CN110457814A - Multi-modal cross entropy optimization algorithm based on mixed Gauss model - Google Patents
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Abstract
The multi-modal cross entropy optimization algorithm based on mixed Gauss model that the invention discloses a kind of.The algorithm includes the following steps: to determine objective function according to satellite master-plan demand;Based on objective function, multiple random samples are generated, and set loop termination condition;It is clustered using the niche algorithm based on exclusion, multiple random samples is divided into multiple microhabitats;The optimal sample in each microhabitat is picked out, mixed Gauss model is constructed, and Cross-Entropy Algorithm is used to carry out sampling iteration to obtain multiple solutions;Judge whether loop termination condition meets, if so, carrying out in next step, if it is not, return step 3;Local optimal searching is carried out using Sequential Quadratic Programming method in each Xie Chu of multiple solutions of acquisition, obtains and exports multiple globally optimal solutions.Multi-modal cross entropy optimization algorithm of the invention can be used to solve the challenge such as satellite master-plan one kind, be effectively reduced algorithm calculation amount, can obtain multiple globally optimal solutions, to provide more multi-design methods.
Description
Technical Field
The invention relates to the technical field of satellite overall design, in particular to a multi-mode cross entropy optimization algorithm based on a Gaussian mixture model.
Background
The multidisciplinary design optimization technology is a complex system design method for performing optimization design from the whole system perspective by fully exploring and utilizing a cooperative mechanism of interaction in an engineering system and considering the interaction between each discipline (each subsystem). For the overall design of the satellite, the traditional design optimization algorithm mainly comprises a gradient method and an intelligent optimization algorithm, but the gradient method is easy to fall into local optimization, and most of the intelligent optimization algorithms can only obtain one solution. In order to overcome the defects, a multi-modal optimization algorithm is adopted to process the multi-modal problem of the satellite at present, the existing multi-modal optimization algorithm is divided into three types, the first type is based on a complex model, the model can comprise a plurality of sub-models, and each sub-model can obtain a solution in the problem solving process; the second type is that a clustering idea and an evolutionary algorithm are combined, the clustering idea is used for dividing the population, and then each type corresponds to an area to be solved by using an intelligent optimization algorithm; the third type is to improve the population diversity by combining the group intelligent optimization algorithm and the traditional optimization algorithm, such as particle swarm, and then to perform local optimization by the traditional optimization algorithm, such as a gradient method.
However, as technology advances, satellite systems become more complex, such as more constraints, more targets, and higher dimensions. For the overall design of more complex satellites, in the existing three types of multi-modal optimization methods, when the first type of algorithm is used for optimizing the overall design of the satellites, a complex model is difficult to establish, and a large amount of computing resources are consumed when an Expectation Maximization (EM) algorithm is used for computing; when the second type of algorithm is used for satellite overall design optimization, division is mainly carried out based on a niche (niche) technology, the algorithm is sensitive to parameter setting, and the niche radius or the population number greatly affects the solving effect of the algorithm; when the third type of algorithm is used for overall design optimization of the satellite, the more complex the structure and the system of the satellite are, the more complex the parameters of the algorithm are, and for the complex system, the solving effect of the algorithm is poor in the actual solving process.
Disclosure of Invention
In order to solve the technical problems in the prior art, the invention provides a multi-modal cross entropy optimization algorithm based on a Gaussian mixture model.
Therefore, the invention discloses a multi-mode cross entropy optimization algorithm based on a Gaussian mixture model, which comprises the following steps:
1) determining an objective function according to the overall design requirements of the satellite;
2) generating a plurality of random samples based on the objective function, and setting a cycle termination condition;
3) clustering by using a niche algorithm based on displacement to divide the plurality of random samples into a plurality of niches;
4) selecting an optimal sample in each niche, constructing a Gaussian mixture model, and performing sampling iteration by adopting a cross entropy algorithm to obtain a plurality of solutions;
5) judging whether the circulation termination condition is met, if so, carrying out the next step, and if not, returning to the step 3;
6) and performing local optimization at each solution of the obtained multiple solutions by using a sequential quadratic programming method, and obtaining and outputting multiple global optimal solutions.
Further, in the multi-modal cross entropy optimization algorithm based on the Gaussian mixture model, clustering is carried out by the niche algorithm based on displacement according to formula 1;
wherein S isiDenotes the ith niche, P denotes the population excluding individuals in the first i-1 niche, xjDenotes the jth individual in the population P, xseedRepresents the optimal individual in the ith niche, riDenotes the niche radius of the ith niche, and D denotes the dimension of the design variable.
Further, in the multi-modal cross entropy optimization algorithm based on the Gaussian mixture model, the extrusion-based niche algorithm comprises the following steps:
A1) setting the population as PiSetting the radius of the niche as ri;
A2) And (5) setting i to 1, and determining a population P according to the objective function and the design requirementiInitial value P of1And a small habitat radius riInitial value r of1;
A3) Using the population PiTarget function value pair population P corresponding to inner sampleiSorting is carried out;
A4) judging the population PiIf the set is empty, performing step A7; if not, carrying out the next step;
A5) selecting a population PiBest sample R in (1)iFor clustering centers, a population P is searchediAll and best samples R iniThe distance is less than the radius r of the nicheiAnd aggregating the searched individuals into a niche Si;
A6) Increase the value of i by 1 from the population PiExcluding the searched individuals, and setting the radius r of the nicheiReturns to step a 4;
A7) and finishing the circulation and outputting all the obtained niches.
Further, in the multi-modal cross entropy optimization algorithm based on the Gaussian mixture model, the optimal individual of each ecological niche is selected as a mean center, three times of the radius of each ecological niche is selected as a mean square error, and the Gaussian mixture model is constructed.
Further, in the multi-modal cross entropy optimization algorithm based on the Gaussian mixture model, a probability model of the Gaussian mixture model is shown as formula 2 and formula 3:
wherein c represents the number of Gaussian distributions, n represents the dimension of the design variable, μi,jAnd σi,jDistribution parameter, mu, representing the j-th design variable in the i-th Gaussian distributioni,jDenotes the mean value, σi,jDenotes mean square error, piRepresenting the weight corresponding to the ith Gaussian distribution, e representing a natural constant, f (x) representing a probability distribution function, xi,jAnd the value of the j dimension design variable in the ith Gaussian distribution is shown.
Further, in the multi-modal cross entropy optimization algorithm based on the Gaussian mixture model, the parameter mui,j、σi,jAnd piPerforming iteration by using the formulas 4, 5 and 6 respectively;
wherein epsiloniRepresenting the number of all samples in the ith habitat, FiThe optimal function value in the ith habitat is shown,representing the sum of the optimal function values, p, in all nichesiThe probability that the gaussian distribution corresponding to the ith niche corresponds to the generated sample at the next iteration is shown,anda distribution parameter representing the ith gaussian distribution,the mean value is represented by the average value,denotes mean square error, xi,bestRepresenting the sample with the largest objective function in the ith gaussian distribution.
Further, in the multi-modal cross entropy optimization algorithm based on the Gaussian mixture model, the cross entropy algorithm comprises the following steps:
B1) setting a termination condition according to the objective function;
B2) setting sample capacity N, a quantile point parameter rho and a distribution parameter vector of an initial sampling probability density function according to a design space range;
B3) judging whether the termination condition is met, if so, performing step B7; if not, carrying out the next step;
B4) generating mutually independent samples according to the probability density function and calculating objective function values corresponding to the samples;
B5) sorting the objective function values in the order from small to large, and selecting NeAn elite sample, Ne=ρN;
B6) According to NeUpdating the distribution parameter vector by each elite sample, and then returning to the step B3;
B7) and ending the loop and outputting a solution.
Further, in the multi-modal cross entropy optimization algorithm based on the Gaussian mixture model, the distribution parameter vector of the probability density function comprises a mean vector and a mean square error vector.
Further, in the multi-modal cross entropy optimization algorithm based on the Gaussian mixture model, updating the mean vector and the mean square error vector by using a formula 7 and a formula 8;
wherein,anda distribution parameter vector representing a probability density function,the mean value vector is represented by a mean value vector,represents the mean square error vector, N
Indicating the volume of a sample, i.e. the number of samples, xkDenotes the kth sample, XkIndicates the value of the objective function corresponding to the kth sample, NeRepresenting the number of elite samples, gamma representing the threshold, I { S (X)k≧ γ) } represents an indicative function,
the mixed Gaussian model-based multi-modal cross entropy optimization algorithm can be used for solving complex engineering problems such as overall satellite design by fusing the extrusion-based niche algorithm, the mixed Gaussian model updating mechanism and the cross entropy algorithm and fusing a sequence quadratic programming method local optimization mechanism in the algorithm, can effectively reduce the calculated amount of the algorithm, and can obtain a plurality of global optimal solutions to provide more design schemes.
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The accompanying drawings, which are included to provide a further understanding of embodiments of the invention and are incorporated in and constitute a part of this specification, illustrate embodiments of the invention and together with the description serve to explain the invention and not to limit the invention. In the drawings:
FIG. 1 is a flow chart of a multi-modal cross entropy optimization algorithm based on a Gaussian mixture model according to an embodiment of the present invention;
FIG. 2a, FIG. 2b and FIG. 2c are schematic diagrams of an iterative process of the cross-entropy algorithm according to an embodiment of the present invention;
fig. 3a and fig. 3b respectively show an iterative process schematic diagram and a result schematic diagram of a multi-modal cross entropy optimization algorithm based on a gaussian mixture model according to an embodiment of the present invention.
Detailed Description
In order to make the objects, technical solutions and advantages of the present invention more apparent, the technical solutions of the present invention will be clearly and completely described below with reference to the specific embodiments of the present invention and the accompanying drawings. It is to be understood that the described embodiments are merely a few embodiments of the invention, and not all embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments of the present invention without making any creative effort, shall fall within the protection scope of the present invention.
The technical scheme provided by the embodiment of the invention is described in detail below with reference to the accompanying drawings.
As shown in fig. 1, an embodiment of the present invention provides a multi-modal cross entropy optimization algorithm based on a gaussian mixture model, which includes the following steps:
1) determining an objective function according to the overall design requirements of the satellite;
2) generating a plurality of random samples based on the target function, and setting a cycle termination condition;
3) clustering by using a niche algorithm based on displacement to divide a plurality of random samples into a plurality of niches;
4) selecting an optimal sample in each niche, constructing a Gaussian mixture model, and performing sampling iteration by adopting a cross entropy algorithm to obtain a plurality of solutions;
5) judging whether the circulation termination condition is met, if so, carrying out the next step, and if not, returning to the step 3;
6) and local optimization is carried out at each solution of the obtained multiple solutions by using a sequence quadratic programming method, and a plurality of global optimal solutions are obtained and output.
The principle and steps of the multi-modal cross entropy optimization algorithm based on the Gaussian mixture model provided by the embodiment of the invention are specifically described below.
Specifically, in step 3, the niche algorithm based on displacement may perform clustering using the following formula 1;
wherein S isiDenotes the ith niche, P denotes the population excluding individuals in the first i-1 niche, xjDenotes the jth individual in the population P, xseedRepresents the optimal individual in the ith niche, riDenotes the niche radius of the ith niche, and D denotes the dimension of the design variable.
Based on the above setting, in step 3 of the embodiment of the present invention, the niche algorithm based on displacement includes the following steps:
A1) setting the population as PiSetting the radius of the niche as ri;
A2) Let i equal to 1, and determine the population P according to the objective function and the design requirementiInitial value P of1And a small habitat radius riInitial value r of1;
A3) According to the population PiTarget function value pair population P corresponding to inner sampleiSorting is carried out;
A4) judging the population PiIf the set is empty, performing step A7; if not, carrying out the next step;
A5) selecting a population PiBest sample R in (1)iFor clustering centers, a population P is searchediAll and best samples R iniThe distance is less than the radius r of the nicheiAnd aggregating the searched individuals into a niche Si;
A6) Increase the value of i by 1 from the population PiExcluding the searched individuals, and setting the radius r of the nicheiReturns to step a 4;
A7) and finishing the circulation and outputting all the obtained niches.
Among them, the best sample RiTo be in a population PiThe sample having the largest objective function value among all the samples.
In an embodiment of the present invention, the population P isiInitial value P of1Corresponding to the random samples generated in step 2.
In this manner, by utilizing a crowd-sourcing-based niche algorithm, random samples generated according to an objective function can be divided into a plurality of niches for subsequent processing.
Further, in the embodiment of the present invention, after dividing the population (random sample) into a plurality of niches, a gaussian mixture model is constructed based on the niches, and iterative evolution is performed using the gaussian mixture model.
How to construct the gaussian mixture model and perform iterative population evolution by using the gaussian mixture model is described in detail below.
Specifically, the optimal individual of each of the generated niches is selected as a mean center, three times of the radius of the niche of each niche is selected as a mean square error, and a Gaussian mixture model is constructed for sampling iteration.
Based on the above settings of mean center and mean square error, the probability model of the Gaussian mixture model is shown in equations 2 and 3:
wherein c represents the number of Gaussian distributions, n represents the dimension of the design variable, μi,jAnd σi,jDistribution parameter, mu, representing the j-th design variable in the i-th Gaussian distributioni,jDenotes the mean value, σi,jDenotes mean square error, piRepresents the weight corresponding to the ith Gaussian distribution, e represents a natural constant, f (x) represents the probabilityDistribution function, xi,jAnd the value of the j dimension design variable in the ith Gaussian distribution is shown.
In the process of sampling iteration by using the Gaussian mixture model, the parameter mui,j、σi,jAnd piPerforming iteration by using the formulas 4, 5 and 6 respectively;
wherein epsiloniRepresenting the number of all samples in the ith habitat, FiThe optimal function value in the ith habitat is shown,representing the sum of the optimal function values, p, in all nichesiThe weight corresponding to the ith Gaussian distribution is shown, namely the probability of the generated sample corresponding to the Gaussian distribution corresponding to the ith niche in the next iteration,anda distribution parameter representing the ith gaussian distribution,the mean value is represented by the average value,denotes mean square error, xi,bestThe sample representing the maximum objective function in the ith Gaussian distribution is p for each iterationi、Andthe three distribution parameters are updated.
As can be seen from the above formulas 2 to 6, the gaussian mixture model provided in the embodiment of the present invention does not employ the maximum Expectation (EM) algorithm, so that the calculation amount in the iterative process can be greatly reduced, and by introducing each iterative clustering and using the feature of updating each gaussian distribution probability in the gaussian mixture model, the bad peaks can be automatically eliminated, so as to reduce the unnecessary calculation amount.
Further, after the iterative clustering of a certain number of steps based on the extrusion niche algorithm and the Gaussian mixture model, sampling iteration is performed by using a cross entropy algorithm to obtain a plurality of solutions of the objective function.
How to use the cross-entropy algorithm to obtain multiple solutions of the objective function is described in detail below.
Specifically, in the embodiment of the present invention, the cross entropy algorithm includes the following steps:
B1) setting a termination condition according to the objective function;
B2) setting sample capacity N, a quantile point parameter rho and a distribution parameter vector of an initial sampling probability density function according to a design space range;
B3) judging whether the termination condition is met, if so, performing step B7; if not, carrying out the next step;
B4) generating mutually independent samples according to the probability density function and calculating objective function values corresponding to the samples;
B5) sorting the objective function values in the order from small to large, and selecting NeAn elite sample, Ne=ρN;
B6) According to NeUpdating the distribution parameter vector by each elite sample, and then returning to the step B3;
B7) and ending the loop and outputting a solution.
The distribution parameter vector of the probability density function comprises a mean vector and a mean square error vector.
Specifically, when the cross entropy algorithm is applied specifically, a reasonable initial central point and an initial standard deviation, namely an initial mean vector and an initial mean square error vector, are determined according to a target function and a design space range, and then a first-generation sample is generated by using the initial central point and the initial standard deviation; then based on the first generation sample, updating a mean vector and a mean square error vector by using a formula 7 and a formula 8, generating a next generation sample by using a new mean vector and a mean square error vector to perform iterative computation, and updating the mean vector and the mean square error vector by using the new sample; and (4) circularly iterating until the final result reaches the set condition.
Wherein,anda distribution parameter vector representing a probability density function,the mean value vector is represented by a mean value vector,representing the mean square error vector, N representing the sample capacity, i.e. number of samples, xkDenotes the kth sample, XkIndicates the value of the objective function corresponding to the kth sample, NeRepresenting the number of elite samples, gamma representing the threshold, I { S (X)k≧ γ) } represents an indicative function,
in particular, the amount of the solvent to be used,by an objective functionFor example, a cross entropy algorithm provided by the embodiment of the present invention is explained; as shown in fig. 2a to fig. 2c, fig. 2a may represent a first iteration of the cross entropy algorithm, fig. 2b represents a second iteration of the cross entropy algorithm, and fig. 2c represents a third iteration of the cross entropy algorithm, it can be known that, in the process of the iteration of the cross entropy algorithm, the distribution of the sample points shows that the sample points are continuously converged to the position of the optimal solution, in the diagram, the black points represent the sample points, and the wire frame plane represents the threshold plane.
Further, after iteration of the above steps, each solution in the multiple solutions is obtained to be substantially close to the optimal solution region, but the precision may be relatively low, and in order to further obtain the global optimal solution, local optimization is performed by using a Sequential Quadratic Programming (SQP) method at each solution of the multiple solutions obtained, so that the multiple global optimal solutions of the objective function can be obtained, and the purpose of multi-modal optimization is achieved.
As shown in fig. 3a and fig. 3b, in the multi-modal cross entropy optimization algorithm based on a gaussian mixture model provided in the embodiment of the present invention, samples are first divided into multiple groups, a gaussian mixture model is constructed, for example, group1, group2, group3, group4, and group5, an inferior peak (for example, group5) is automatically eliminated, sampling iteration is performed by using a cross entropy algorithm to obtain multiple solutions, local optimization is performed at each solution of the obtained multiple solutions by using a sequential quadratic programming method, and finally multiple globally optimal solutions of an objective function can be obtained.
Therefore, the multi-modal cross entropy optimization algorithm based on the Gaussian mixture model provided by the embodiment of the invention can be used for solving complex engineering problems such as overall satellite design by fusing the extrusion-based niche algorithm, the Gaussian mixture model updating mechanism and the cross entropy algorithm and integrating the sequence quadratic programming method local optimization mechanism into the algorithm, the calculated amount of the algorithm can be effectively reduced, and a plurality of global optimal solutions can be obtained to provide more design schemes.
It is noted that, in this document, relational terms such as "first" and "second," and the like, may be used solely to distinguish one entity or action from another entity or action without necessarily requiring or implying any actual such relationship or order between such entities or actions. Also, the terms "comprises," "comprising," or any other variation thereof, are intended to cover a non-exclusive inclusion, such that a process, method, article, or apparatus that comprises a list of elements does not include only those elements but may include other elements not expressly listed or inherent to such process, method, article, or apparatus. In addition, "front", "rear", "left", "right", "upper" and "lower" in this document are referred to the placement states shown in the drawings.
Finally, it should be noted that: the above examples are only for illustrating the technical solutions of the present invention, and not for limiting the same; although the present invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical solutions described in the foregoing embodiments may still be modified, or some technical features may be equivalently replaced; and such modifications or substitutions do not depart from the spirit and scope of the corresponding technical solutions of the embodiments of the present invention.
Claims (9)
1. A multi-modal cross entropy optimization algorithm based on a Gaussian mixture model is characterized by comprising the following steps:
1) determining an objective function according to the overall design requirements of the satellite;
2) generating a plurality of random samples based on the objective function, and setting a cycle termination condition;
3) clustering by using a niche algorithm based on displacement to divide the plurality of random samples into a plurality of niches;
4) selecting an optimal sample in each niche, constructing a Gaussian mixture model, and performing sampling iteration by adopting a cross entropy algorithm to obtain a plurality of solutions;
5) judging whether the circulation termination condition is met, if so, carrying out the next step, and if not, returning to the step 3;
6) and performing local optimization at each solution of the obtained multiple solutions by using a sequential quadratic programming method, and obtaining and outputting multiple global optimal solutions.
2. The hybrid gaussian model-based multi-modal cross-entropy optimization algorithm of claim 1, wherein the displacement-based niche algorithm is clustered using equation 1;
wherein S isiDenotes the ith niche, P denotes the population excluding individuals in the first i-1 niche, xjDenotes the jth individual in the population P, xseedRepresents the optimal individual in the ith niche, riDenotes the niche radius of the ith niche, and D denotes the dimension of the design variable.
3. The hybrid gaussian model-based multi-modal cross-entropy optimization algorithm of claim 2, wherein the displacement-based niche algorithm comprises the steps of:
A1) setting the population as PiSetting the radius of the niche as ri;
A2) And (5) setting i to 1, and determining a population P according to the objective function and the design requirementiInitial value P of1And a small habitat radius riInitial value r of1;
A3) Using the population PiTarget function value pair population P corresponding to inner sampleiSorting is carried out;
A4) judging the population PiIf the set is empty, performing step A7; if not, carrying out the next step;
A5) selecting a population PiBest sample R in (1)iFor clustering centers, a population P is searchediAll and best samples R iniThe distance is smallIn a small habitat radius riAnd aggregating the searched individuals into a niche Si;
A6) Increase the value of i by 1 from the population PiExcluding the searched individuals, and setting the radius r of the nicheiReturns to step a 4;
A7) and finishing the circulation and outputting all the obtained niches.
4. The multi-modal cross-entropy optimization algorithm based on the Gaussian mixture model of claim 3, wherein the optimal individual of each niche is selected as a mean center, three times of the radius of each niche is selected as a mean square error, and the Gaussian mixture model is constructed.
5. The cross-entropy algorithm of claim 4, wherein the probability model of the Gaussian mixture model is as shown in equations 2 and 3:
wherein c represents the number of Gaussian distributions, n represents the dimension of the design variable, μi,jAnd σi,jDistribution parameter, mu, representing the j-th design variable in the i-th Gaussian distributioni,jDenotes the mean value, σi,jDenotes mean square error, piRepresenting the weight corresponding to the ith Gaussian distribution, e representing a natural constant, f (x) representing a probability distribution function, xi,jAnd the value of the j dimension design variable in the ith Gaussian distribution is shown.
6. The cross-entropy hybrid Gaussian model-based optimization algorithm of claim 5, wherein the parameter μi,j、σi,jAnd piBy using the formulae 4, 5 and 6 respectivelyIteration is carried out;
wherein epsiloniRepresenting the number of all samples in the ith habitat, FiThe optimal function value in the ith habitat is shown,representing the sum of the optimal function values, p, in all nichesiThe probability that the gaussian distribution corresponding to the ith niche corresponds to the generated sample at the next iteration is shown,anda distribution parameter representing the ith gaussian distribution,the mean value is represented by the average value,denotes mean square error, xi,bestRepresenting the sample with the largest objective function in the ith gaussian distribution.
7. The cross-entropy hybrid-gaussian-model-based optimization algorithm of claim 6, comprising the following steps:
B1) setting a termination condition according to the objective function;
B2) setting sample capacity N, a quantile point parameter rho and a distribution parameter vector of an initial sampling probability density function according to a design space range;
B3) judging whether the termination condition is met, if so, performing step B7; if not, carrying out the next step;
B4) generating mutually independent samples according to the probability density function and calculating objective function values corresponding to the samples;
B5) sorting the objective function values in the order from small to large, and selecting NeAn elite sample, Ne=ρN;
B6) According to NeUpdating the distribution parameter vector by each elite sample, and then returning to the step B3;
B7) and ending the loop and outputting a solution.
8. The cross-entropy algorithm of claim 7, wherein the distribution parameter vector of the probability density function comprises a mean vector and a mean square error vector.
9. The cross-entropy hybrid-gaussian-model-based optimization algorithm of claim 8, wherein the mean vector and the mean-squared error vector are updated by using equations 7 and 8;
wherein,andrepresenting a probability density functionThe distribution parameter vector of (a) is,the mean value vector is represented by a mean value vector,representing the mean square error vector, N representing the sample capacity, i.e. number of samples, xkDenotes the kth sample, XkIndicates the value of the objective function corresponding to the kth sample, NeRepresenting the number of elite samples, gamma representing the threshold, I { S (X)k≧ γ) } represents an indicative function,
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CN114327859A (en) * | 2021-11-18 | 2022-04-12 | 西安电子科技大学 | Source model cluster selection method for cloud computing environment large-scale problem agent optimization |
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CN114327859A (en) * | 2021-11-18 | 2022-04-12 | 西安电子科技大学 | Source model cluster selection method for cloud computing environment large-scale problem agent optimization |
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Application publication date: 20191115 |