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

Abstract

The invention provides a decomposition-based high-dimensional multi-target evolution method, which comprises the steps of generating a reference vector, decomposing a high-dimensional multi-target optimization problem into a plurality of single-target optimization subproblems, constructing a subproject of the single-target optimization subproblem based on the reference vector, distributing individuals for the subproject by using a distribution mechanism, constructing a neighborhood subproject, selecting the individuals to carry out genetic evolution by using the constructed neighborhood subproject, selecting the individuals with excellent performance in the population to enter a next generation of population by using a designed local and global selection strategy, and repeatedly executing an evolution process until the evolution process is terminated and a Pareto solution set of the high-dimensional multi-target optimization problem is obtained. The invention effectively reduces the solving complexity of the problem, solves the problem that the multi-objective optimization algorithm is difficult to ensure good balance between population convergence and diversity, obtains Pareto solution sets with good diversity and convergence, effectively improves the algorithm efficiency, and can effectively ensure the global convergence and population diversity of the algorithm.

Description

High-dimensional multi-target evolution method based on decomposition

Technical Field

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

Background

Many of the problems in real life are actually multi-objective optimization problems (MOPs), i.e. with two or more objective optimization problems. Since the multi-objective optimization problem has multiple conflicting objectives, the final solution is not an optimal solution but a set of Pareto solutions. With the increase of the number of the targets, the optimization problem is increased to 4 or more from the original 2-3 targets, and the problem becomes a high-dimensional multi-target optimization problem (MaOPs). The number of non-dominant individuals in the population increases dramatically due to the increase in the number of targets, and when the targets are increased to a certain number, almost all individuals in the population are non-dominant, thus rapidly lowering the selection pressure of the evolving population, i.e. the pressure at which the population converges towards the Pareto Frontier (PF) when the dominant relationship is used as a criterion for selecting individuals of a limited population size. When the existing multi-objective evolutionary algorithm is used for solving the high-dimensional multi-objective optimization problem, great challenges are provided for the evolutionary algorithm in order to ensure good balance of population diversity and convergence. Although the evolutionary algorithm shows excellent performance when solving the multi-objective optimization problem, for the high-dimensional multi-objective optimization problem, the existing method has the difficulties that the target dimension is difficult to expand, the Pareto domination relation cannot distinguish evolution individuals, the diversity maintenance strategy is invalid and the like.

The high-dimensional multi-objective optimization algorithm proposed at present is mainly divided into three categories:

1) based on the Pareto dominant method. The method enhances the selection pressure by modifying a Pareto domination relation or a diversity maintenance strategy, so that a high-dimensional multi-objective optimization problem is solved.

2) Index-based methods. The method guides the evolution of the population through some evaluation indexes, so that the high-dimensional multi-target optimization problem is solved.

3) A decomposition-based approach. The method decomposes a high-dimensional target optimization problem into a plurality of multi-target optimization sub-problems or a series of single-target optimization sub-problems by introducing a polymerization function, and then evolves the decomposed multi-target or single-target optimization sub-problems simultaneously, thereby solving the high-dimensional multi-target optimization problem.

Disclosure of Invention

In order to overcome the defects of the prior art and overcome the defect that the convergence and diversity are difficult to balance effectively when the existing multi-target evolutionary algorithm is used for solving the high-dimensional target optimization problem, a decomposition-based high-dimensional multi-target evolutionary algorithm is provided and mainly comprises reference vector generation, sub-population construction, population initialization, individual fitness evaluation, ideal point and extreme point calculation and updating, target vector normalization, target vector and reference vector association, population non-dominant sorting, neighborhood sub-population construction, cross variation individual selection, offspring population generation and environment selection strategies. The method comprises the steps of generating a series of uniformly distributed reference vectors, decomposing a high-dimensional multi-objective optimization problem into a plurality of single-objective optimization subproblems by using the reference vectors, constructing a subprogram of the single-objective optimization subproblems based on the reference vectors, distributing individuals for the subprogram by using a distribution mechanism, constructing a neighborhood subprogram based on the subprogram, selecting the individuals to carry out genetic evolution by using the constructed neighborhood subprogram, selecting the individuals with excellent performance in the subprogram to enter a next generation of subprogram by using designed local and global selection strategies, and repeatedly executing an evolution process until an algorithm is terminated and a Pareto solution set of the high-dimensional multi-objective optimization problem is obtained. By means of the method, good balance between population convergence and diversity of the algorithm in solving the high-dimensional multi-target optimization problem is guaranteed through the concept based on decomposition, the selection strategy of cross variation individuals, the self-adaptive adjustment of cross and variation probability, the environment selection strategy and the like, and the defect that the performance of the high-dimensional multi-target optimization algorithm based on Pareto domination relation is rapidly deteriorated in solving the high-dimensional multi-target optimization problem is effectively overcome.

The technical scheme adopted by the invention for solving the technical problem comprises 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 { RP on a unit hyperplane L by using a normal boundary Intersection (PBI) 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

h represents that each target is divided into H equal parts, and the number of generated reference vectors is N, wherein

Step 3, randomly generating an initial population P with the size of N_gWhen the individual fitness is 0, the individual fitness evaluation is performed:

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 algebra of the current population;

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 k is argminX_i(θ_ij|W_jJ ═ 1, 2.. times.n), subject i 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_NIn the order of (1), individual ones are sequentially selected and non-dominated sorting is performed, and r is set in advance to 0 non-dominated layers { F ═ 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. 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_iThe nearest T reference vectors, denoted by 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 sorted non-dominantly using step 5.1.2(b), even moreNew non-dominated layer F₁,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: in the environment selection process, N individuals with excellent performance are selected from 2N individuals in the R to serve as parent populations of next generations, and the environment selection strategy comprises a local environment selection strategy and a global environment selection strategy;

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 each sub-population SP when performing step 6.6 individual assignments_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 of population for use

Representing non-dominant layers after execution of a local environment selection policy

If 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

K and i ≠ j, order

Representing a 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

Indicating adjacent 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.

The invention has the beneficial effects that:

1. aiming at the high-dimensional target optimization problem, a decomposition-based high-dimensional multi-target evolutionary algorithm is provided, the high-dimensional multi-target optimization problem is decomposed into a plurality of single-target optimization sub-problems, and the plurality of single-target optimization sub-problems are evolved, so that the problem solving complexity is effectively reduced, the problem that the existing multi-target optimization algorithm is difficult to ensure good balance between population convergence and diversity is solved, and a Pareto solution set with good diversity and convergence is obtained;

2. aiming at a non-dominant sorting method, the dominant relationship between the non-ranked individuals and the ranked individuals is adopted to compare the individuals, so that the algorithm efficiency can be effectively improved;

3. the individual for executing genetic operation is selected based on the neighborhood sub-population, so that the global convergence and population diversity of the algorithm can be effectively ensured;

4. aiming at individuals subjected to genetic operation, for individuals with excellent fitness values, the cross probability and the mutation probability are small in order to maintain good performance of the individuals, and for individuals with poor fitness values, the cross probability and the mutation probability are large in order to improve the performance of the individuals, and the convergence speed and diversity of a population can be effectively improved by adaptively adjusting the cross probability and the mutation probability through the fitness values of the individuals;

5. aiming at the environment selection strategy, the problem of failure of a diversity maintenance strategy is effectively solved by fusing a local environment selection strategy and a global environment selection strategy, the performance of a high-dimensional multi-target optimization algorithm based on a Pareto domination relation in solving the high-dimensional target optimization problem is improved, and the convergence and the diversity are well balanced.

Drawings

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

FIG. 2 is a block diagram of the algorithm 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 an illustration of reference point generation and reference vector sample construction in a three-dimensional target space according to the present invention.

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

FIG. 6 is an illustration of a neutron population sample in a two-dimensional target space according to the present invention.

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

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

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

FIG. 10 is a sample diagram of a global environment selection strategy in a two-dimensional target space according to the present invention.

Detailed Description

The invention is further illustrated with reference to the following figures and examples.

Example 1: as shown in fig. 1-10, a decomposition-based high-dimensional multi-objective evolutionary algorithm includes steps 1-8.

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.

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

Step 2.1, generating uniformly distributed reference points RP ═ RP { RP on a unit hyperplane L by using a normal boundary intersection Point (PBI) 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_N. The reference vector satisfies the following condition:

wherein

Represents a reference vector W_iAnd component value of j-th dimension of

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

If M is 3 and H is 4, the number of uniformly distributed reference vectors generated on the unit hyperplane is

Step 3, randomly generating an initial population P with the size of N_gWhen the individual fitness is 0, the individual fitness evaluation is performed:

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_NIn which X is_i＝{x₁,x₂,...,x_DN, D denotes a decision variable X_iG represents the algebra of the current population;

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, so based on the current population P_gThe target vector of the medium body continuously updates the ideal point Z^*Worst point Z^worstAnd extreme point Z^nad：

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

Is carried out, wherein f_i(X) denotes the actual target value, f' of the individual X_i(X) represents the target vector of the individual X after normalization, wherein i ∈ 1, 2.. multidot.m. the target vectors mentioned later are all 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. When an object has a value range of [0,100 ] in the object space]The range of values of another object in the target space is [0,10000 ]]The value ranges of different targets can be greatly different, and after normalization processing, the value ranges of all targets can be converted into the same interval for comparison without affecting the optimization process.

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, the reference vectors are associated with the sub-populations. 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_ijIs largeSmall;

(c) if it is not

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

(d) Repeating a-c until the population P_gAll individuals X in_iAre assigned to a sub-population.

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_NIn the order of (1), individual ones are sequentially selected and non-dominated sorting is performed, and r is set in advance to 0 non-dominated layers { F ═ 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. 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) k is k +1 and steps b-c are repeated, otherwise the individual X_mCreate a new rank r +1, Individual X_mBelonging to the (r +1) th non-dominant layer F_r+1. Sorting the results X in step 5.1.1(c)₁,X₂,...,X_NSequentially selects individuals to perform non-dominated sorting, first selects the first individual X with a sorting of 1₁The individual X₁Put into the first non-dominated layer F₁Rank 1; then select the second individual X with rank 2₂Comparing individuals X₂And the first individual X that has been ranked₁The target value of (2), judging the individual X₁And individual X₂If the individual X₁Dominating individual X₂Then X will be₂Placing a second non-dominated layer F₂Rank 2 if individual X₁And individual X₂If they do not dominate each other, then X will be₂Placing a first non-dominated layer F₁In, X₂Rank of 1; then selectSelecting an individual X with rank 3₃Comparing individuals X₃And a first non-dominated layer F₁The dominant relationship of all individuals in (1), and then the non-dominant layer F₂、F₃、…、F_rIf the individual X dominates the relationship₃If no ranking is assigned, the rank is the individual X₃Creating a new rank r +1 until all individuals are assigned a corresponding rank;

(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_iThe nearest T reference vectors, denoted by 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_iTIs randomly selected. The individual is selected through the neighborhood for crossing and variation, so that the individual can be ensured to inherit the property of a solution with better convergence performance, and the global search capability of the algorithm can be ensured, thereby ensuring the diversity and convergence of the population;

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'_jAnd N sub-generation individuals, namely a new population Q, are generated after N/2 times of repeated execution. 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,

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：

And 6.1, evaluating the fitness of all the individuals in the new population Q by adopting the step 3.2. 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^*Worst and sumPoint 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 performed, the population P is actually selected_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 distribution method to distribute 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＝g+1: in the environment selection process, N individuals with excellent performance are selected from 2N individuals in the R to serve as parent populations of next generations, and the environment selection strategy comprises a local environment selection strategy and a global environment selection strategy;

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_iI denotes the reference directionQuantity 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 each sub-population SP when performing step 6.6 individual assignments_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_iOnly one of themSelecting the individual if 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:

and 7.2.1, deleting all individuals obtained after the local environment selection strategy is executed by the global environment selection strategy based on non-dominant ranking, and selecting N individuals to enter a next generation population. By using

Representing non-dominant layers after execution of a local environment selection policy

If 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 adjacent individuals, if the ith individual is adjacent to the jth individual, the Euclidean distance between the ith individual and the jth individualA distance of

K and i ≠ j, order

Representing a 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

Performing s as s +1 and comparing

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 an initial individual, and let i ═ i + s, calculate the euclidean distance between the ith individual and the (i + s) th individual in ascending order until the number of selected individuals reaches K ″, where

Indicating adjacent distance

Greater than average distance

Is the minimum of all adjacent distances, δ is the parameter controlling the selection range.

Step 8, G is less than or equal to algebraic G_maxCircularly executing the step 4 to the step 7; otherwise, the algorithm is terminated and a Pareto solution set of the high-dimensional multi-objective optimization problem is output.

While the present invention has been described in detail with reference to the embodiments shown in the drawings, the present invention is not limited to the embodiments, and various changes can be made without departing from the spirit of the present invention within the knowledge of those skilled in the art.

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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