence radial basis agent model, and belongs to the technical field of multidisciplinary optimization in engineering design.

Background

Today, engineering optimization problems are more and more complex, and many hybrid analysis and simulation software are applied to design and research, but most of the analysis and simulation problems are high-precision analysis models, such as Finite Element Analysis (FEA) models used in structural analysis, Computational Fluid Dynamics (CFD) analysis models used in pneumatic analysis, and the like. The high-precision analysis model also brings the problem of time consumption in calculation while improving the analysis precision and the reliability, although the computer software and hardware technology has been developed sufficiently at present, the calling of the high-precision analysis model to complete one analysis still consumes very much time, for example, the CFD model needs several hours or even tens of hours to complete one pneumatic simulation analysis; second, modern engineering problems often involve multiple disciplines coupled to one another. For example, the design of an aircraft relates to the disciplines of pneumatics, structures, power, stealth, control and the like, the disciplines influence each other and restrict each other, and the performance of the aircraft is the comprehensive embodiment of the coupling of the disciplines. Due to the coupling relationships between disciplines, the systematic analysis of engineering design problems appears as a multidisciplinary analysis. Essentially, the multidisciplinary analysis process is a typical nonlinear solving process, each multidisciplinary analysis needs to be iterated for many times, the calculation is time-consuming, and if each discipline adopts a high-precision analysis model, the calculation amount is huge. Thirdly, in the process of engineering optimization design, a local or global optimal solution can be converged through repeated iteration, and each iteration needs multidisciplinary analysis of the engineering design problem for many times, so that the calculation cost is further increased. The traditional gradient algorithm can only find the local optimal solution of the analysis problem, and does not have the global searching capability.

To obtain a globally optimal solution to the engineering design problem, optimization algorithms with global search capability, such as Genetic Algorithms (GA), simulated annealing algorithms (SA), etc., are often used directly. However, the global optimization algorithm requires a larger amount of computation than the conventional gradient algorithm. For example, optimizing an analytical model using genetic algorithms typically requires calling hundreds or thousands of analytical systems. For a large number of complex modern engineering design problems adopting high-precision disciplinary analysis models, the traditional global optimization strategy has too high or even unacceptable calculation cost. In addition, most of the current high-precision analysis models adopt a Black-box Model (Black-box Model) established by commercial CAE software, and the interface between the Black-box Model and an optimization algorithm (optimizer) is quite difficult.

In order to reduce the high computation amount involved in the optimization process of the traditional global optimization algorithm on the existing engineering design problem, the optimization method based on the proxy model is gradually researched by people. It is essentially a mathematical proxy model that is similar to the high-precision analytical model, but less computationally expensive, to construct and use for optimization. Because the time required by the high-precision analysis model for calculating once is in the order of hours, and the time required by the agent model for calculating once is only in the order of seconds or even milliseconds, compared with the calculation time of the high-precision analysis model, the calculation time for constructing the agent model and optimizing the agent model is usually negligible. In the last 10 years, many companies have started to research and promote the application of approximate proxy model techniques in the field of design and optimization, such as: the software company Enginous developed iSIGHT, Visual DOC developed by Vanderplaats R & D, Optimus developed by LMS International, Modelcenter developed by Phoenix, and Design Explorer developed by Boeing.

The method can be divided into a static agent model and a dynamic agent model according to the use mode of the agent model in the optimization process. The static agent model adopts enough sample points once, then constructs the agent model, and the agent model is kept unchanged in the optimization process; and the dynamic proxy model is to take sample points in sequence, and then gradually improve and update the proxy model according to known information in each optimization iteration process until optimization converges. Compared with a static agent model, the dynamic agent model has more advantages in the aspects of optimization efficiency and result precision.

A Radial Basis Function (RBF) is one of the most commonly used proxy model methods, and has the advantages that for a high-order nonlinear optimization problem, the Radial Basis Function has higher global approximation precision; and with the increase of the number of the sample points, the approximation precision of the constructed radial basis function model can be improved; in the vicinity of the sample point, the approximation accuracy is high. However, in order to make the approximation accuracy of the radial basis function model constructed by designing the sample points once meet the design requirement, more design sample points are required, and thus, the number of times of calling the analysis model is required.

In order to better explain the technical scheme of the invention, the following is a detailed description of the related methods that may be applied:

1) calculation test design method

The method comprises design methods such as uniform design, Latin hypercube design, sample point minimum distance maximum design and the like, and accurate experimental design sample points can be obtained by adopting the methods.

Wherein, the uniform design is provided by the professor of Fangkaitai, has very good one-dimensional projection uniformity, and the general solving process is as follows:

a) suppose that n is to be selected in the experimental design space_sFor each sample point, a set P is first determined. P is all in the interval [1, n_s]In and n_s+1 is a set of prime numbers for each other.

b) For the

A set of vectors A can be constructed by the formula (1)_i

<math><mrow><msub><mover><mi>A</mi><mo>&RightArrow;</mo></mover><mi>i</mi></msub><mo>=</mo><mo>{</mo><msub><mi>a</mi><mi>ij</mi></msub><mo>|</mo><msub><mi>a</mi><mi>ij</mi></msub><mo>=</mo><mrow><mo>(</mo><mi>j</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><msub><mi>p</mi><mi>i</mi></msub><mi>mod</mi><mrow><mo>(</mo><msub><mi>n</mi><mi>s</mi></msub><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mo>,</mo><mi>j</mi><mo>=</mo><mn>0</mn><mo>,</mo><mo>·</mo><mo>·</mo><mo>·</mo><mo>,</mo><msub><mi>n</mi><mi>s</mi></msub><mo>-</mo><mn>1</mn><mo>}</mo><mo>-</mo><mo>-</mo><mo>-</mo><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></mrow></math>

All elements in the set P are constructed into corresponding vectors through the formula (1), and then all vectors can construct a matrix

Each of which

Corresponding to the ith column in matrix B.

c) If the design space is an s-dimensional space, then selecting any s columns from the matrix B to form a new matrixThenIs n_sA set of sample points, which

Each row vector representing a sample point.

Hikernelll (1998) proposesCan reflect the center L of the spatial equipartition degree of the calculation test design₂Deviation (centered L)₂-discrepancy，CL₂) The analytical expression is shown as formula (2). A calculation test design corresponding to a CL₂Value when CL is₂The smaller the size, the better the space uniformity of the test design.

(2)

Where s denotes the dimension of the design sample space P, x_k＝(x_k1，…，x_ks)∈P。

Now take two-dimensional space as an example, adopt uniform design in space

Designing 7 test points in the interior, and respectively calculating CL of optimal uniform design and general uniform design₂The value is obtained. As shown in FIG. 1, a) in FIG. 1 is an optimally uniform design, with CL₂0.0058; b) for a generally uniform design, its CL₂0.0074. As can be easily found from the figure, the one-dimensional projection uniformity of the general uniform design is the same as that of the optimal uniform design, but the optimal uniform design has better space uniformity.

2) Radial Basis Function (RBF) proxy model

The basic form of the radial basis function is:

the basis function in the formula (3)

Vector of weight coefficients

And beta is_rShould satisfy the condition of difference

(f_r)_i＝y_i，i＝1，2，…，n_s(4)

Wherein, y_iTo a precise value, (f)_r)_iIs a predicted value. Then there are

A_rβ_r＝y (5)

In the formula

Phi is a radial function.

3) Sequential Radial Basis Function (SRBF) proxy model description

SRBF is a high-precision analytical and simulation model for computing time consuming, including objective and constraint functions, for near-optimal solution. The standard non-linear optimization problem is of the form:

minf(x)x＝[x₁，…，x_s]

s.t.h_j(x)＝0，(j＝1，…，J)(8)

g_k(x)≤0，(k＝1，…，K)

x_L，i≤x_i≤x_U，i (i＝1，…，s)

to reduce the number of times to compute high-precision analysis and simulation models, the objective function and the constraint are respectively appliedThe function constructs a radial base proxy model. Wherein x_iUpper and lower boundaries x of_LiAnd x_U，iA design initial boundary is selected. The optimization problem becomes

x_L，i≤x_i≤x_U，i (i＝1，…，s)

Wherein the equation with the symbol- "in formula (9) is the radial basis agent model of the corresponding equation in formula (8). The SRBF is the optimal value for solving these real analytical simulation models.

For a general proxy model, the more sample points, the closer the constructed proxy model is to the real model, but the more times the real model is calculated. For engineering problems, users often care about the optimal solution of the real model, and thus the main task of the proxy model is to find a global optimal solution close to the real model.

Disclosure of Invention

The invention provides an efficient global optimization method based on a sequence radial basis agent model, aiming at the defects that the calculation time is consumed in the optimization process of a high-precision analysis model by using a traditional global optimization algorithm and more sample points are needed by constructing the agent model once by using a radial basis agent model technology.

The invention is suitable for various complicated modern engineering optimization design problems, and for many engineering design and analysis software, such as: the method is used for approximating the real model in the engineering design and analysis software, and the approximation model only needs several seconds or dozens of seconds, so that the period of the engineering optimization design is greatly shortened, the design cost is greatly saved, and the design efficiency is remarkably improved.

The invention relates to a global optimization method based on a sequence radial basis function proxy model, which adopts a proxy model based on a radial basis function and distributed points for multiple times, and has the design principle that: in the process of optimizing the engineering design problem, a key sampling space is constructed according to known information in each iteration process, and sample points are added in the key sampling space, so that the approximation precision of the proxy model in the area near the optimal point of the real model is improved; and then updating the radial basis agent model of the real model, and optimizing the agent model by using a global optimization algorithm until the optimal solution of the real model is obtained. The searching method adopted by the invention has global searching capability.

The invention relates to an efficient global optimization method based on a sequence radial basis agent model, which comprises the following specific implementation steps of:

step 1, selecting a sample point in a primary iteration design space according to an initial condition given by a user.

The method comprises the steps of taking an analysis and simulation model which needs to be researched and is given by a user as a real model, taking design variables and target function constraint conditions related in the real model and the whole design space as initial conditions, enabling an iteration counting parameter k to be 1, and carrying out initial iteration. In the whole design space, sample points are selected by using a calculation experiment design method. Number n of selected sample points_sIs composed of

n_{s} = \frac{(n_{v} + 1) (n_{v} + 2)}{2} - - - (10)

Wherein n is_vRepresenting the dimensions of the design space.

And 2, calculating a response value of the real model.

And when k is equal to 1, calculating the response value of the real model corresponding to each sample point selected in the step 1 by calling the real model in the step 1, and storing the response values of the real models corresponding to the sample points into a design sample point database.

And when k is more than or equal to 2, calling the real model in the step 1, calculating the response values of the real model corresponding to the sample points newly added to the designed sample point database in the step 8, and storing the new sample points and the corresponding real model response values thereof in the designed sample point database.

And 3, constructing a Radial Basis Function (RBF) proxy model.

And when k is equal to 1, extracting all the sample points in the design sample point database obtained in the step 2 and the response values of the corresponding real models, and reconstructing a Radial Basis (RBF) proxy model by adopting a construction method of the RBF proxy model.

And when k is more than or equal to 2, extracting response values of all newly added and existing sample points in the designed sample point database and the corresponding real models thereof, and reconstructing a Radial Basis Function (RBF) proxy model by adopting a construction method of the RBF proxy model.

And 4, solving the current approximate optimal solution of the Radial Basis Function (RBF) agent model.

And (3) solving a current iteration approximate optimal solution for the Radial Basis Function (RBF) agent model obtained in the step (3) by adopting an optimization algorithm with global search capability.

And 5, calculating a response value of the current iteration approximate optimal solution in the real model.

And (4) substituting the current iteration approximate optimal solution obtained in the step (4) into the real model, solving a response value of the current approximate optimal solution corresponding to the real model, and storing the response value into an optimal solution response value set.

And 6, judging whether the global optimization method meets the convergence criterion.

If it is the first calculation, i.e. k is 1, the process proceeds directly to step 7.

If not, namely k is more than or equal to 2, calculating relative errors of real model response values corresponding to approximate optimal solutions of the RBF proxy model constructed by the current kth iteration and the kth-1 iteration in the optimal solution response value set by calling, and judging whether the relative errors meet a given convergence standard epsilon or not. If so, stopping circulation, obtaining the optimal solution which is the optimal value of the real model in the step 4, and ending the process of the global optimization method; if not, go to step 7.

And 7, determining a key sampling space of the next iteration. The construction method of the key sampling space is as follows:

firstly, determining the position and size of an important sampling space, namely constructing a set B of a k-th important sampling space_k＝[B_L，k，B_U，K]. The selected key sampling space of the invention is positioned near the approximate optimal solution of the current proxy model obtained in the step 4. Set B_k＝[B_L，k，B_U，k]Is represented by the formula (11). B is_kThe ith row in the drawing shows the value range of the ith dimension.

B_{L, k} = x_{k - 1}^{*} - \frac{1}{n_{s}} {BL}_{k - 1}

(11)

B_{U, k} = x_{k - 1}^{*} + \frac{1}{n_{s}} {BL}_{k - 1}

Wherein, B_L，kVector representing the lower bound of the k-th key-sampled space, B_U，kA vector representing the upper bound of the k-th key-sample space,

representing the optimal solution for the k-1 th proxy model.

BL_k-1A vector representing the size of the k-1 th key sample space,

represents the size of the s-dimension in the k-1 th key sampling space and has the expression

BL_k-1＝B_U，k-1-B_L，k-1

(12)

Secondly, if the key sampling space B obtained in the step I is obtained_k＝[B_L，K，B_U，k]If the radial base proxy model is too small, the newly added sample points are dense, and the effect of improving the approximate accuracy of the radial base proxy model near the optimal solution of the real model is not obvious. Therefore, the invention gives a minimum key sampling space, so that the key sampling space can not cause the sample points to be dense because of too small contraction, and the searching method can jump out the local optimum point to enable the local optimum point to have the global optimizing capability. The specific implementation method comprises the following steps: when in use

Then, it is ordered

Wherein,

is selected to have the size of the whole design space given in step 1

It is related.

Step three, determining the key sampling space B_k＝[B_L，k，B_U，k]And (3) when the whole design space given in the step (1) is exceeded, taking the intersection of the whole design space and the key sampling space as a new key sampling space.

And 8, adding new sample points in the key sampling space constructed in the step 7 by a calculation test design method, and storing the new sample points in a design sample point database.

And (4) in the key sampling space constructed in the step (7), adding new sample points by adopting a calculation test design method, and storing the new sample points into a design sample point database. The number of the newly added sample points is determined by the formula (10).

In order to ensure the projection uniformity of the newly added sample point in the key sampling space, the newly added sample point and the existing sample point in the key sampling space are not overlapped in projection on each dimension.

If the projection of a certain newly added sample point and the projection of the existing sample point on a certain dimension coincide, the newly added sample point is translated to the right side or the left side of the dimension, and the translation principle is as follows: in the same global optimization process, the translation directions are consistent, namely, the translation is performed to the right (left) during the first coincidence, and then the translation is performed to the right (left) every time. Meanwhile, in order to ensure that the projection of the new sample point obtained after translation does not coincide with other sample points, the translation distance is 1/2n with the size of the dimension coinciding with the original projection in the key sampling space_sAnd (4) doubling.

And 9, on the basis of the step 8, making k equal to k +1, and shifting to the step 3 for the next iteration.

Advantageous effects

Compared with the general one-time sampling construction proxy model technology, if the same number of sample points are distributed in the design space, the one-time sampling uniformly distributes all the sample points in the whole design space, and focuses on the whole design space, so that the approximation precision of the proxy model and the real analysis model is improved; and the SRBF distributes most of sample points in the vicinity of the optimal solution of the real analysis model in a key way, so that the constructed proxy model has high approximate precision with the real analysis model in the vicinity of the optimal solution, and the optimal solution can be solved through global search. If the RBF technique by one sampling also distributes as many sample points near the optimal solution, the number of distributed sample points in the entire design space is multiplied, which also results in an increase in the number of times the real analysis model is called. Therefore, compared with the RBF technology of one-time sampling, the SRBF technology has obvious improvement on the efficiency.

The invention overcomes the defect of time consumption of calculation in the engineering optimization design problem of the traditional global optimization method, and compared with a static agent model, the SRBF can find the global optimum point of the analysis model by calling the analysis model for a few times, thereby effectively reducing the calculation cost, improving the optimization efficiency and being beneficial to shortening the engineering optimization design period. The method has good application prospect in engineering designs such as aerodynamic optimization design and wing optimization design of the aircraft.

Drawings

FIG. 1 is a schematic diagram of a prior art optimized uniform design and a generally uniform design;

FIG. 2 is a flow chart of a global optimization method based on a Sequence Radial Basis (SRBF) proxy model according to the present invention;

FIG. 3 is a schematic diagram illustrating a translation of a newly added sample point in an embodiment;

FIG. 4 is a three-dimensional grid diagram of a BR function in a design space according to an embodiment;

fig. 5 is a schematic diagram of an i-beam optimization design problem in a specific embodiment.

Detailed Description

The invention provides and realizes an efficient global optimization method based on a Sequence Radial Basis Function (SRBF), which is suitable for complex engineering optimization problems, is beneficial to improving the optimization efficiency and further can compress the design period and the cost.

In order to better illustrate the objects and advantages of the invention, the invention is further explained by a standard analytical function test example and an I-beam optimization design example in combination with the attached drawings, and the comprehensive performance of the invention is verified and analyzed by comparing with the technical result of constructing the RBF proxy model by one-time sampling.

Analytical function optimization example

Assuming that a Branin function (BR) function is a high-precision analysis model for calculating time consumption in engineering design, the performance of the SRBF is verified by solving the minimum value of the BR function in a design space. The design objective of this embodiment is to improve the efficiency of the optimization design, i.e. to reduce the number of times the BR function is solved.

The given initial condition BR function is shown as the formula (13)

x₁∈[-5，10]x₂∈[0，15]

Objective function f_BR(x) In the design space x₁∈[-5，10]x₂∈[0，15]Three-dimensional grid map of (1) such as error! No reference source is found. As shown, the global optimum point of the BR function in the design space is

x_{opt}^{*} = (3.1415,2.2749),

f_{BR}^{*} (x_{opt}^{*}) = 0.3979 .

In order to obtain a global optimal solution, an optimization method with global optimization capability is used. The SRBF agent model method of the present invention and the prior art method for constructing the RBF agent model by adopting one-time points are used herein. The RBF proxy model construction method based on one-time sampling in the prior art adopts two methods with different sampling points to respectively describe: the number of the primary sampling points of the method I is the same as that of the SRBF agent model method; the number of sampling points in one time of the method II is 100 sample points. The above three proxy models are solved using genetic algorithms for analytical comparison.

As shown in fig. 2, the global optimization method of the present invention includes the following specific steps:

step 1, the real model of this embodiment is a BR functionThe optimized mathematical model of (2) is shown as formula (13); the embodiment adopts unconstrained optimization, namely, no constraint condition exists; the whole design space is x₁∈[-5，10]，x₂∈[0，15]The iteration number k is 1, the method for calculating the experimental design in this embodiment is a uniform design method, and since the design space is a two-dimensional model, the number of sample points is designed

And (4) respectively.

And 2, calculating a response value of the real model.

And when k is equal to 1, calculating response values of the real models corresponding to the 6 sample points selected in the step 1 by calling the real models in the step 1, and storing the 6 sample points and the corresponding real model response values thereof in a design sample point database.

And when k is larger than or equal to 2, calling the real model in the step 1, calculating the response values of the real model corresponding to the sample points newly added to the designed sample point database in the step 8, and storing the response values of the real model corresponding to the new sample points into the designed sample point database.

And 3, constructing a radial base proxy model.

And (3) extracting newly added and existing all sample points in the designed sample point database obtained in the step (2) and response values of real models corresponding to the sample points, and reconstructing a Radial Basis Function (RBF) proxy model by adopting a construction method of the RBF proxy model.

The radial function selected in this embodiment is a Gaussian function (Gaussian function), and the expression is shown as the following formula

Where φ is a radial function, r is the Euclidean distance between the predicted point and any sample point, and c is a normal real number. Empirically, c typically takes positive real numbers between [0, 10 ].

And 4, solving the optimal solution of the Radial Basis Function (RBF) agent model.

The global optimization algorithm selected in this embodiment is a Genetic Algorithm (GA), and a current iteration approximate optimal solution is obtained for the Radial Basis Function (RBF) agent model obtained in step 3.

If not, namely k is more than or equal to 2, calculating relative errors of real model response values corresponding to approximate optimal solutions of the RBF proxy models constructed by the current kth iteration and the kth-1 iteration in the optimal solution response value set by calling, and judging whether the relative errors meet a given convergence standard epsilon of 0.01. If so, stopping circulation, obtaining the optimal solution which is the optimal value of the real model in the step 4, and ending the process of the global optimization method; if not, go to step 7.

And 7, determining a key sampling space of the next iteration. The construction of the emphasized sampling space is as follows:

(1) firstly, determining the position and size of the key sampling space, namely constructing a set B of the k-th key sampling space_k＝[B_L，k，B_U，k]. The selected key sampling space of the invention is positioned near the approximate optimal solution of the current proxy model obtained in the step 4. Set B_k＝[B_Lk，B_U，k]Is represented by the formula (11). B is_kThe ith row in the drawing shows the value range of the ith dimension.

(2) If the key sampling space B obtained in step 7(1)_k＝[B_L，k，B_U，k]If the radial base proxy model is too small, the newly added sample points are dense, and the effect of improving the approximate accuracy of the radial base proxy model near the optimal solution of the real model is not obvious. Therefore, the invention gives a minimum key sampling space, so that the key sampling space can not cause the sample points to be dense because of too small contraction, and the searching method can jump out the local optimum point to enable the local optimum point to have the global optimizing capability. For example, the upper bound of the k-th sample space in the i-th dimension is calculated by (11)

If it is

Then order

If it is

Then order

Based on experimental experience, in this example

This ensures that

The small space is not too small or too large, so that the invention has global searching capability.

(3) If the emphasized sampling space B obtained in the steps 7(1) and 7(2)_k＝[B_L，k，B_U，k]And (3) exceeding the whole design space given in the step 1, taking the intersection of the whole design space and the key sampling space as a new key sampling space.

And 8, adding new sample points in the key sampling space constructed in the step 7 by using a uniform design method, and storing the new sample points in a design sample point database. The number of the newly added sample points is 6.

In this embodiment, the principle of translation is: if the projection of a new sample point and the projection of the existing sample point on a certain dimension coincide, the new sample point is translated to the right side of the dimension. Meanwhile, in order to ensure that the projection of the new sample point obtained after translation does not coincide with other sample points, the distance of translation is 1/12 times of the size of the dimension in which the original projections coincide in the key sampling space.

The total number of sample points needed for solving BR examples by the SRBF surrogate model method is 36, a method I (selecting 36 sample points) and a method II (selecting 100 sample points) are adopted for sampling once in the same design space, RBF surrogate models corresponding to the RBF surrogate models are constructed, and then the surrogate models obtained by the two methods I and II are optimized respectively by using a Genetic Algorithm (GA). Furthermore, the BR function is optimized directly using a Genetic Algorithm (GA). The results of the SRBF, method I, method II and GA calculations were compared as shown in table 1.

TABLE 1 comparison of BR function optimization results

In the aspect of optimization effect, as can be seen from table 1, the SRBF optimization method and the GA optimization algorithm of the present invention can completely find the global optimal solution of the BR function, and have good global optimization capability. However, the method I and the method II can not find out the global optimal solution, but the optimization result of the method II is better than that of the method I, because the number of sample points distributed by the method II in the design space is far more than that of the method I, the approximation accuracy of the proxy model and the real model constructed by the method II is higher than that of the proxy model and the real model constructed by the method I.

In terms of efficiency, as can be seen from table 1, the number of times that the SRBF optimization method of the present invention calls the BR function analysis model is 42, which is reduced by about 58.4% compared to method II; compared with the GA algorithm, the frequency of SRBF calling the analysis model is only 4% of GA; compared with the method I, although the number of times of calling the analysis model is 5 times more by the SRBF, the response value corresponding to the current optimal solution needs to be solved by calling the real analysis model in each iteration process by the SRBF, the optimization effect of the method I is far inferior to that of the SRBF, and the number of times of calling the analysis model even for a few times is negligible.

The specific implementation of the analysis function optimization example shows that the SRBF global optimization method is beneficial to reducing the times of calling a high-precision model in the process of processing the engineering optimization design problem, so that the aim of improving the efficiency is fulfilled, and the SRBF global optimization method also has good global optimization capability.

I-beam optimization design example

And the performance of the SRBF is verified through an I-beam optimization design example. The engineering problem is aimed at minimizing the vertical deflection of the i-beam, given the cross-sectional area and pressure constraints, such as errors! No reference source is found. As shown. The basic parameters of the design problem are:

● Beam maximum bending stress was 6kN/cm²

● Young's modulus of 2X 10⁴kN/cm²

● maximum bending pressure P600 kN and Q50 kN

● Beam Length L200 cm

Then, depending on the design parameters, the mathematical description of the engineering optimization problem can be:

\min f (x) = \frac{5000}{\frac{1}{12} x_{3} {(x_{1} - 2 x_{4})}^{3} + \frac{1}{6} x_{2} x_{4}^{3} + 2 x_{2} x_{4} {(\frac{x_{1} - x_{4}}{2})}^{2}}

s.t.g₁(x)＝2x₂x₄+x₃(x₁-2x₄)≤300 (15)

wherein the objective function f (x) is the deformation amount of the I-beam, g₁(x) Is the cross-sectional area, g₂(x) Is a bending pressure.

The problem is a nonlinear constraint problem, the problem is optimized and solved by adopting the method, the target function is solved and optimized by adopting SRBF, and the constraint condition uses a real analysis model; meanwhile, a genetic algorithm carried by MATLAB is selected for solving, and results of the two algorithms are compared. The specific implementation steps for solving the i-beam optimization problem by SRBF are as follows. The calculation results are shown in table 2.

Step 1, since the present embodiment is constraint optimization, and the target function is optimized by using the SRBF proxy model, the real model uses the target function, as shown in (16). The whole design space is more than or equal to x and is more than or equal to 10₁≤80，10≤x₂≤50，0.9≤x₃≤5，0.9≤x₄The iteration number k is not more than 5, the iteration number k is 1, the method for calculating the experimental design in the embodiment is a uniform design method, and the number of the designed sample points is designed because the design space is a four-dimensional model

And (4) respectively.

f (x) = \frac{5000}{\frac{1}{12} x_{3} {(x_{1} - 2 x_{4})}^{3} + \frac{1}{6} x_{2} x_{4}^{3} + 2 x_{2} x_{4} {(\frac{x_{1} - x_{4}}{2})}^{2}} - - - (16)

And 2, calculating a response value of the real model.

And when k is equal to 1, calculating response values of the real models corresponding to the 15 sample points selected in the step 1 by calling the real models in the step 1, and storing the response values of the real models corresponding to the 15 sample points in a design sample point database.

And 3, constructing a radial base proxy model.

The radial function used in this embodiment is a Gaussian function (Gaussian function).

And (4) substituting the current iteration approximate optimal solution obtained in the step (4) into the real model, solving a response value of the current approximate optimal solution corresponding to the real model, and storing the response value into an optimal solution set.

If not, namely k is more than or equal to 2, calculating relative errors of real model response values corresponding to approximate optimal solutions of the RBF proxy models constructed by the current kth iteration and the kth-1 iteration in the optimal solution set by calling the real model response values, and judging whether the relative errors meet a given convergence criterion epsilon of 0.01. If so, stopping circulation, obtaining the optimal solution which is the optimal value of the real model in the step 4, and ending the process of the global optimization method; if not, go to step 7.

And 7, determining a key sampling space of the next iteration.

1) Firstly, determining the position and size of the key sampling space, namely constructing a set B of the k-th key sampling space_k＝[B_L，k，B_U，k]. The selected key sampling space of the invention is positioned near the approximate optimal solution of the current proxy model obtained in the step 4. Set B_k＝[B_L，k，B_U，k]Is represented by the formula (11). B is_kThe ith row in the drawing shows the value range of the ith dimension.

2) If the key sampling space B obtained in step 7(1)_k＝[B_L，k，B_U，k]If the radial base proxy model is too small, the newly added sample points are dense, and the effect of improving the approximate accuracy of the radial base proxy model near the optimal solution of the real model is not obvious. Therefore, the invention gives a minimum key sampling space, so that the key sampling space can not cause the sample points to be dense because of too small contraction, and the searching method can jump out the local optimum point to enable the local optimum point to have the global optimizing capability. For example, the upper bound of the k-th sample space in the i-th dimension is calculated by (11)

If it is

Then order

If it is

Then order

Obtained from experimental experience, in this example

Therefore, the minimum space is not too small or too large, and the invention has global searching capability.

3) If the emphasized sampling space B obtained in the steps 7(1) and 7(2)_k＝[B_L，k，B_U，k]And (3) exceeding the whole design space given in the step 1, taking the intersection of the whole design space and the key sampling space as a new key sampling space.

And 8, adding new sample points in the key sampling space constructed in the step 7 by using a uniform design method, and storing the new sample points in a design sample point database. The number of the newly added sample points is 15.

In this embodiment, the principle of translation is: if the projection of a new sample point and the projection of the existing sample point on a certain dimension coincide, the new sample point is translated to the right side of the dimension. Meanwhile, in order to ensure that the projection of the new sample point obtained after translation does not coincide with other sample points, the distance of translation is 1/30 times the size of the dimension of the key sampling space coinciding with the original projection.

TABLE 2I-Beam optimization problem result comparison

The data in table 2 show that the optimal result obtained by the SRBF optimization method proposed by the present invention is 0.0137, and satisfies both the cross-sectional area constraint and the bending pressure constraint, and the optimization results are almost the same compared with the genetic algorithm with MATLAB. In the aspect of optimization efficiency, compared with the genetic algorithm, the frequency of calling the analysis model by the SRBF is only 0.46% of that of the genetic algorithm, and the optimization efficiency is far higher than that of the genetic algorithm. Obviously, the SRBF can obtain the optimal solution of the original engineering optimization problem on the premise of greatly reducing the calculation amount.

Therefore, the method basically achieves the expected purpose, and compared with the method for constructing the RBF proxy model by one-time point sampling and directly using the genetic algorithm, the method has the advantage that the optimization efficiency is greatly improved on the premise of obtaining the optimal solution of the optimization problem. Therefore, on one hand, the SRBF can effectively reduce the calculation cost, improve the optimization efficiency and contribute to shortening the period of engineering design; on the other hand, the method has strong global optimization capability, improves the global optimization capability in the engineering optimization design problem, and is favorable for improving the quality of engineering design.

The above detailed description is intended to illustrate the objects, aspects and advantages of the present invention, and it should be understood that the above detailed description is only exemplary of the present invention, and is not intended to limit the scope of the present invention, and any modifications, equivalents, improvements, etc. made within the spirit and principle of the present invention should be included in the scope of the present invention.

Claims

1. An efficient global optimization method based on a sequence radial basis function model is characterized in that: the method comprises the following steps:

step 1, selecting sample points in a primary iteration design space according to initial conditions given by a user

Taking an analysis and simulation model which needs to be researched and is given by a user as a real model, taking design variables, target function constraint conditions and the whole design space which are related in the real model as initial conditions, and enabling an iteration counting parameter k to be 1 to carry out initial iteration; in the whole design space, the utilization meterCalculating a test design method and selecting a sample point; number n of selected sample points_SIs composed of

n_{s} = \frac{(n_{v} + 1) (n_{v} + 2)}{2}

Wherein n is_vA dimension representing a design space;

step 2, calculating the response value of the real model

When k is equal to 1, calculating a response value of the real model corresponding to each sample point selected in the step 1 by calling the real model in the step 1, and storing the response values of the real models corresponding to the sample points into a design sample point database;

when k is larger than or equal to 2, calling the real model in the step 1, calculating the response values of the real model corresponding to the sample points newly added to the designed sample point database in the step 8, and storing the new sample points and the corresponding real model response values into the designed sample point database;

step 3, constructing a radial basis function agent model

When k is equal to 1, extracting all sample points in the designed sample point database obtained in the step 2 and response values of the real models corresponding to the sample points, and reconstructing a radial basis agent model by adopting a construction method of the radial basis agent model;

when k is larger than or equal to 2, newly adding all the existing sample points in the designed sample point database and extracting the response values of the corresponding real models of the sample points, and adopting a construction method of a radial basis agent model to reconstruct the radial basis agent model;

step 4, solving the current approximate optimal solution of the Radial Basis Function (RBF) agent model

Obtaining a current iteration approximate optimal solution for the radial basis function model obtained in the step 3 by adopting an optimization algorithm with global search capability;

step 5, calculating the response value of the current iteration approximate optimal solution in the real model

Substituting the current iteration approximate optimal solution obtained in the step 4 into the real model, obtaining a response value of the current approximate optimal solution corresponding to the real model, and storing the response value into an optimal solution response value set;

step 6, judging whether the global optimization method meets the convergence criterion

If the calculation is the first time, namely k is 1, directly proceeding to step 7;

if not, namely k is more than or equal to 2, calculating relative errors of real model response values corresponding to approximate optimal solutions of the radial basis agent model constructed by the current kth iteration and the kth-1 iteration in the optimal solution response value set by calling, and judging whether the relative errors meet a given convergence standard epsilon or not; if so, stopping circulation, obtaining an approximate optimal solution which is the optimal value of the real model in the step 4, and ending the process of the global optimization method; if not, turning to step 7;

step 7, determining the key sampling space of the next iteration

The construction method of the key sampling space is as follows:

firstly, determining the position and size of the key sampling space, namely constructing a set B of the k-th key sampling space_k＝[B_L，k，B_U，k](ii) a The selected key sampling space is positioned near the approximate optimal solution of the current proxy model obtained in the step 4; set B_k＝[B_L，k，B_L，k]The expression of (a) is shown as follows; b is_kThe ith row in the drawing shows the value range of the ith dimension;

B_{L, k} = x_{k - 1}^{*} - \frac{1}{n_{s}} {BL}_{k - 1}

B_{U, k} = x_{k - 1}^{*} + \frac{1}{n_{s}} {BL}_{k - 1}

expressing the optimal solution of the k-1 th proxy model;

BL_k-1a vector representing the size of the k-1 th key sample space,

BL_k-1＝B_U，k-1-B_L，k-1

② a minimum key sampling space is given

When in use

Then, it is ordered

Wherein,

is selected to have the size of the whole design space given in step 1

(ii) related;

Step 8, in the key sampling space constructed in the step 7, adding new sample points by a calculation test design method, and storing the new sample points into a design sample point database;

in the key sampling space constructed in the step 7, adding new sample points by adopting a calculation test design method, and storing the new sample points into a design sample point database; the number of the newly added sample points is determined by the formula in the step 1;

in order to ensure the projection uniformity of the newly added sample points in the key sampling space, the newly added sample points and the existing sample points in the key sampling space are not overlapped in projection on each dimension;

if the projection of a certain newly added sample point and the projection of the existing sample point on a certain dimension coincide, the newly added sample point is translated to the right side or the left side of the dimension, and the translation principle is as follows: in the same global optimization process, the translation directions are consistent, namely the translation is performed to the right or left during the first coincidence, and the translation is performed to the right or left every time; meanwhile, in order to ensure that the projection of the new sample point obtained after translation does not coincide with other sample points, the translation distance is 1/2n with the size of the dimension coinciding with the original projection in the key sampling space_sDoubling;