CN111369000A - High-dimensional multi-target evolution method based on decomposition - Google Patents

Abstract

The invention provides a decomposition-based high-dimensional multi-objective evolution method, generating a reference vector, decomposing a high-dimensional multi-objective optimization problem into a plurality of single-objective optimization sub-problems, and constructing a sub-population of the single-objective optimization sub-problems based on the reference vector, Use the allocation mechanism to assign individuals to the sub-population, and construct a neighborhood sub-population, use the constructed neighborhood sub-population to select individuals for genetic evolution, and use the designed local and global selection strategies to select individuals with excellent performance in the population to enter the next generation of populations , repeat the evolution process until termination and obtain the Pareto solution set of the high-dimensional multi-objective optimization problem. The invention effectively reduces the solving complexity of the problem, solves the problem that the multi-objective optimization algorithm is difficult to ensure a good balance between population convergence and diversity, obtains a Pareto solution set with good diversity and convergence, effectively improves the algorithm efficiency, and can effectively Ensure the global convergence and population diversity of the algorithm.

Description

A high-dimensional multi-objective evolution method based on decomposition

technical field

The invention relates to the field of intelligent optimization algorithms, in particular to a multi-objective evolution method.

Background technique

Many real-life problems are actually multi-objective optimization problems (MOPs), i.e. optimization problems with two or more objectives. Since the multi-objective optimization problem has multiple conflicting objectives, the final result is not an optimal solution but a set of Pareto solutions. With the increase of the number of objectives, the optimization problem increases from the original 2-3 objectives to 4 or more, and the problem becomes a high-dimensional multi-objective optimization problem (MaOPs). Due to the increase in the number of targets, the number of non-dominated individuals in the population increases sharply. When the target increases to a certain number, almost all individuals in the population are non-dominated, so that the selection pressure of the evolutionary population decreases rapidly, that is, when the dominance relationship is used as the selection The pressure on the population to converge towards the Pareto Front (PF) when the criterion of finite population size individuals. When the existing multi-objective evolutionary algorithms solve high-dimensional multi-objective optimization problems, in order to ensure a good balance between population diversity and convergence, the evolutionary algorithms pose a huge challenge. Although evolutionary algorithms have shown excellent performance in solving multi-objective optimization problems, for high-dimensional multi-objective optimization problems, existing methods have difficulties such as difficulty in expanding the objective dimension, inability to distinguish evolutionary individuals by Pareto dominance, and failure of diversity maintenance strategies. .

The proposed high-dimensional multi-objective optimization algorithms are mainly divided into three categories:

1) A method based on Pareto domination. This kind of method enhances the selection pressure by modifying the Pareto dominance relation or the diversity maintenance strategy, so as to solve the high-dimensional multi-objective optimization problem.

2) Indicator-based approach. This kind of method guides the evolution of the population through some evaluation indicators, so as to solve the high-dimensional multi-objective optimization problem.

3) Decomposition-based approach. This kind of method decomposes the high-dimensional objective optimization problem into several multi-objective optimization sub-problems or a series of single-objective optimization sub-problems by introducing aggregation functions, and then simultaneously evolves the decomposed multi-objective or single-objective optimization sub-problems, so as to solve high-dimensional optimization problems. Multi-objective optimization problem.

SUMMARY OF THE INVENTION

In order to overcome the deficiencies of the existing technology, a high-dimensional multi-objective evolutionary algorithm based on decomposition is proposed, which mainly includes reference Vector generation, subpopulation construction, population initialization, individual fitness evaluation, ideal point and extreme point calculation and update, target vector normalization, target vector and reference vector association, population non-dominated sorting, neighborhood subpopulation construction, crossover Variant individual selection, offspring population generation, and environmental selection strategies. The invention generates a series of uniformly distributed reference vectors, uses the reference vectors to decompose the high-dimensional multi-objective optimization problem into multiple single-objective optimization sub-problems, builds sub-populations of the single-objective optimization sub-problems based on the reference vectors, and uses an allocation mechanism to create sub-populations. Allocate individuals, build neighborhood subpopulations based on subpopulations, use the constructed neighborhood subpopulations to select individuals for genetic evolution, and use the designed local and global selection strategies to select individuals with excellent performance in the population to enter the next generation population, and repeat the execution The evolution process is performed until the algorithm terminates and the Pareto solution set of the high-dimensional multi-objective optimization problem is obtained. Through the idea of decomposition, the selection strategy of crossover and mutation individuals, the adaptive adjustment of crossover and mutation probability, and the environment selection strategy, the algorithm ensures a good balance between population convergence and diversity when solving high-dimensional multi-objective optimization problems. The shortcomings of the high-dimensional multi-objective optimization algorithm based on Pareto dominance relation are effectively improved when solving high-dimensional multi-objective optimization problems.

The technical scheme adopted by the present invention to solve its technical problem comprises the following steps:

Step 1, parameter setting: algebra g=0, maximum number of iterations G _max , target number M≥4, population size N;

Step 2, generate N reference vectors W={W ₁ , W ₂ ,...,W _N }:

Step 2.1, using the normal boundary intersection (Penalty Boundary Intersection, PBI) method to generate a uniformly distributed reference point RP={RP ₁ ,RP ₂ ,...,RP _N } on a unit hyperplane L;

Step 2.2, based on the reference points RP ₁ , RP ₂ ,...,RP _N , construct uniformly distributed reference vectors W ₁ , W ₂ ,..., W _N in the target space, and the reference vectors satisfy the following conditions:

表示参考向量W_i的第j维度的分量值，且

H表示将每个目标划分为H等份，生成参考向量的数量为N，其中

in

represents the component value of the _jth dimension of the reference vector Wi, and

H means that each target is divided into H equal parts, and the number of generated reference vectors is N, where

Step 3, randomly generate an initial population P _g = 0 of size N, and perform individual fitness evaluation:

Step 3.1, under the condition that the constraints are satisfied, randomly generate an initial population P _g = {X ₁ , X ₂ ,...,X _i ...,X _N } with a population size of N from the decision space, where X _i = {x ₁ , x ₂ ,...,x _D }, i=1,2,...,N, D represents the dimension of the decision variable X _i , and g represents the algebra of the current population;

Step 3.2, calculate the target vector F(X)=(f ₁ (X), f ₂ (X),..., f _M (X)) of all individuals in the population P _g , X∈P _g , and let the individual The fitness value of X is Fit(F(X))=F(X);

Step 3.3, based on the target vector F(X) of all individuals, calculate the ideal point Z ^* , the extreme point Z ^nad and the worst point Z ^worst , but in the actual multi-objective optimization process, the ideal point Z ^* , the worst point Z ^worst and The extreme point Z ^nad cannot be predicted in advance. Based on the target vector of the individuals in the current population P _g , the ideal point Z ^* , the worst point Z ^worst and the extreme point Z ^nad are calculated as:

其中

任意X∈P_g；(1) Ideal point

in

any _X∈Pg ;

其中

任意X∈P_g；(2) Worst point

in

any _X∈Pg ;

其中

而极值点

表示极小值点对应的自变量，即

(3) Extreme point

in

the extreme point

represents the independent variable corresponding to the minimum point, that is

Step 3.4, normalize the target vectors of all individuals in the population P _g ;

进行，其中f_i(X)表示个体X实际目标值，f′_i(X)表示个体X执行归一化后的目标向量，其中i∈1,2,...,M，后续提到的目标向量若不具体说明，则都是经过归一化处理之后的目标向量，且令f_i(X)＝f′_i(X)，即用f_i(X)表示归一化之后的目标向量；Transform the range of target values for each target to the interval [0,1] by normalizing by

carry out, where f _i (X) represents the actual target value of individual X, and f′ _i (X) represents the normalized target vector of individual X, where i∈1,2,...,M, mentioned later If the target vector is not specified, it is the target vector after normalization, and let f _i (X)=f′ _i (X), that is, f _i (X) represents the normalized target vector ;

Step 4, based on the reference vectors W ₁ , W ₂ ,...,W _N , assign individuals in the population P _g to the subpopulation SP={SP _i , i∈1,2,...,N}:

Step 4.1, reference vector associated subpopulations: Initialize N reference vectors W ₁ , W ₂ ,...,W _N corresponding to N empty associated subpopulations SP ₁ , SP ₂ , ..., SP _N , that is, the reference vector W _i associated subpopulation SP _i ;

Step 4.2, assign individuals to each subpopulation SP _i :

(a) First calculate the angle between the target vector F(X _i ) of each individual in the population P _g and all the reference vectors W _j

(b) Compare the size of θ _ij ;

(c) If k=argminX _i (θ _ij |W _j ,j=1,2,...,N), associate the target vector F(X _i ) of the i-th individual X _i with the j-th reference vector W _j , that is, assign the i-th individual X _i to the sub-population SP _{j constructed by the j-th reference vector W j ;}

(d) Repeat steps ac until all individuals X _i in the population P _g are assigned to the sub-population;

Step 5, perform the selection operation based on the domain subpopulation to select the parent individual to perform the genetic operation:

f_i(a)≤f_i(b)且

f_i(a)＜f_i(b)，那么个体aPareto支配个体b；Step 5.1, perform non-dominated sorting on all individuals in the population P _g according to the following rules: for any two individuals a and b in the population, if and only if for

f _i (a) ≤ f _i (b) and

f _i (a) < f _i (b), then individual aPareto dominates individual b;

Step 5.1.1, Presort:

(a) Define an ordering in advance for the M objectives of the multi-objective optimization problem;

(b) Sort all individuals in the population P _g in ascending order according to the target value f ₁ (X) of the first target. If the first target value f ₁ (X) of the two individuals is equal, then according to the second target The target value f ₂ (X) of the two individuals is sorted in ascending order. If all the target values of the two individuals are equal, the two individuals will be sorted arbitrarily;

(c) Step b is repeated until all individuals in the population P _g are sorted according to the target, and the sorted results are set as X ₁ , X ₂ ,...,X _N ;

Step 5.1.2, non-dominated sorting:

(a) Select individuals in the order of sorting X ₁ , X ₂ ,..., X _N to perform non-dominated sorting, pre-set r=0 non-dominated layers {F ₁ , F ₂ ,...}, F _i is The i-th non-dominated layer, i∈1,2,...,r, let k=0;

(b) For any individual X _m , m∈1,2,...,N, first check the dominance relationship between individual _{X m} and all individuals in the non-dominated layer F _k , if there is no any If the individual dominates the individual X _m , the rank of the individual X _m is set to k, that is, the individual X _m belongs to the kth non-dominated layer F _k . When the individual X _m is compared with the individuals in the non-dominated layer F _k , the individual X _m is first compared with the last individual in the non-dominated layer F _k , then with the penultimate individual in the non-dominated layer F _i , and finally with the non-dominated layer F i. The first individual comparison in the dominant layer F _k , that is, the reverse order comparison is performed in the non-dominated layer;

(c) if the individual X _m is dominated by at least one individual in the non-dominated layer F _k and k <r;

(d) then add 1 to k, and repeat step bc, otherwise create a new ranking r+1 for the individual X _m , that is, the individual X _m belongs to the (r+1)th non-dominated layer _Fr+1 ;

(e) After the non-dominated sorting is performed, the ranked non-dominated layers are denoted as F ₁ , F ₂ ,...,F _r ;

Step 5.2, select individuals to perform crossover and mutation operations:

Step 5.2.1, determine the neighborhood population:

且i≠j，

表示W_i和W_j之间的夹角；(a) First define the neighborhood reference vector of the reference vector, and calculate the Euclidean distance between any two reference vectors

and i≠j,

represents the angle between _{Wi and W j ;}

(b) Then calculate the T reference vectors closest to each reference vector Wi, and use B( _i )={i ₁ , i ₂ , . . . , _{i T} } to represent the T neighborhood reference vectors close to Wi The set of , each element in the set is

所构建的T个子种群，记为SP_i1,SP_i2,...,SP_iT；(c) Correspondingly, the T neighborhood subpopulations closest to the subpopulation SP _i are the T neighborhood reference vectors of the corresponding reference vector W _i

The constructed T subpopulations are denoted as SP _i1 , SP _i2 ,..., SP _iT ;

Step 5.2.2, select two parent individuals from the neighborhood population:

(a) Individual-X _i randomly selects a non-dominated individual from the sub-population SP _i with the highest ranking;

(b) Individual two X _j are randomly selected from any neighborhood population SP _i1 , SP _i2 ,..., SP _iT of the _{subpopulation} SP i;

变异概率为

K₁和K₂是两个常系数，k表示第k个目标，k∈1,2,...,M，

f_k(X_i)表示个体X_i的第k个目标值，f_k(X_j)表示个体X_j的第k个目标值；Step 5.2.3, perform genetic operations on the two parent individuals: for the two selected individuals X _i and X _j , perform genetic operations, that is, simulate binary crossover and polynomial mutation, and generate two individuals X′ _i and X′ _j , After repeating N/2 times, N offspring individuals are generated, that is, a new population Q; in order to improve the diversity and convergence speed of the population, the crossover probability and mutation probability are adaptively adjusted according to the fitness value of the selected individual, where the crossover probability is

The probability of mutation is

K ₁ and K ₂ are two constant coefficients, k represents the kth target, k∈1,2,...,M,

f _k (X _i ) represents the k-th target value of the individual X _i , and f _k (X _j ) represents the k-th target value of the individual X _j ;

Step 6: Update the subpopulations SP ₁ , SP ₂ , . . . , SP _N based on the individuals in the sub-population Q:

Step 6.1, adopt step 3.2 to evaluate the fitness of all individuals in the new population Q: calculate the target value F(X) and fitness value Fit(F(X)) of all individuals in the new population Q;

Step 6.2, based on the new population Q, adopt step 3.3 to update the ideal point Z ^* and the worst point Z ^nad ;

Step 6.3, use step 3.4 to normalize all individuals in the new population Q;

Step 6.4, based on step _5.1.2 ( _e ), the non-dominated layers F ₁ , F ₂ , . Individuals perform non-dominated sorting, update the individuals in the non-dominated layers F ₁ , F ₂ ,..., _Fr , and compare the dominance of the unranked individuals in Q with the ranked individuals in the non-dominated layers F ₁ , F ₂ ,..., _Fr relationship, which can effectively improve the efficiency of the algorithm;

Step 6.5, after performing step 6.4, is to combine the population P _g and the population Q to obtain a combined population R=P _g ∪Q and update the non-dominated layers F ₁ , F ₂ ,...,F _r , where the scale of the combined population R is 2N ;

Step 6.6, update the subpopulations SP ₁ , SP ₂ ,..., SP _N : adopt the individual assignment mechanism of step 4.2, assign the individuals in the new population Q to the sub population SP _i , i∈{1,2,...,N }.

Step 7: Perform environment selection on the combined population R, and update the next generation population P _g+1 : The environment selection process is to select N individuals with excellent performance from the 2N individuals in R as the parent population of the next generation. The environment selection strategy includes local environment. Selection strategy and global environment selection strategy;

Step 7.1, local environment selection strategy:

即目标向量F(X_i)到参考向量W_i的垂直距离，其中||F(X_i)||表示目标向量F(X_i)的模长，||W_i||表示参考向量W_i的模长，<F(X_i)，W_i>表示个体的目标向量F(X_i)和参考向量W_i之间的夹角，sin<F(X_i)，W_i>表示夹角<F(X_i)，W_i>的正弦值；Step 7.1.1, according to the updated sub-populations SP ₁ , SP ₂ ,..., SP _N in step 6.4, first calculate the Euclidean relationship between the target vector F(X _{i )} of the individual in each sub-population SP _i to the reference vector Wi distance

That is, the vertical distance from the target vector F(X _i ) to the reference vector Wi, where ||F(X _i )|| _{represents the modulo length of the target vector F(X i )} , and ||W _i || _represents the reference vector Wi The modulo length of , <F(X _i ), Wi > _{represents the angle between the individual target vector F(X i )} and the reference vector Wi, sin<F(X _{i )} , Wi > _represents the angle< F(X _i ) _, the sine of Wi >;

其中||F(X_i)-W_i||表示向量(F(X_i)-W_i)的模长，即目标向量F(X_i)的位置到参考向量W_i的参考点的距离，如果子种群SP_i中某个体位于构建的超平面L之下，即该个体支配第i个参考点，此时

为负；Step 7.1.2, calculate the distance from the position of the target vector F(X _{i )} of the individual in each subpopulation SP _i to the reference point for constructing the reference vector Wi

where ||F(X _i )-W _i || represents the modular length of the vector (F(X _i )-W _i ), that is, the distance from the position of the target vector F(X _i ) to the reference point of the reference vector Wi _, If an individual in the subpopulation SP _i is located under the constructed hyperplane L, that is, the individual dominates the i-th reference point, then

is negative;

和

选择个体，选择标准为

表示在每个子种群SP_i中选择两个距离

之和最小的前min(2，|SP_i|)两个个体。但在执行步骤6.6个体分配时，每个子种群SP_i中可能没有个体、只有一个个体或有多个个体，如果子种群SP_i中没有个体，不进行选择；如果子种群SP_i中只有一个个体，则选择该个体；否则选择两个距离

和

之和最小的前两个个体；Step 7.1.3, in each subpopulation SP _i , synthesize the two distances

and

Select individuals, the selection criteria are

means choosing two distances in each subpopulation SP _i

The first min(2, |SP _i |) two individuals with the smallest sum. However, when performing step 6.6 individual allocation, there may be no individual, only one individual or multiple individuals in each sub-population SP _i . If there is no individual in the sub-population SP _i , no selection is made; if there is only one individual in the sub-population SP _i , select the individual; otherwise, select two distances

and

The first two individuals with the smallest sum;

Step 7.2, Global Environment Selection Policy:

表示执行局部环境选择策略后非支配层

中的个体数，如果所有前r个非支配层中的个体数量之和大于种群规模N，而前r-1个非支配层中的个体数量之和小于种群规模N，即

且

则执行全局环境选择策略从第r非支配层

中选择

个个体进入下一代种群；Step 7.2.1, the global environment selection strategy is to delete all individuals obtained after executing the local environment selection strategy based on non-dominant rankings, select N individuals to enter the next generation population, and use

Represents the non-dominated layer after executing the local environment selection strategy

If the sum of the number of individuals in all the first r non-dominated layers is greater than the population size N, and the sum of the number of individuals in the first r-1 non-dominated layers is less than the population size N, that is

and

Then execute the global environment selection strategy from the rth non-dominated layer

choose

individuals into the next generation;

中相邻个体的欧式距离，若第i个个体和第j个个体相邻，则第i个个体和第j个个体的欧式距离为

K且i≠j，令

表示非支配层

中个体的数量；Step 7.2.2, Calculate the non-dominated layer

The Euclidean distance of adjacent individuals in the

K and i≠j, let

represents the non-dominated layer

the number of individuals in the

中所有相邻个体的平均距离

并执行全局选择：Step 7.2.3, Calculate the non-dominated layer

The average distance of all adjacent individuals in

and perform a global selection:

大于平均距离

的个体数量大于

选择相邻距离

最大的前

个个体进入下一代种群；(a) If the adjacent distance

greater than average distance

The number of individuals is greater than

select neighbor distance

biggest ex

individuals into the next generation;

大于平均距离

的个体数量为K′且

首先选择相邻距离

大于平均距离

的K′个个体进入下一代种群；(b) If the adjacent distance

greater than average distance

The number of individuals is K' and

First select the adjacent distance

greater than average distance

K' individuals enter the next generation population;

小于平均距离

的个体中选择K″个个体进入下一代种群，其中

选择过程如下：(c) Then from the adjacent distance

less than average distance

Select K″ individuals from the individuals to enter the next generation population, where

The selection process is as follows:

小于平均距离

的个体按照某一目标进行升序排序，首先选择排序中的第一个个体进入下一代种群，因为往往排序后的第一个个体是极值点所对应的目标向量，对维护种群多样性具有重要的意义，然后计算

即排序后的第i个个体和第(i+s)个个体的距离，初始s＝2；1) For the adjacent distance

less than average distance

The individuals are sorted in ascending order according to a certain target, and the first individual in the sorting is selected to enter the next generation population, because the first individual after sorting is often the target vector corresponding to the extreme point, which is important for maintaining the diversity of the population. meaning, then calculate

That is, the distance between the i-th individual and the (i+s)-th individual after sorting, the initial s=2;

选择第(i+s)个个体X_i+s进入下一代种群；2) If

Select the (i+s)th individual Xi _+s to enter the next generation population;

执行s加1，再比较

和

的大小，直到

然后选择第(i+s)个个体X_i+s进入下一代种群；3) If

Execute s plus 1, and then compare

and

size until

Then select the (i+s)th individual Xi _+s to enter the next generation population;

表示相邻距离

大于平均距离

的所有相邻距离的最小值，δ是控制选择范围的参数；4) Then take the (i+s) th individual X _i+s as the initial individual, and set i equal to i+s, and calculate the difference between the ith individual and the (i+s) th individual in ascending order. Euclidean distance until the number of selected individuals reaches K″, where

Indicates the adjacent distance

greater than average distance

The minimum value of all adjacent distances of , δ is a parameter that controls the selection range;

Step 8: Steps 4-7 are executed cyclically until the algebra g>G _max , the calculation is terminated and the Pareto solution set of the high-dimensional multi-objective optimization problem is output.

The beneficial effects of the present invention are:

1. For the high-dimensional objective optimization problem, a high-dimensional multi-objective evolutionary algorithm based on decomposition is proposed. The high-dimensional multi-objective optimization problem is decomposed into multiple single-objective optimization sub-problems, and multiple single-objective optimization sub-problems are evolved at the same time. , effectively reduce the complexity of solving the problem and solve the problem that the existing multi-objective optimization algorithm is difficult to ensure a good balance between population convergence and diversity, and obtain a Pareto solution set with good diversity and convergence;

2. For the non-dominated sorting method, the efficiency of the algorithm can be effectively improved by comparing the dominance relationship between the unranked individuals and the ranked individuals;

3. Select individuals who perform genetic operations based on neighborhood subpopulations, which can effectively ensure the global convergence and population diversity of the algorithm;

4. For individuals with genetic manipulations, for individuals with excellent fitness values, in order to maintain their good performance, the crossover probability and mutation probability should be small, while for individuals with poor fitness values, in order to improve their performance, crossover probability and mutation probability The probability should be large, and the adaptive adjustment of the crossover probability and mutation probability through the fitness value of the individual can effectively improve the convergence speed and diversity of the population;

5. For the environment selection strategy, by integrating the local environment selection strategy and the global environment selection strategy, the problem of the failure of the diversity maintenance strategy is effectively solved, and the high-dimensional multi-objective optimization algorithm based on the Pareto dominance relationship is improved in solving the high-dimensional objective optimization problem. performance, a good balance of convergence and diversity.

Description of drawings

FIG. 1 is a flow chart of the present invention.

FIG. 2 is an algorithm frame diagram of the present invention.

FIG. 3 is a schematic diagram of an ideal point, a maximum point and a worst point of the present invention.

FIG. 4 is a sample diagram of reference point generation and construction of reference vectors in the three-dimensional target space of the present invention.

FIG. 5 is a schematic diagram of individual allocation in the two-dimensional target space of the present invention.

FIG. 6 is an example diagram of a subpopulation in the two-dimensional target space of the present invention.

Figure 7 is a flow chart of the selection of genetically manipulated individuals of the present invention.

FIG. 8 is a flow chart of the environment selection strategy of the present invention.

FIG. 9 is a sample diagram of a local environment selection strategy in a two-dimensional target space of the present invention.

FIG. 10 is an example diagram of the global environment selection strategy in the two-dimensional target space of the present invention.

Detailed ways

The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

Embodiment 1: As shown in Figure 1-10, a decomposition-based high-dimensional multi-objective evolutionary algorithm includes steps 1-8.

Step 1, parameter setting: algebra g=0, maximum number of iterations G _max , target number M≥4, population size N.

Step 2, generate N reference vectors W={W ₁ , W ₂ ,...,W _N }:

Step 2.1, using the normal boundary intersection (PBI) method to generate a uniformly distributed reference point RP={RP ₁ ,RP ₂ ,...,RP _N } on a unit hyperplane L;

Step 2.2, based on the reference points RP ₁ , RP ₂ ,...,RP _N , construct uniformly distributed reference vectors W ₁ , W ₂ ,..., W _N in the target space. The reference vector satisfies the following conditions:

表示参考向量W_i的第j维度的分量值且

H表示将每个目标划分为H等份，生成参考向量的数量为N，其中

如M＝3，H＝4，在单位超平面上生成的均匀分布参考向量数量为

in

represents the component value of the _jth dimension of the reference vector Wi and

H means that each target is divided into H equal parts, and the number of generated reference vectors is N, where

For example, M=3, H=4, the number of uniformly distributed reference vectors generated on the unit hyperplane is

Step 3, randomly generate an initial population P _g = 0 of size N, and perform individual fitness evaluation:

Step 3.1, under the condition that the constraints are satisfied, randomly generate an initial population P _g ={X ₁ ,X ₂ ,...,X _N } with a population size of N from the decision space, where X _i ={x ₁ ,x ₂ ,...,x _D }, i=1,2,...,N, D represents the dimension of the decision variable X _i , and g represents the algebra of the current population;

Step 3.2, calculate the target vector F(X)=(f ₁ (X), f ₂ (X),..., f _M (X)) of all individuals in the population P _g , X∈P _g , and let the individual The fitness value of X is Fit(F(X))=F(X);

Step 3.3, based on the target vector F(X) of all individuals, calculate the ideal point Z ^* , the extreme point Z ^nad and the worst point Z ^worst , but in the actual multi-objective optimization process, the ideal point Z ^* , the worst point Z ^worst and The extreme point Z ^nad cannot be predicted in advance, so the ideal point Z ^* , the worst point Z ^worst and the extreme point Z ^nad are continuously updated based on the target vector of individuals in the current population P _g :

其中

任意X∈P_g；(1) Ideal point

in

any _X∈Pg ;

其中

任意X∈P_g；(2) Worst point

in

any _X∈Pg ;

其中

而极值点

表示极小值点对应的自变量，即

(3) Extreme point

in

the extreme point

represents the independent variable corresponding to the minimum point, that is

进行，其中f_i(X)表示个体X实际目标值，f＇_i(X)表示个体X执行归一化后的目标向量，其中i∈1,2,...,M。后续提到的目标向量若不具体说明，则都是经过归一化处理之后的目标向量，且令f_i(X)＝f′_i(X)，即用f_i(X)表示归一化之后的目标向量。当一个目标在目标空间中的取值范围是[0,100]，另一个目标在目标空间中的取值范围是[0,10000]，不同目标的取值范围可能有很大区别，通过归一化处理后，可以将所有目标的取值范围转换到同一个区间进行比较而不影响优化过程。Step 3.4, normalize the target vectors of all individuals in the population _Pg . Convert the target value range for each target to the interval [0,1] by normalization. normalized by

, where f _i (X) represents the actual target value of individual X, and f' _i (X) represents the normalized target vector of individual X, where i ∈ 1,2,...,M. If the target vectors mentioned later are not specified, they are all target vectors after normalization, and let f _i (X)=f′ _i (X), that is, f _i (X) is used to denote normalization The target vector after that. When the value range of one target in the target space is [0, 100], and the value range of another target in the target space is [0, 10000], the value range of different targets may be very different. By normalizing After processing, the value ranges of all objectives can be converted to the same interval for comparison without affecting the optimization process.

Step 4, based on the reference vectors W ₁ , W ₂ ,...,W _N , assign individuals in the population P _g to the subpopulation SP={SP _i , i∈1,2,...,N}:

Step 4.1, the reference vector is associated with subpopulations. Initialize N reference vectors W ₁ , W ₂ ,..., W _N corresponding to N empty associated subpopulations SP ₁ , SP ₂ , ..., SP _{N , namely the reference vector Wi associated sub population SP i ;}

Step 4.2, assign individuals to each subpopulation SP _i :

(a) First calculate the angle between the target vector F(X _i ) of each individual in the population P _g and all the reference vectors W _j

(b) Compare the size of θ _ij ;

将第i个体X_i的目标向量F(X_i)与第k＝j参考向量W_j关联，即将第i个个体X_i分配给由第j个参考向量W_j构建的子种群SP_j；(c) if

Associating the target vector F(X _i ) of the _ith individual Xi with the k=j reference vector W _j , that is, assigning the _ith individual Xi to the subpopulation SP _{j constructed by the j th reference vector W j ;}

(d) Repeat ac until all individuals X _i in the population P _g are assigned to the subpopulation.

Step 5, perform the selection operation based on the domain subpopulation to select the parent individual to perform the genetic operation:

f_i(a)≤f_i(b)且

f_i(a)＜f_i(b)，那么个体aPareto支配个体b；Step 5.1, perform non-dominated sorting on all individuals in the population P _g according to the following rules: for any two individuals a and b in the population, if and only if for

f _i (a) ≤ f _i (b) and

f _i (a) < f _i (b), then individual aPareto dominates individual b;

Step 5.1.1, Presort:

(a) Define an ordering in advance for the M objectives of the multi-objective optimization problem;

(b) Sort all individuals in the population P _g in ascending order according to the target value f ₁ (X) of the first target. If the first target value f ₁ (X) of the two individuals is equal, then according to the second target The target value f ₂ (X) of the two individuals is sorted in ascending order. If all the target values of the two individuals are equal, the two individuals will be sorted arbitrarily;

(c) Step b is repeated until all individuals in the population P _g are sorted according to the target, and the sorted results are set as X ₁ , X ₂ ,...,X _N ;

Step 5.1.2, non-dominated sorting:

(a) Select individuals in the order of sorting X ₁ , X ₂ ,..., X _N to perform non-dominated sorting, pre-set r=0 non-dominated layers {F ₁ , F ₂ ,...}, F _i is The i-th non-dominated layer, i∈1,2,...,r, let k=0;

(b) For any individual X _m , m∈1,2,...,N, first check the dominance relationship between individual _{X m} and all individuals in the non-dominated layer F _k , if there is no any If the individual dominates the individual X _m , the rank of the individual X _m is set to k, that is, the individual X _m belongs to the kth non-dominated layer F _k . When the individual X _m is compared with the individuals in the non-dominated layer F _k , the individual X _m is first compared with the last individual in the non-dominated layer F _k , then with the penultimate individual in the non-dominated layer F _i , and finally with the non-dominated layer F i. The first individual comparison in the dominant layer F _k , that is, the reverse order comparison is performed in the non-dominated layer;

(c) if the individual X _m is dominated by at least one individual in the non-dominated layer F _k and k <r;

(d) Then let k=k+1 and repeat step bc, otherwise create a new rank r+1 for the individual X _m , that is, the individual X _m belongs to the (r+1)th non-dominated layer _Fr+1 . Specifically, according to the order of step 5.1.1(c) sorting results X ₁ , X ₂ ,..., X _N , select individuals in turn to perform non-dominated sorting, first select the first individual X ₁ ranked as 1, and put the individual X ₁ Put it into the first non-dominated layer F ₁ , and the ranking is 1; then select the second individual X _{2 ranked 2} , compare the target value of the individual X ₂ and the already ranked first individual X ₁ , and judge the individual X ₁ and the Dominance relationship between individual X ₂ , if individual X ₁ dominates individual X ₂ , then put X ₂ into the second non-dominated layer F ₂ , and the ranking is 2, if individual X ₁ and individual X ₂ do not dominate each other, then Put X ₂ into the first non-dominated layer F ₁ , and X ₂ is ranked 1; then select the individual X ₃ ranked as 3, and compare the dominance of individual X ₃ and all individuals in the first non-dominated layer F ₁ Then compare the dominance relationship with the individuals in the non-dominant layers F ₂ , F ₃ , ..., _Fr in turn, if the individual X ₃ is not assigned a rank, create a new rank r+1 for the individual X ₃ until all individuals are assign a corresponding ranking;

(e) After the non-dominated sorting is performed, the ranked non-dominated layers are denoted as F ₁ , F ₂ ,...,F _r ;

Step 5.2, select individuals to perform crossover and mutation operations:

Step 5.2.1, determine the neighborhood population:

且i≠j，

表示W_i和W_j之间的夹角；(a) First define the neighborhood reference vector of the reference vector, and calculate the Euclidean distance between any two reference vectors

and i≠j,

represents the angle between _{Wi and W j ;}

(b) Then calculate the T reference vectors closest to each reference vector Wi, and use B( _i )={i ₁ , i ₂ , . . . , _{i T} } to represent the T neighborhood reference vectors close to Wi The set of , each element in the set is

所构建的T个子种群，记为SP_i1,SP_i2,...,SP_iT；(c) Correspondingly, the T neighborhood subpopulations closest to the subpopulation SP _i are the T neighborhood reference vectors of the corresponding reference vector W _i

The constructed T subpopulations are denoted as SP _i1 , SP _i2 ,..., SP _iT ;

Step 5.2.2, select two parent individuals from the neighborhood population:

(a) Individual-X _i randomly selects a non-dominated individual from the sub-population SP _i with the highest ranking;

(b) Individual two X _j are randomly selected from any neighborhood population SP _i1 , SP _i2 ,..., SP _iT of the _{subpopulation} SP i. Selecting individuals in the neighborhood for crossover and mutation can not only ensure that individuals inherit the properties of solutions with better convergence performance, but also ensure the global search ability of the algorithm, thereby ensuring the diversity and convergence of the population;

变异概率为

K₁和K₂是两个常系数，k表示第k个目标，k∈1,2,...,M，

表示个体X_i的第k个目标值，f_k(X_j)表示个体X_j的第k个目标值。Step 5.2.3, perform genetic operations on the two parent individuals: for the two selected individuals X _i and X _j , perform genetic operations, that is, simulate binary crossover and polynomial mutation, and generate two individuals X′ _i and X′ _j , After repeated execution N/2 times, N offspring individuals are generated, that is, a new population Q. In order to improve the diversity and convergence speed of the population, the crossover probability and mutation probability are adaptively adjusted according to the fitness value of the selected individual, where the crossover probability is

The probability of mutation is

K ₁ and K ₂ are two constant coefficients, k represents the kth target, k∈1,2,...,M,

represents the k-th target value of the individual X _i , and f _k (X _j ) represents the k-th target value of the individual X _j .

Step 6: Update the subpopulations SP ₁ , SP ₂ , . . . , SP _N based on the individuals in the sub-generation population Q:

In step 6.1, step 3.2 is used to evaluate the fitness of all individuals in the new population Q. Calculate the target value F(X) and fitness value Fit(F(X)) of all individuals in the new population Q;

Step 6.2, based on the new population Q, adopt step 3.3 to update the ideal point Z ^* and the worst point Z ^nad ;

Step 6.3, use step 3.4 to normalize all individuals in the new population Q;

Step 6.4, based on step _5.1.2 ( _e ), the non-dominated layers F ₁ , F ₂ , . Individuals perform non-dominated sorting, update the individuals in the non-dominated layers F ₁ , F ₂ ,..., _Fr , and compare the dominance of the unranked individuals in Q with the ranked individuals in the non-dominated layers F ₁ , F ₂ ,..., _Fr relationship, which can effectively improve the efficiency of the algorithm;

In step 6.5, after step 6.4 is executed, the population P _g and the population Q are actually combined to obtain a combined population R=P _g ∪Q and update the non-dominated layers F ₁ , F ₂ ,...,F _r , where the scale of the combined population R is 2N;

Step 6.6, update the subpopulations SP ₁ , SP ₂ ,..., SP _N : adopt the individual assignment method of step 4.2, assign the individuals in the new population Q to the sub population SP _i , i∈{1,2,... ,N}.

Step 7: Perform environmental selection on the combined population R, and update the next generation population P _g=g+1 : The process of environmental selection is to select N individuals with excellent performance from 2N individuals in R as the parent population of the next generation. The environmental selection strategy Including local environment selection strategy and global environment selection strategy;

Step 7.1, local environment selection strategy:

即目标向量F(X_i)到参考向量W_i的垂直距离，其中||F(X_i)||表示目标向量F(X_i)的模长，||W_i||表示参考向量W_i的模长，<F(X_i)，W_i>表示个体的目标向量F(X_i)和参考向量W_i之间的夹角，sin<F(X_i)，W_i>表示夹角<F(X_i)，W_i>的正弦值；Step 7.1.1, according to the updated sub-populations SP ₁ , SP ₂ ,..., SP _N in step 6.4, first calculate the Euclidean relationship between the target vector F(X _{i )} of the individual in each sub-population SP _i to the reference vector Wi distance

That is, the vertical distance from the target vector F(X _i ) to the reference vector Wi, where ||F(X _i )|| _{represents the modulo length of the target vector F(X i )} , and ||W _i || _represents the reference vector Wi The modulo length of , <F(X _i ), Wi > _{represents the angle between the individual target vector F(X i )} and the reference vector Wi, sin<F(X _{i )} , Wi > _represents the angle< F(X _i ) _, the sine of Wi >;

其中||F(X_i)-W_i||表示向量(F(X_i)-W_i)的模长，即目标向量F(X_i)的位置到参考向量W_i的参考点的距离，如果子种群SP_i中某个体位于构建的超平面L之下，即该个体支配第i个参考点，此时

为负；Step 7.1.2, calculate the distance from the position of the target vector F(X _{i )} of the individual in each subpopulation SP _i to the reference point for constructing the reference vector Wi

where ||F(X _i )-W _i || represents the modular length of the vector (F(X _i )-W _i ), that is, the distance from the position of the target vector F(X _i ) to the reference point of the reference vector Wi _, If an individual in the subpopulation SP _i is located under the constructed hyperplane L, that is, the individual dominates the i-th reference point, then

is negative;

和

选择个体，选择标准为

表示在每个子种群SP_i中选择两个距离

之和最小的前min(2，|SP_i|)两个个体。但在执行步骤6.6个体分配时，每个子种群SP_i中可能没有个体、只有一个个体或有多个个体，如果子种群SP_i中没有个体，不进行选择；如果子种群SP_i中只有一个个体，则选择该个体；否则选择两个距离

和

之和最小的前两个个体；Step 7.1.3, in each subpopulation SP _i , synthesize the two distances

and

Select individuals, the selection criteria are

means choosing two distances in each subpopulation SP _i

The first min(2, |SP _i |) two individuals with the smallest sum. However, when performing step 6.6 individual allocation, there may be no individual, only one individual or multiple individuals in each sub-population SP _i . If there is no individual in the sub-population SP _i , no selection is made; if there is only one individual in the sub-population SP _i , select the individual; otherwise, select two distances

and

The first two individuals with the smallest sum;

Step 7.2, Global Environment Selection Policy:

表示执行局部环境选择策略后非支配层

中的个体数，如果所有前r个非支配层中的个体数量之和大于种群规模N，而前r-1个非支配层中的个体数量之和小于种群规模N，即

且

则执行全局环境选择策略从第r非支配层

中选择

个个体进入下一代种群；Step 7.2.1, the global environment selection strategy is to delete all individuals obtained after executing the local environment selection strategy based on the non-dominant ranking, and select N individuals to enter the next generation population. use

Represents the non-dominated layer after executing the local environment selection strategy

If the sum of the number of individuals in all the first r non-dominated layers is greater than the population size N, and the sum of the number of individuals in the first r-1 non-dominated layers is less than the population size N, that is

and

Then execute the global environment selection strategy from the rth non-dominated layer

choose

individuals into the next generation;

中相邻个体的欧式距离，若第i个个体和第j个个体相邻，则第i个个体和第j个个体的欧式距离为

K且i≠j，令

表示非支配层

中个体的数量；Step 7.2.2, Calculate the non-dominated layer

The Euclidean distance of adjacent individuals in the

K and i≠j, let

represents the non-dominated layer

the number of individuals in the

中所有相邻个体的平均距离

并执行全局选择。Step 7.2.3, Calculate the non-dominated layer

The average distance of all adjacent individuals in

and perform a global selection.

大于平均距离

的个体数量大于

选择相邻距离

最大的前

个个体进入下一代种群；(a) If the adjacent distance

greater than average distance

The number of individuals is greater than

select neighbor distance

biggest ex

individuals into the next generation;

大于平均距离

的个体数量为K′且

首先选择相邻距离

大于平均距离

的K′个个体进入下一代种群；(b) If the adjacent distance

greater than average distance

The number of individuals is K' and

First select the adjacent distance

greater than average distance

K' individuals enter the next generation population;

小于平均距离

的个体中选择K″个个体进入下一代种群，其中

选择过程如下：(c) Then from the adjacent distance

less than average distance

Select K″ individuals from the individuals to enter the next generation population, where

The selection process is as follows:

小于平均距离

的个体按照某一目标进行升序排序，首先选择排序中的第一个个体进入下一代种群，因为往往排序后的第一个个体是极值点所对应的目标向量，对维护种群多样性具有重要的意义，然后计算

即排序后的第i个个体和第(i+s)个个体的距离，初始s＝2；1) For the adjacent distance

less than average distance

The individuals are sorted in ascending order according to a certain target, and the first individual in the sorting is selected to enter the next generation population, because the first individual after sorting is often the target vector corresponding to the extreme point, which is important for maintaining the diversity of the population. meaning, then calculate

That is, the distance between the i-th individual and the (i+s)-th individual after sorting, the initial s=2;

选择第(i+s)个个体X_i+s进入下一代种群；2) If

Select the (i+s)th individual Xi _+s to enter the next generation population;

执行s＝s+1，再比较

和

的大小，直到

然后选择第(i+s)个个体X_i+s进入下一代种群；3) If

Execute s=s+1, and then compare

and

size until

Then select the (i+s)th individual Xi _+s to enter the next generation population;

表示相邻距离

大于平均距离

的所有相邻距离的最小值，δ是控制选择范围的参数。4) Then take the (i+s) th individual X _i+s as the initial individual, and let i=i+s, calculate the relationship between the i th individual and the (i+s) th individual in ascending order. Euclidean distance until the number of selected individuals reaches K″, where

Indicates the adjacent distance

greater than average distance

The minimum value of all adjacent distances of , δ is a parameter that controls the selection range.

Step 8, the algebra g≤G _max , execute step 4-step 7 in a loop; otherwise, the algorithm terminates and outputs the Pareto solution set of the high-dimensional multi-objective optimization problem.

The specific embodiments of the present invention have been described in detail above in conjunction with the accompanying drawings, but the present invention is not limited to the above-mentioned embodiments, and within the scope of knowledge possessed by those of ordinary skill in the art, it can also be done without departing from the purpose of the present invention. various changes.

Claims (1)

1. A high-dimensional multi-target evolution method based on decomposition is characterized by comprising the following steps:

step 1, setting parameters: algebraic G being 0, maximum number of iterations G_maxThe target quantity M is more than or equal to 4, and the population size N is obtained;

step 2, generating N reference vectors W ═ { W ═ W₁,W₂,...,W_N}：

Step 2.1, generating uniformly distributed reference points RP ═ RP on a unit hyperplane L by using a normal boundary intersection method₁,RP₂,...,RP_N}；

Step 2.2, based on reference point RP₁,RP₂,...,RP_NConstructing uniformly distributed reference vectors W in a target space₁,W₂,...,W_NThe reference vector satisfies the following condition:

wherein

Represents a reference vector W_iA component value of the j-th dimension of (a), and

i 1, 2., N, j 1, 2., M, H denotes dividing each target into H equal parts, and generating reference vectors in N number, where N is the number of reference vectors

Step 3, randomly generating an initial population P with the size of N_g＝0And carrying out individual fitness evaluation:

step 3.1, randomly generating an initial population P with the population size N from the decision space under the condition of satisfying the constraint condition_g＝{X₁,X₂,...,X_i...，X_NIn which X is_i＝{x₁,x₂,...,x_DN, D denotes a decision variable X_iG represents the generation of the current populationCounting;

step 3.2, calculate population P_gTarget vectors f (x) ═ f of all individuals in (c)₁(X),f₂(X),...,f_M(X))，X∈P_gAnd let the fitness value of individual X be Fit (f (X)) ═ f (X);

step 3.3, calculating ideal points Z based on target vectors F (X) of all individuals^*Extreme point Z^nadSum worst point Z^worstHowever, in the actual multi-objective optimization process, the ideal point Z^*Worst point Z^worstAnd extreme point Z^nadCannot be predicted in advance, based on the current population P_gTarget vector calculation ideal point Z of the medium individual^*Worst point Z^worstAnd extreme point Z^nadComprises the following steps:

(1) ideal point

Wherein

Any X ∈ P_g；

(2) Worst point

Wherein

Any X ∈ P_g；

(3) Extreme point

Wherein

And extreme point

Representing the argument of the point correspondence of the minima, i.e.

Step 3.4, normalization of population P_gTarget vectors of all individuals in the group;

converting the target value range of each target to the interval [0,1 ] by normalization]Normalized by

Is carried out, wherein f_i(X) represents the actual target value of individual X, f'_i(X) represents the target vector of the individual X after normalization, wherein i ∈ 1, 2.. times.M, the target vectors mentioned later are the target vectors after normalization, if not specifically stated, and let f_i(X)＝f′_i(X), namely f_i(X) represents the target vector after normalization;

step 4, based on the reference vector W₁,W₂,...,W_NThe population P_gIs assigned to a sub-population SP ═ SP_i，i∈1,2,...,N}：

Step 4.1, reference vector association sub-population: initializing N reference vectors W₁,W₂,...,W_NCorresponding to N empty associated sub-populations SP₁,SP₂,...,SP_NI.e. the reference vector W_iAssociation sub-population SP_i；

Step 4.2, assign individuals to each sub-population SP_i：

(a) First, calculate the population P_gTarget vector F (X) of each individual_i) With all reference vectors W_jAngle therebetween

(b) Comparison of θ_ijThe size of (d);

(c) if it is not

The ith individual X_iTarget vector F (X)_i) With the jth reference vector W_jAssociating, i.e. the ith individual X_iTo the jth reference vector W_jConstructed sub-population SP_j；

(d) Repeating steps a-c until the population P_gAll individuals X in_iAre all distributed into sub-populations;

and 5, performing selection operation based on the field sub-population to select parent individuals to perform genetic operation:

step 5.1, for population P_gAll individuals in (a) are non-dominated ranked according to the following rules: for any two individuals a and b in the population, if and only if

f_i(a)≤f_i(b) And is

f_i(a)＜f_i(b) Then individual a pareto dominates individual b;

step 5.1.1, pre-sequencing:

(a) defining a sequence for M targets of the multi-target optimization problem in advance;

(b) target value f according to the first target₁(X) combining the population P_gAll individuals in the two groups are sorted in ascending order if the first target value f of the two individuals₁(X) equal, then according to the target value f of the second target₂(X) sorting the two individuals in ascending order, and if all the target values of the two individuals are equal, randomly sorting the two individuals;

(c) repeating step b until the population P_gAll the individuals in the system are sorted according to the target, and the sorted result is set as X₁,X₂,…,X_N；

Step 5.1.2, non-dominated sorting:

(a) according to the order X₁,X₂,...,X_NSequentially selects individual execution non-dominant orderingIn advance, r is set to 0 non-dominant layers { F₁,F₂,…}，F_iIs the ith non-dominant layer, i ∈ 1, 2.., r, let k be 0;

(b) for any one individual X_mM ∈ 1, 2.., N, individual X is first examined_mAnd a non-dominant layer F_kDominant relationship of all individuals in (1), if the non-dominant layer F_kIn the absence of any individual dominating individual X_mThen subject X_mIs set to k, i.e. the individual X_mBelonging to the kth non-dominant layer F_k(ii) a Individual X_mAnd a non-dominant layer F_kWhen comparing individuals in (1), individual X_mFirst and second non-dominant layer F_kIs compared with the last individual in (1) and then with the non-dominant layer F_iComparison of penultimate individuals, final and non-dominated layer F_kThe first individual comparison, i.e., performing a reverse order comparison in the non-dominant layer;

(c) if the individual X_mDominated layer F_kAnd k < r;

(d) then add 1 to k and repeat steps b-c, otherwise, for the individual X_mCreate a new rank r +1, Individual X_mBelonging to the (r +1) th non-dominant layer F_r+1；

(e) After the non-dominant sorting is executed, the ranked non-dominant layer is marked as F₁,F₂,…,F_r；

And 5.2, selecting individuals for performing crossover and mutation operations:

step 5.2.1, determining neighborhood population:

(a) firstly, defining neighborhood reference vectors of reference vectors, and calculating Euclidean distance between any two reference vectors

And i is not equal to j,

represents W_iAnd W_jThe included angle between them;

(b) then calculate the distance to each reference vector W_iMost recent T referencesVector, using B (i) ═ i₁，i₂，...，i_TDenotes approaching W_iA set of T neighborhood reference vectors, each element in the set being

(c) Correspondingly, the closest sub-population SP_iIs the corresponding reference vector W_iT neighborhood reference vectors

The constructed T sub-populations are marked as SP_i1,SP_i2,...,SP_iT；

Step 5.2.2, two parent individuals are selected from the neighborhood population:

(a) individual one X_iFor random slave sub-population SP_iSelecting one individual with a non-dominant ranking;

(b) individual II X_jSlave sub-population SP_iOf any neighborhood population SP_i1,SP_i2,...,SP_iTSelecting randomly;

step 5.2.3, performing genetic operations on the two parent individuals: for two selected individuals X_iAnd X_jPerforming genetic operations, i.e. simulating binary crosses and polynomial variations, yielding two individuals X'_iAnd X'_jRepeatedly executing N/2 times to generate N sub-generation individuals, namely a new population Q; in order to improve the diversity and convergence speed of the population, the cross probability and the mutation probability are adaptively adjusted according to the fitness value of the selected individual, wherein the cross probability is

The probability of variation is

K₁And K₂Are two constant coefficients, k representing the kth target, k ∈ 1, 2.., M,

f_k(X_i) Representing an individual X_iOf the kth target value, f_k(X_j) Representing an individual X_jThe kth target value of (1);

step 6, updating the sub-population SP based on the individuals in the sub-population Q₁,SP₂,...,SP_N：

Step 6.1, adopting step 3.2 to evaluate the fitness of all individuals in the new population Q: calculating target values f (x) and fitness values Fit (f (x)) of all individuals in the new population Q;

step 6.2, based on the new population Q, adopting step 3.3 to update the ideal point Z^*Sum worst point Z^nad；

Step 6.3, normalizing all individuals in the new population Q by adopting the step 3.4;

step 6.4, population P is treated based on step 5.1.2(e)_gNon-dominant layer F after medium individual ranking₁,F₂,…,F_rOn the basis, all individuals in the new population Q are subjected to non-dominant sorting by adopting the step 5.1.2(b), and a non-dominant layer F is updated₁,F₂,…,F_rBy the non-ranked ones of Q and non-dominated layer F₁,F₂,…,F_rCompared with the dominant relationship of the ranked individuals, the efficiency of the algorithm can be effectively improved;

step 6.5, after step 6.4 is executed, the population P is divided_gCombining with the population Q to obtain a combined population R ═ P_g∪ Q and updates the non-dominated layer F₁,F₂,…,F_rWherein the size of the combo population R is 2N;

step 6.6, update sub-population SP₁,SP₂,...,SP_N: adopting a step 4.2 individual allocation mechanism to allocate the individuals in the new population Q to the sub-population SP_i，i∈{1,2,...,N}；

Step 7, selecting the execution environment of the combined population R and updating the next generation population P_g+1: the environment selection process is to select N individuals with excellent performance from 2N individuals in R as a parent population of a next generation, and the environment selection strategy comprises a local environment selection strategy and a global environment selection strategyA little bit;

step 7.1, local environment selection strategy:

step 7.1.1, update of sub-population SP according to step 6.4₁,SP₂,...,SP_NFirst, each sub-population SP is calculated_iTarget vector F (X) of the middle individual_i) To a reference vector W_iEuclidean distance of

I.e. the target vector F (X)_i) To a reference vector W_iWherein F (X)_i) I represents the target vector F (X)_i) Is of a length, | W_i| | denotes a reference vector W_iThe length of the die (c) is,<F(X_i)，W_i>target vector F (X) representing an individual_i) And a reference vector W_iAngle between them sin<F(X_i)，W_i>Indicating included angle<F(X_i)，W_i>The sine value of (d);

step 7.1.2, calculate each sub-population SP_iTarget vector F (X) of the middle individual_i) To construct a reference vector W_iDistance from the reference point of

Wherein | | | F (X)_i)-W_iI represents a vector (F (X)_i)-W_i) Is the target vector F (X)_i) To the reference vector W_iIf the sub-population SP_iOne is located below the constructed hyperplane L, i.e. the individual dominates the ith reference point, at this time

Is negative;

step 7.1.3, in each sub-population SP_iIn, synthesize two distances

And

selecting individuals according to the selection criteria

Indicated in each sub-population SP_iTwo distances are selected

Minimum sum of pre-min (2, | SP)_i|) two individuals, but when step 6.6 individual assignment is performed, each sub-population SP_iMay have no individual, only one individual or a plurality of individuals, if the sub-population SP_iNo individual in the population, no selection; if the sub-population SP_iIf only one individual is present, the individual is selected; otherwise two distances are selected

And

the first two individuals with the smallest sum;

step 7.2, global environment selection strategy:

step 7.2.1, the global environment selection strategy is to delete all individuals obtained after the local environment selection strategy is executed based on non-dominant ranking, select N individuals to enter the next generation population, and use | F_i ^gI represents the non-dominant layer F after execution of the local environment selection policy_i ^gIf the sum of the number of individuals in all the first r non-dominant layers is greater than the population size N and the sum of the number of individuals in the first r-1 non-dominant layers is less than the population size N, i.e., the population size N is smaller than the population size N

And is

Then the global environment selection policy is executed from the r-th non-dominant layer

In selection

Entering individual into next generation population;

step 7.2.2, calculate non-dominated layer

The Euclidean distance between the ith individual and the jth individual is equal to

And i ≠ j, let K ═ F_i ^gI denotes the non-dominant layer

The number of individuals;

step 7.2.3, calculate non-dominated layer

Average distance of all neighboring individuals in

And performs global selection:

(a) if adjacent distance

Greater than average distance

Greater than

Selecting adjacent distances

Maximum front

Entering individual into next generation population;

(b) if adjacent distance

Greater than average distance

Has a number of individuals of K' and

first select the adjacent distance

Greater than average distance

The K' individuals enter the next generation population;

(c) then from adjacent distance

Less than the average distance

Selecting K' individuals from the individuals of (1) into a next generation population, wherein

The selection process is as follows:

1) for adjacent distance

Less than the average distance

The individuals are sorted in ascending order according to a certain target, the first individual in the sorting is selected to enter the next generation of population, because the first individual after the sorting is often the target vector corresponding to the extreme point, the significance for maintaining the diversity of the population is achieved, and then the calculation is carried out

The distance between the ith individual and the (i + s) th individual after sorting, wherein the initial s is 2;

2) if it is not

(i + s) th individual X is selected_i+sEntering a next generation population;

3) if it is not

Perform s plus 1 and compare again

And

up to

Then selecting the (i + s) th individual X_i+sEntering a next generation population;

4) then the (i + s) th individual X_i+sAs initial individuals, and let i equal i + s, the Euclidean distance between the ith individual and the (i + s) th individual is calculated in ascending order until the number of selected individuals reaches K ″, wherein

To representAdjacent distance

Greater than average distance

δ is a parameter controlling the selection range;

step 8, circularly executing the step 4 to the step 7 until the algebra G is more than G_maxAnd terminating the calculation and outputting a Pareto solution set of the high-dimensional multi-objective optimization problem.

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