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

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

A kind of efficient global optimization method based on sequence radial base agent model

Technical field

The present invention relates to a kind of efficient global optimization method, belong to multidisciplinary optimisation technique field in the engineering design based on sequence radial base agent model.

Background technology

Current engineering optimization problem becomes increasingly complex, many hybrid analyses and simulation software all are applied in design and the research, but these are analyzed and simulation problems all is the high accuracy analysis model mostly, for example Fluid Mechanics Computation (CFD) analytical model of using in the finite element analysis of adopting in the structure analysis (FEA) model, the aerodynamic analysis etc.The high accuracy analysis model has also brought the problem consuming time of calculating when improving analysis precision and confidence level, though the computer nowadays software and hardware technology has had significant progress, yet, call the high accuracy analysis model and finish once and to analyze still extremely consuming timely, for example using the CFD model to finish an aerodynamics simu1ation analysis needs a few hours even tens of hours; Secondly, the modern project design problem often relates to a plurality of subjects that intercouple.For example, subjects such as that Flight Vehicle Design relates to is pneumatic, structure, power, stealthy, control, each subject influences each other, and mutual restriction, the performance of aircraft are the comprehensive embodiments of each subject coupling.Because interdisciplinary coupled relation, the systematic analysis of engineering design problem shows as multidisciplinary analysis.In essence, the multidisciplinary analysis process is a typical nonlinear solution procedure, and each multidisciplinary analysis all needs to carry out repeatedly iteration, calculates consuming timely, if each subject all adopts the high accuracy analysis model, its calculated amount will be very huge.Once more, need in the Optimum design of engineering structure process can converge to part or globally optimal solution through iterating, and each iteration all needs to carry out the multidisciplinary analysis of repeatedly engineering design problem, as seen assessing the cost further to increase.Traditional gradient algorithm often can only find the locally optimal solution of problem analysis, does not possess ability of searching optimum.

In order to obtain the globally optimal solution of engineering design problem, usually directly use the optimized Algorithm with ability of searching optimum, for example genetic algorithm (GA), simulated annealing (SA) etc.But, to compare with traditional gradient algorithm, the required calculated amount of global optimization approach is bigger.For example, use genetic algorithm that an analytical model is optimized common needs and call analytic system hundreds and thousands of times.For the complicated modern project design problem of a large amount of employing high precision subject analytical models, the assessing the cost excessive or even be difficult to accept of traditional global optimization strategy.In addition, the black-box model (Black-box Model) that present most high accuracy analysis models all adopt commercial CAE software to set up, quite difficult with the interface of optimized Algorithm (optimizer).

In order to reduce traditional global optimization approach to related high calculated amount in the present engineering design problem optimizing process, based on the optimization method of agent model by being studied gradually by people.It is exactly structure and high accuracy analysis model approximation in essence, but the mathematics agent model that assesses the cost lower, and this model is used for optimizing.Because the magnitude of a required time of high accuracy analysis Model Calculation is hour, and agent model calculate once the magnitude of used time only be second in addition millisecond, therefore compare structure agent model and based on often ignoring the computing time of agent model optimization with the computing time of high accuracy analysis model.In nearest 10 years, many companies all begin one's study and have promoted the application of approximate agent model technology in design and optimization field, for example: the research and development iSIGHT of Engineous software company, Vanderplaats R﹠amp; The Visual DOC of D company research and development, the Optimus of LMS international organization research and development, the Design Explorer of the ModelCenter of Phoenix company research and development and Boeing's research and development.

Can be divided into static agent model and dynamic proxy model according to the use-pattern of agent model in optimizing process.Static agent model is constructed agent model then by once taking abundant sample point, and agent model remains unchanged in optimizing process; And dynamic proxy model to be sequence take sample point is progressively improving and the update agent model according to Given information in the iterative process each optimization, then until optimizing convergence.Compare with static agent model, the dynamic proxy model is being optimized efficient and is being had more advantage aspect the precision as a result.

(Radial Basis Function is one of the most frequently used agent model method RBF) to radial basis function, and its advantage is that for the high-order nonlinear optimization problem, radial basis function is higher at overall approximation quality; And along with the increase of sample point number, the approximation quality of the radially basic agent model of being constructed can improve; Near sample point, approximation quality is higher.But, make radially basic its approximation quality of agent model of constructing will satisfy design requirement by design sample point, so needed design sample point is more, and the required like this number of times that calls analytical model is more.

For technical scheme of the present invention better is described, below the correlation technique that may be applied to is done concrete introduction:

1) calculates test design method

This method comprises methods for designing such as uniform Design, Latin hypercube design, sample point minor increment design maximum, adopts these methods can obtain accurate experimental design sample point.

Wherein, uniform Design is to be proposed by professor Fang Kaitai, has good one dimension projection homogeneity, and its general solution procedure is:

A) hypothesis will be selected n in the test design space _sIndividual sample point is at first determined set P.P be all interval [1, n _s] in n _s+ 1 set of the array one-tenth of prime number each other.

B) for

Through type (1) can be constructed one group of vector A _i

${\stackrel{\→}{A}}_i=\left\{a_{\mathrm{ij}}\right|a_{\mathrm{ij}}=(j+1)p_i\mathrm{mod}(n_s+1),j=0,\·\·\·,n_s-1\}---\left(1\right)$

To gather among the P all elements all through type (1) construct pairing vector, then institute's directed quantity can construct a matrix

Wherein each

Be listed as corresponding to the i in the matrix B.

C), from matrix B, select any s row to constitute a new matrix so if the design space is the s dimension space So Be exactly n _sThe set that individual sample point constitutes, its

In each row vector representation sample point.

Hickernell (1998) has proposed to reflect calculating test design space uniform property degree center L ₂Deviation (centered L ₂-discrepancy, CL ₂), its analytical expression is as the formula (2).A kind of calculating test design is corresponding to a CL ₂Value is worked as CL ₂More hour, show that the uniform property in space of test design is good more.

${\mathrm{CL}}_2{\left(P\right)}^2={\left(\frac{13}{12}\right)}^s-\frac{2}{n}\underset{k=1}{\overset{n_s}{\mathrm{\Σ}}}\underset{j=1}{\overset{s}{\mathrm{\Π}}}(1+\frac{1}{2}|x_{\mathrm{kj}}-0.5|-\frac{1}{2}{|x_{\mathrm{kj}}-0.5|}^2)$

(2)

$\frac{1}{x_{\mathrm{<figure-callout\; id="2"\; label="s"\; filenames="HSA00000195088400011.png,HSA00000195088400021.png"\; state="\{\{state\}\}">s</figure-callout>}}^2}\underset{k=1}{\overset{n_s}{\mathrm{\&Sigma;}}}\underset{j=1}{\overset{n_s}{\mathrm{\&Sigma;}}}\underset{i=1}{\overset{s}{\mathrm{\&Pi;}}}[1+\frac{1}{2}|x_{\mathrm{ki}}-0.5|+\frac{1}{2}|x_{\mathrm{ji}}-0.5|-\frac{1}{2}|x_{\mathrm{ki}}-x_{\mathrm{ji}}\left|\right]$

Wherein, s represents the dimension of design sample space P, x _k=(x _K1..., x _Ks) ∈ P.

Now be example, adopt uniform Design in the space with the two-dimensional space

In 7 testing sites of design, calculate the CL of optimal uniform design and general uniform Design respectively ₂Value.As shown in Figure 1, a) design its CL among Fig. 1 for optimal uniform ₂,=0.0058; B) be general uniform Design, its CL ₂=0.0074.Be not difficult to find that general uniform Design is all identical with the one dimension projection homogeneity of optimal uniform design by figure, but the optimal uniform design has the uniform property in better space.

2) radial basis function (RBF) agent model

The citation form of radial basis function is:

Basis function in the formula (3)

Power is to coefficient vector

And β _rShould satisfy the difference condition

(f _r) _i＝y _i，i＝1，2，…，n _s(4)

Wherein, y _iBe exact value, (f _r) _iBe predicted value.So have

A _rβ _r＝y (5)

${\mathrm{\&beta;}}_r=A_r^{-1}y---\left(6\right)$

In the formula

φ is a radial function.

3) the sequence radial base (Sequential Radial Basis Function, SRBF) describe by agent model

SRBF is used for carrying out the approximate solution optimum to calculating high accuracy analysis consuming time and realistic model comprising objective function and constraint function.The form of the nonlinear optimal problem of standard is:

minf(x)x＝[x ₁，…，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)

In order to reduce the number of times that calculates high accuracy analysis and realistic model, respectively to objective function and the radially basic agent model of constraint construction of function.X wherein _iUp-and-down boundary x _LiAnd x _{U, i}Select the design initial boundary for use.Then optimization problem becomes

$\min \tilde{f}\left(x\right)x=[x_1,\&CenterDot;\&CenterDot;\&CenterDot;,x_s]$

$s.t.{\tilde{h}}_j\left(x\right)=0,(j=1,\&CenterDot;\&CenterDot;\&CenterDot;,J)---\left(9\right)$

${\tilde{g}}_k\left(x\right)\&le;0,(k=1,\&CenterDot;\&CenterDot;\&CenterDot;,K)$

x _L，i≤x _i≤x _U，i (i＝1，…，s)

The equation of tape symbol "～" is the radially basic agent model of institute's corresponding equation in the formula (8) in its Chinese style (9).SRBF is used to find the solution these true optimal values of analyzing realistic model.

For general agent model, sample point is many more, and the agent model of being constructed approaches true model more, and still, the number of times that calculates true model also can increase.For engineering problem, what the user often was concerned about is the optimum solution of true model, thereby the main task of agent model is to find a globally optimal solution that approaches true model.

Summary of the invention

The present invention is directed to and use traditional global optimization approach in to high accuracy analysis model optimization process, to calculate time-consuming, and use radially basic agent model technology by once constructing the more defective of the required sample point of agent model, a kind of efficient global optimization method based on sequence radial base agent model has been proposed.

The present invention is applicable to the modern project optimal design problem of various complexity, for many engineering designs and analysis software, as: aircraft aerodynamic analysis software CFD, structural analysis of flight vehicle software Nastrane, finite element analysis software FEA etc. in the Flight Vehicle Structure, in carrying out the Optimum design of engineering structure process, often optimal design need consume several hrs even several days, and adopt the present invention that the true model in engineering design and the analysis software is similar to, such approximate model is optimized design only needs several seconds or tens seconds, shortened the cycle of Optimum design of engineering structure so greatly, design cost is saved greatly, and design efficiency significantly improves.

Global optimization method based on sequence radial base agent model of the present invention adopts a kind of agent model based on radial basis function of repeatedly layouting, its design concept is: in to engineering design problem optimizing process, by in each iterative process, constructing the important sampling space according to Given information, and in the important sampling space, increase sample point, improve the approximation quality of agent model thus at the optimum point near zone of true model; Upgrade the radially basic agent model of true model then, and use global optimization approach that agent model is optimized, until the optimum solution that obtains true model.The searching method that the present invention adopts has ability of searching optimum.

A kind of efficient global optimization method of the present invention based on sequence radial base agent model, the specific implementation step is as follows:

Step 1 by the given starting condition of user, is selected the sample point in the first Iterative Design space.

As true model, and with design variable related in the true model, objective function constraint condition, and whole design space makes iteration count parameter k=1 as starting condition, carries out first iteration with the given analysis that needs research of user and realistic model.In whole design space, utilize and calculate test design method selection sample point.Selected sample point number n _sFor

$n_s=\frac{(n_v+1)(n_v+2)}{2}---\left(10\right)$

Wherein, n _vThe dimension of expression design space.

Step 2 is calculated the response of true model.

When k=1, by the true model in the invocation step 1, the response of the selected pairing true model of each sample point in the calculation procedure 1, and the pairing true model response value of these sample points is saved in the design sample point data base.

When k 〉=2, true model in the invocation step 1, increase the response of the pairing true model of sample point in the design sample point data base in the calculation procedure 8 newly, and these new sample point and pairing true model response value thereof are saved in the design sample point data base.

Step 3, structure be base (RBF) agent model radially.

When k=1, all sample point and the response of pairing true model thereof extract in the design sample point data base that step 2 is obtained, and adopt the radially building method of base (RBF) agent model, re-construct radially base (RBF) agent model.

When k 〉=2, the sample points newly-increased and existing whole in the design sample point data base and the response of pairing true model thereof are extracted, adopt the radially building method of base (RBF) agent model, re-construct radially base (RBF) agent model.

Step 4 is found the solution the radially current approximate optimal solution of base (RBF) agent model.

Employing has the optimized Algorithm of ability of searching optimum, and radially base (RBF) agent model that step 3 obtains is tried to achieve the current iteration approximate optimal solution.

Step 5 is calculated the response of current iteration approximate optimal solution in true model.

The current iteration approximate optimal solution that calculation procedure 4 is obtained is updated in the true model, tries to achieve the response of current approximate optimal solution corresponding to true model, and is saved in the set of optimum solution response.

Step 6 judges whether global optimization method satisfies convergence criterion.

If calculate for the first time, promptly k=1 then directly changes step 7 over to.

If not calculating for the first time, be k 〉=2, then by calling the pairing true model response value of approximate optimal solution of the RBF agent model that current the k time iteration and the k-1 time iteration are constructed in the optimum solution response set, calculate their relative error, judge whether this relative error satisfies given convergence criterion epsilon.If satisfy, then stop circulation, and the resultant optimum solution of step 4 is the optimal value of true model, the flow process end of global optimization method of the present invention; If do not satisfy, then change step 7 over to.

Step 7 is determined the important sampling space of next iteration.The building method in important sampling space is as follows:

1. at first determine the position and the size in important sampling space, promptly construct the set B in the k time important sampling space _k=[B _{L, k}, B _{U, K}].The selected important sampling space of the present invention is positioned near the approximate optimal solution of the current agent model that step 4 obtains.Set B _k=[B _{L, k}, B _{U, k}] expression formula as the formula (11).B _kIn the span of i line display i dimension.

$B_{L,k}=x_{k-1}^{*}-\frac{1}{n_s}{\mathrm{BL}}_{k-1}$

(11)

$B_{U,k}=x_{k-1}^{*}+\frac{1}{n_s}{\mathrm{BL}}_{k-1}$

Wherein, B _{L, k}The vector of representing the k time important sampling space lower bound, B _{U, k}The vector of representing the upper bound, the k time important sampling space,

The optimum solution of representing the k-1 time agent model.

BL _K-1The vector of representing the k-1 time important sampling space size,

Represent the size of s dimension in the k-1 time important sampling space, its expression formula is

BL _k-1＝B _U，k-1-B _L，k-1

(12)

${\mathrm{BL}}_{k-1}=\{{\mathrm{BL}}_{k-1}^{\left(1\right)},\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{BL}}_{k-1}^{\left(s\right)}\}$

If the 2. important sampling space B that 1. obtains of step _k=[B _{L, K}, B _{U, k}] too small, can cause newly-increased sample point intensive, to improving near the DeGrain of the approximation quality of radially basic agent model true model optimum solution.So the given minimum important sampling space of the present invention so both can also can make it have global optimizing ability so that this searching method can be jumped out the local optimum point so that the important sampling space can not cause sample point intensive because of shrinking too little.The specific implementation method is: when

The time, then order

Wherein,

Choose size with the given whole design space of step 1

Relevant.

3. when the 1. 2. definite important sampling space B of step with step _k=[B _{L, k}, B _{U, k}] when having exceeded in the step 1 given whole design space, then with the common factor in whole design space and important sampling space as new important sampling space.

Step 8 in the important sampling space that step 7 is constructed, increases new sample point by calculating test design method, and it is preserved into the design sample point data base.

In the important sampling space that step 7 is constructed, adopt and calculate the new sample point of test design method increase, and it is preserved into the design sample point data base.Newly-increased sample point number is determined by (10) formula.

In order to guarantee the newly-increased projection homogeneity of sample point in the important sampling space, must make newly-increased sample point not overlap with the projection of the already present sample point in important sampling space on each dimension.

If certain newly-increased sample point overlaps with the projection of existing sample point on certain one dimension, then make this newly-increased sample point to this dimension right side or left side translation, the translation principle is: in the same global optimization procedure, the translation direction unanimity, (left side) translation to the right when promptly overlapping for the first time, then (left side) translation all to the right each time later on.Simultaneously, in order to guarantee that the projection of resulting new sample point does not overlap with other sample point after the translation, the 1/2n of that one dimension size that the distance that makes translation overlaps for the former projection in the important sampling space _sDoubly.

Step 9 on the basis of step 8, makes k=k+1, changes step 3 over to and carries out next iteration.

Beneficial effect

SRBF is than general once sampling structure agent model technology, if the sample point number of distribution similar number in the design space, so, once sampling is that all sample points are uniformly distributed in whole design space, pay attention in whole design space, improve the approximation quality of agent model and true analytical model; SRBF is distributed in most sample point emphasis near the optimum solution of true analytical model, so near Gou Zao agent model approximation quality with true analytical model optimum solution is very high thus, can solve this optimum solution by global search like this.If, near the identical many sample points that also optimum solution, distribute of the RBF technology by once sampling, so, the increase that the sample point number that is distributed in whole design space will be at double, the number of times increase that so also can cause calling true analytical model.So the SRBF technology is compared and the RBF technology of once sampling, and is significantly improved on efficient.

The present invention has overcome the calculating shortcoming consuming time that traditional global optimization method exists when the Optimum design of engineering structure problem, and compare with static agent model, SRBF can be by calling global optimum's point that less number of times analytical model finds analytical model, can effectively reduce and assess the cost, improve and optimize efficient, help to shorten the cycle of Optimum design of engineering structure.In engineering designs such as aircraft Aerodynamic optimization design, wing optimal design, application promise in clinical practice is arranged.

Description of drawings

Fig. 1 is the optimization uniform Design and the general uniform Design synoptic diagram of prior art;

Fig. 2 is the process flow diagram of the global optimization method based on sequence radial base (SRBF) agent model of the present invention;

Fig. 3 is a newly-increased sample point translation synoptic diagram in the embodiment;

Fig. 4 is BR function three dimensional network trrellis diagram in the design space in the embodiment;

Fig. 5 is the synoptic diagram of I-beam optimal design problem in the embodiment.

Embodiment

The present invention proposes and has realized a kind of efficient global optimization method based on sequence radial base agent model (SRBF), and this method is applicable to the complex engineering optimization problem, help to improve to optimize efficient, and then can compression design cycle and cost.

For purpose of the present invention and advantage better are described, analytical function test example and an I-beam optimum design example below by a standard, the present invention will be further described in conjunction with the accompanying drawings, and, combination property of the present invention is carried out check analysis by comparing with the structure RBF agent model technical result of once sampling.

(1) analytical function is optimized example

Suppose that Branin function (BR) function is to calculate high accuracy analysis model consuming time in the engineering design, by finding the solution the minimum value of BR function in the design space, the performance of checking SRBF.The purpose of design of present embodiment is for improving optimal design efficient, promptly reducing the number of times of finding the solution the BR function.

Given starting condition BR function is shown in (13) formula

$f_{\mathrm{BR}}\left(x\right)={(x_2-\frac{5.1}{4{\mathrm{\&pi;}}^2}{x}_1^2+\frac{5}{\mathrm{\&pi;}}{x}_1-6)}^2+10(1-\frac{1}{8\mathrm{\&pi;}})\cos x_1+10---\left(13\right)$

x ₁∈[-5，10]x ₂∈[0，15]

Objective function f _BR(x) at design space x ₁∈ [5,10] x ₂Three dimensional network trrellis diagram such as a mistake among the ∈ [0,15]! Do not find Reference source.Shown in, the global optimum point of BR function in this design space is $x_{\mathrm{opt}}^{*}=\left(\mathrm{3.1415,2.2749}\right),$ $f_{\mathrm{BR}}^{*}\left(x_{\mathrm{opt}}^{*}\right)=0.3979.$

In order to obtain globally optimal solution, use optimization method with global optimizing ability.Use the structure RBF agent model method of once adopting of SRBF agent model method of the present invention and prior art at this.Wherein, the structure RBF agent model method of once adopting of prior art adopts two kinds to adopt the diverse ways of counting and describe respectively: once the adopting of method I counted identical with SRBF agent model method of the present invention; It is 100 sample points that once the adopting of method II counted.Use genetic algorithm that above three kinds of agent models are found the solution, so that analyze comparison.

As shown in Figure 2, the specific implementation step of global optimization method of the present invention is as follows:

Step 1, the true model of present embodiment is the optimization mathematical model of BR function, as the formula (13); Present embodiment adopts unconstrained optimization, does not promptly have constraint condition; Whole design space is x ₁∈ [5,10], x ₂∈ [0,15] makes iterations k=1, and the method for the calculating test design of present embodiment is the uniform Design method, because the design space is a two dimensional model, so design sample point number

Individual.

Step 2 is calculated the response of true model.

When k=1, by the true model in the invocation step 1, the response of 6 selected pairing true models of sample point in the calculation procedure 1, and these 6 sample points and pairing true model response value thereof are saved in the design sample point data base.

When k 〉=2, true model in the invocation step 1, increase the response of the pairing true model of sample point in the design sample point data base in the calculation procedure 8 newly, and the pairing true model response value of these new sample point is saved in the design sample point data base.

Step 3 is constructed radially basic agent model.

Newly-increased and the existing whole sample point and the response of pairing true model thereof extract in the design sample point data base that step 2 is obtained, and adopt the radially building method of base (RBF) agent model, re-construct radially base (RBF) agent model.

The radial function that present embodiment is selected for use is Gaussian function (Gaussian function), and expression formula is shown below

$\mathrm{\&phi;}(r,c)=\exp (-\frac{1}{2}({\mathrm{<figure-callout\; id="2"\; label="r"\; filenames="HSA00000195088400011.png,HSA00000195088400021.png"\; state="\{\{state\}\}">r</figure-callout>}}^2/c^2))---\left(14\right)$

Wherein, φ is a radial function, and r is the Euclidean distance between future position and the arbitrary sample point, and c is normal real number.Rule of thumb, c gets the arithmetic number between [0,10] usually.

Step 4 is found the solution the radially optimum solution of base (RBF) agent model.

The global optimization approach that present embodiment is selected for use is genetic algorithm (GA), and radially base (RBF) agent model that step 3 obtains is tried to achieve the current iteration approximate optimal solution.

Step 5 is calculated the response of current iteration approximate optimal solution in true model.

The current iteration approximate optimal solution that calculation procedure 4 is obtained is updated in the true model, tries to achieve the response of current approximate optimal solution corresponding to true model, and is saved in the set of optimum solution response.

Step 6 judges whether global optimization method satisfies convergence criterion.

If calculate for the first time, promptly k=1 then directly changes step 7 over to.

If not calculating for the first time, be k 〉=2, then by calling the pairing true model response value of approximate optimal solution of the RBF agent model that current the k time iteration and the k-1 time iteration are constructed in the optimum solution response set, calculate their relative error, judge whether this relative error satisfies given convergence criterion epsilon=0.01.If satisfy, then stop circulation, and the resultant optimum solution of step 4 is the optimal value of true model, the flow process end of global optimization method of the present invention; If do not satisfy, then change step 7 over to.

Step 7 is determined the important sampling space of next iteration.Being constructed as follows of important sampling space:

(1) at first determines the position and the size in important sampling space, promptly construct the set B in the k time important sampling space _k=[B _{L, k}, B _{U, k}].The selected important sampling space of the present invention is positioned near the approximate optimal solution of the current agent model that step 4 obtains.Set B _k=[B _Lk, B _{U, k}] expression formula as the formula (11).B _kIn the span of i line display i dimension.

(2) if the important sampling space B that step 7 (1) obtains _k=[B _{L, k}, B _{U, k}] too small, can cause newly-increased sample point intensive, to improving near the DeGrain of the approximation quality of radially basic agent model true model optimum solution.So the given minimum important sampling space of the present invention so both can also can make it have global optimizing ability so that this searching method can be jumped out the local optimum point so that the important sampling space can not cause sample point intensive because of shrinking too little.For example, the k time sample space calculated by (11) in the upper bound of i dimension

If

Then order

If

Then order

According to the test experience, get in the present embodiment

Can guarantee the most like this

Little space is unlikely to too little, also is unlikely to too big, makes the present invention have ability of searching optimum.

(3) if the important sampling space B that step 7 (1) and step 7 (2) obtain _k=[B _{L, k}, B _{U, k}] exceeded whole design space given in the step 1, then with the common factor in whole design space and important sampling space as new important sampling space.

Step 8 in the important sampling space that step 7 is constructed, increases new sample point by the uniform Design method, and it is saved in the design sample point data base.Newly-increased sample point number is 6.

In order to guarantee newly-increased sample point projection homogeneity in the important sampling space, must make newly-increased sample point not overlap with the projection of the already present sample point in important sampling space on each dimension.

In the present embodiment, the translation principle is: if certain newly-increased sample point overlaps with the projection of existing sample point on certain one dimension, then make this newly-increased sample point to this dimension right side translation.Simultaneously, in order to guarantee that the projection of resulting new sample point does not overlap with other sample point after the translation, 1/12 times of that one dimension size that the distance that makes translation overlaps for the former projection in the important sampling space.

Step 9 on the basis of step 8, makes k=k+1, changes step 3 over to and carries out next iteration.

Find the solution the required sample point of BR example by SRBF agent model method of the present invention and add up to 36, now adopt method I (selecting 36 sample points for use) and method II (selecting 100 sample points for use) in same design space, once to sample respectively, and construct its pairing RBF agent model, use genetic algorithm (GA) that the agent model of two method I and method II gained is optimized respectively then.In addition, directly use genetic algorithm (GA) that the BR function is optimized.The result of calculation of SRBF, method I, method II and GA is compared, as shown in table 1.

Table 1BR function optimization result contrast

Aspect the optimization effect, by table 1 as can be known, SRBF optimization method of the present invention and GA optimized Algorithm can find the globally optimal solution of BR function fully, have good global optimizing ability.The globally optimal solution and method I and method II fail to find, but the optimization result of method II is better than the optimization result of method I, this is because the sample point number of method II institute cloth will be far away more than method I in the design space, makes the approximation quality of agent model that method II constructed and true model will be higher than the approximation quality of agent model that method I constructed and true model.

Aspect efficient, as shown in Table 1, it is 42 times that SRBF optimization method of the present invention calls BR Functional Analysis model number of times, and than method II, the required analytical model number of times that calls of SRBF has reduced about 58.4%; And compare algorithm with GA, the number of times that SRBF calls analytical model only is 4% of GA; Than method I, though SRBF call analytical model often 5 times, this is because SRBF needs to call the pairing response of the current optimum solution of true analysis model solution in each iterative process, but because method I is being nothing like SRBF aspect the effect optimizing, so even many number of times that calls analytical model several times also can ignore.

The concrete enforcement that the present invention optimizes example at above-mentioned analytical function shows, the SRBF global optimization method is in treatment project optimal design problem, help reducing the number of times that calls the high precision model, thereby reach the purpose of raising the efficiency, and the SRBF global optimization method also has good global optimizing ability.

(2) I-beam optimum design example

Performance by I-beam optimum design example checking SRBF.The target of this project design problem is under the condition that satisfies the constraint of given cross-sectional area and pressure, makes the vertical missing minimum of I-beam, as mistake! Do not find Reference source.Shown in.The basic parameter of design problem is:

● the maximum stress in bend of beam is 6kN/cm ²

● Young modulus is 2 * 10 ⁴KN/cm ²

● maximum deflection pressure is P=600kN and Q=50kN

● the length of beam is L=200cmm

Then according to design parameter, can be with the mathematical description of this project optimization problem:

$\min f\left(x\right)=\frac{5000}{\frac{1}{12}{x}_3{(x_1-2x_4)}^3+\frac{1}{6}{x}_2{x}_4^3+2x_2{x}_4{\left(\frac{x_1-x_4}{2}\right)}^2}$

s.t.g ₁(x)＝2x ₂x ₄+x ₃(x ₁-2x ₄)≤300 (15)

$g_2\left(x\right)=\frac{180000x_1}{x_3{(x_1-2x_4)}^3+2x_2{x}_4[4x_4^2+3x_1(x_1-2x_4)]}+\frac{15000x_2}{(x_1-2x_4)x_3^3+2x_4{x}_2^3}\&le;6$

$10\&le;x_1\&le;\mathrm{80,10}\&le;x_2\&le;\mathrm{50,0.9}\&le;x_3\&le;\mathrm{5,0.9}\&le;x_4\&le;5$

Wherein, objective function f (x) is the deflection of I-beam, g ₁(x) be cross-sectional area, g ₂(x) be crooked pressure.

The problems referred to above are a non-linear constrain problem, now adopt the present invention that this problem is optimized and find the solution, and adopt SRBF to carry out solving-optimizing to objective function, and constraint condition is used true analytical model; The genetic algorithm of selecting for use MATLAB to carry is simultaneously found the solution, and compares the result of two kinds of algorithms.It is as follows that SRBF finds the solution the concrete implementation step of this I-beam optimization problem.Result of calculation is as shown in table 2.

Step 1 because present embodiment is a constrained optimization, adopts the SRBF agent model to be optimized to objective function, so true model is selected objective function for use, shown in (16).Whole design space is 10≤x ₁≤ 80,10≤x ₂≤ 50,0.9≤x ₃≤ 5,0.9≤x ₄≤ 5, make iterations k=1, the method for the calculating test design of present embodiment is the uniform Design method, because the design space is four-dimensional model, so design sample point number

Individual.

$f\left(x\right)=\frac{5000}{\frac{1}{12}{x}_3{(x_1-2x_4)}^3+\frac{1}{6}{x}_2{x}_4^3+2x_2{x}_4{\left(\frac{x_1-x_4}{2}\right)}^2}---\left(16\right)$

Step 2 is calculated the response of true model.

When k=1, by the true model in the invocation step 1, the response of 15 selected pairing true models of sample point in the calculation procedure 1, and the pairing true model response value of these 15 sample points is saved in the design sample point data base.

When k 〉=2, true model in the invocation step 1, increase the response of the pairing true model of sample point in the design sample point data base in the calculation procedure 8 newly, and these new sample point and pairing true model response value thereof are saved in the design sample point data base.

Step 3 is constructed radially basic agent model.

Newly-increased and the existing whole sample point and the response of pairing true model thereof extract in the design sample point data base that step 2 is obtained, and adopt the radially building method of base (RBF) agent model, re-construct radially base (RBF) agent model.

The radial function that present embodiment is selected for use is Gaussian function (Gaussian function).

Step 4 is found the solution the radially optimum solution of base (RBF) agent model.

The global optimization approach that present embodiment is selected for use is genetic algorithm (GA), and radially base (RBF) agent model that step 3 obtains is tried to achieve the current iteration approximate optimal solution.

Step 5 is calculated the response of current iteration approximate optimal solution in true model.

The current iteration approximate optimal solution that calculation procedure 4 is obtained is updated in the true model, tries to achieve the response of current approximate optimal solution corresponding to true model, and is saved in the optimum solution set.

Step 6 judges whether global optimization method satisfies convergence criterion.

If calculate for the first time, promptly k=1 then directly changes step 7 over to.

If not calculating for the first time, be k 〉=2, then by calling the pairing true model response value of approximate optimal solution of the RBF agent model that current the k time iteration and the k-1 time iteration are constructed in the optimum solution set, calculate their relative error, judge whether this relative error satisfies given convergence criterion epsilon=0.01.If satisfy, then stop circulation, and the resultant optimum solution of step 4 is the optimal value of true model, the flow process end of global optimization method of the present invention; If do not satisfy, then change step 7 over to.

Step 7 is determined the important sampling space of next iteration.

1) at first determines the position and the size in important sampling space, promptly construct the set B in the k time important sampling space _k=[B _{L, k}, B _{U, k}].The selected important sampling space of the present invention is positioned near the approximate optimal solution of the current agent model that step 4 obtains.Set B _k=[B _{L, k}, B _{U, k}] expression formula as the formula (11).B _kIn the span of i line display i dimension.

2) if the important sampling space B that step 7 (1) obtains _k=[B _{L, k}, B _{U, k}] too small, can cause newly-increased sample point intensive, to improving near the DeGrain of the approximation quality of radially basic agent model true model optimum solution.So the given minimum important sampling space of the present invention so both can also can make it have global optimizing ability so that this searching method can be jumped out the local optimum point so that the important sampling space can not cause sample point intensive because of shrinking too little.For example, the k time sample space calculated by (11) in the upper bound of i dimension

If

Then order

If

Then order

Get according to the test experience, get in the present embodiment

Can guarantee that like this minimum space is unlikely to too little, also be unlikely to too big, make the present invention have ability of searching optimum.

3) if the important sampling space B that step 7 (1) and step 7 (2) obtain _k=[B _{L, k}, B _{U, k}] exceeded whole design space given in the step 1, then with the common factor in whole design space and important sampling space as new important sampling space.

Step 8 in the important sampling space that step 7 is constructed, increases new sample point by the uniform Design method, and it is saved in the design sample point data base.Newly-increased sample point number is 15.

In order to guarantee newly-increased sample point projection homogeneity in the important sampling space, must make newly-increased sample point not overlap with the projection of the already present sample point in important sampling space on each dimension.

In the present embodiment, the translation principle is: if certain newly-increased sample point overlaps with the projection of existing sample point on certain one dimension, then make this newly-increased sample point to the translation of the right side of this dimension.Simultaneously, in order to guarantee that the projection of resulting new sample point does not overlap with other sample point after the translation, 1/30 times of make the distance of translation attach most importance to that one dimension size that the sample space overlaps in former projection.

Step 9 on the basis of step 8, makes k=k+1, changes step 3 over to and carries out next iteration.

Table 2 I-beam optimization problem result relatively

Data by table 2 show, the optimal result that SRBF optimization method proposed by the invention is tried to achieve is 0.0137, and satisfy the constraint of cross-sectional area constraint and crooked pressure simultaneously, it is optimized the result and gets over the genetic algorithm that MATLAB carries and compare, and the optimization result much at one.Aspect optimization efficient, to compare with genetic algorithm, the number of times that SRBF calls analytical model only is 0.46% of genetic algorithm, optimizing efficient is to be higher than genetic algorithm far away.Obviously, SRBF can obtain the optimum solution of former engineering optimization problem simultaneously under the prerequisite of a large amount of reduction calculated amount.

This shows that the present invention has realized the goal of the invention of expection substantially, compare and once adopt a structure RBF agent model and directly use genetic algorithm that under the prerequisite that can obtain the optimization problem optimum solution, it is optimized efficient and has obtained improving greatly.So SRBF can effectively reduce on the one hand and assess the cost, and improves and optimizes efficient, helps to shorten the cycle of engineering design; On the other hand, of the present invention have a very strong global optimizing ability, improved the ability of the global optimization in the Optimum design of engineering structure problem, helps improving the quality of engineering design.

Above-described specific descriptions; purpose, technical scheme and beneficial effect to invention further describe; institute is understood that; the above only is specific embodiments of the invention; be used to explain the present invention, and be not intended to limit the scope of the invention, within the spirit and principles in the present invention all; any modification of being made, be equal to replacement, improvement etc., all should be included within protection scope of the present invention.

Claims (1)

1. the efficient global optimization method based on sequence radial base agent model is characterized in that: comprise the steps:

Step 1 by the given starting condition of user, is selected the sample point in the first Iterative Design space

As true model, and with design variable related in the true model, objective function constraint condition, and whole design space makes iteration count parameter k=1 as starting condition, carries out first iteration with the given analysis that needs research of user and realistic model; In whole design space, utilize and calculate test design method selection sample point; Selected sample point number n _SFor

$n_s=\frac{(n_v+1)(n_v+2)}{2}$

Wherein, n _vThe dimension of expression design space;

Step 2 is calculated the response of true model

When k=1, by the true model in the invocation step 1, the response of the selected pairing true model of each sample point in the calculation procedure 1, and the pairing true model response value of these sample points is saved in the design sample point data base;

When k 〉=2, true model in the invocation step 1, increase the response of the pairing true model of sample point in the design sample point data base in the calculation procedure 8 newly, and these new sample point and pairing true model response value thereof are saved in the design sample point data base;

Step 3 is constructed radially basic agent model

When k=1, all sample point and the response of pairing true model thereof extract in the design sample point data base that step 2 is obtained, and adopt the radially building method of basic agent model, re-construct radially basic agent model;

When k 〉=2, the sample points newly-increased and existing whole in the design sample point data base and the response of pairing true model thereof are extracted, adopt the radially building method of basic agent model, re-construct radially basic agent model;

Step 4 is found the solution the radially current approximate optimal solution of base (RBF) agent model

Employing has the optimized Algorithm of ability of searching optimum, and the radially basic agent model that step 3 obtains is tried to achieve the current iteration approximate optimal solution;

Step 5 is calculated the response of current iteration approximate optimal solution in true model

The current iteration approximate optimal solution that calculation procedure 4 is obtained is updated in the true model, tries to achieve the response of current approximate optimal solution corresponding to true model, and is saved in the set of optimum solution response;

Step 6 judges whether global optimization method satisfies convergence criterion

If calculate for the first time, promptly k=1 then directly changes step 7 over to;

If not calculating for the first time, be k 〉=2, then by calling the pairing true model response value of approximate optimal solution of the radially basic agent model that current the k time iteration and the k-1 time iteration are constructed in the optimum solution response set, calculate their relative error, judge whether this relative error satisfies given convergence criterion epsilon; If satisfy, then stop circulation, and the resultant approximate optimal solution of step 4 is the optimal value of true model, the flow process end of global optimization method of the present invention; If do not satisfy, then change step 7 over to;

Step 7 is determined the important sampling space of next iteration

The building method in important sampling space is as follows:

1. determine the position and the size in important sampling space, promptly construct the set B in the k time important sampling space _k=[B _{L, k}, B _{U, k}]; Selected important sampling space is positioned near the approximate optimal solution of the current agent model that step 4 obtains; Set B _k=[B _{L, k}, B _{L, k}] expression formula be shown below; B _kIn the span of i line display i dimension;

$B_{L,k}=x_{k-1}^{*}-\frac{1}{n_s}{\mathrm{BL}}_{k-1}$

$B_{U,k}=x_{k-1}^{*}+\frac{1}{n_s}{\mathrm{BL}}_{k-1}$

Wherein, B _{L, k}The vector of representing the k time important sampling space lower bound, B _{U, k}The vector of representing the upper bound, the k time important sampling space,

The optimum solution of representing k～1 time agent model;

BL _K-1The vector of representing the k-1 time important sampling space size,

Represent the size of s dimension in the k-1 time important sampling space, its expression formula is

BL _k-1＝B _U，k-1-B _L，k-1

${\mathrm{BL}}_{k-1}=\{{\mathrm{BL}}_{k-1}^{\left(1\right)},\&CenterDot;\&CenterDot;\&CenterDot;,{\mathrm{BL}}_{k-1}^{\left(s\right)}\}$

2. a given minimum important sampling space

When

The time, then order

Wherein,

Choose size with the given whole design space of step 1

Relevant;

3. when the 1. 2. definite important sampling space B of step with step _k=[B _{L, k}, B _{U, k}] when having exceeded in the step 1 given whole design space, then with the common factor in whole design space and important sampling space as new important sampling space.

Step 8 in the important sampling space that step 7 is constructed, increases new sample point by calculating test design method, and it is preserved into the design sample point data base;

In the important sampling space that step 7 is constructed, adopt and calculate the new sample point of test design method increase, and it is preserved into the design sample point data base; Newly-increased sample point number is determined by formula in the step 1;

In order to guarantee the newly-increased projection homogeneity of sample point in the important sampling space, must make newly-increased sample point not overlap with the projection of the already present sample point in important sampling space on each dimension;

If certain newly-increased sample point overlaps with the projection of existing sample point on certain one dimension, then make this newly-increased sample point to this dimension right side or left side translation, the translation principle is: in the same global optimization procedure, the translation direction unanimity, when promptly overlapping for the first time to the right or left, then after each time all to the right or left; Simultaneously, in order to guarantee that the projection of resulting new sample point does not overlap with other sample point after the translation, the 1/2n of that one dimension size that the distance that makes translation overlaps for the former projection in the important sampling space _sDoubly;

Step 9 on the basis of step 8, makes k=k+1, changes step 3 over to and carries out next iteration.

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