CN111369000A - High-dimensional multi-target evolution method based on decomposition - Google Patents
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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
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 2.1, generating uniformly distributed reference points RP ═ RP { RP on a unit hyperplane L by using a normal boundary Intersection (PBI) method1,RP2,…,RPN};
Step 2.2, based on reference point RP1,RP2,…,RPNConstructing uniformly distributed reference vectors W in a target space1,W2,…,WNThe reference vector satisfies the following condition:
whereinRepresents a reference vector WiA 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 NgWhen 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 conditiong={X1,X2,...,Xi...,XNIn which X isi={x1,x2,...,xDN, D denotes a decision variable XiG represents the algebra of the current population;
step 3.2, calculate population PgTarget vectors f (x) ═ f of all individuals in (c)1(X),f2(X),...,fM(X)),X∈PgAnd 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 ZnadSum worst point ZworstHowever, in the actual multi-objective optimization process, the ideal point Z*Worst point ZworstAnd extreme point ZnadCannot be predicted in advance, based on the current population PgTarget vector calculation ideal point Z of the medium individual*Worst point ZworstAnd extreme point ZnadComprises the following steps:
(3) Extreme pointWhereinAnd extreme pointRepresenting the argument of the point correspondence of the minima, i.e.
Step 3.4, normalization of population PgTarget vectors of all individuals in the group;
converting the target value range of each target to the interval [0,1 ] by normalization]Normalized byIs carried out, wherein fi(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 fi(X)=f′i(X), namely fi(X) represents the target vector after normalization;
step 4, based on the reference vector W1,W2,...,WNThe population PgIs assigned to a sub-population SP ═ SPi,i∈1,2,...,N}:
Step 4.1, reference vector association sub-population: initializing N reference vectors W1,W2,...,WNCorresponding to N empty associated sub-populations SP1,SP2,…,SPNI.e. the reference vector WiAssociation sub-population SPi;
Step 4.2, assign individuals to each sub-population SPi:
(a) First, calculate the population PgTarget vector F (X) of each individuali) With all reference vectors WjAngle therebetween
(b) Comparison of θijThe size of (d);
(c) if k is argminXi(θij|WjJ ═ 1, 2.. times.n), subject i XiTarget vector F (X)i) With the jth reference vector WjAssociating, i.e. the ith individual XiTo the jth reference vector WjConstructed sub-population SPj;
(d) Repeating steps a-c until the population PgAll individuals X iniAre 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 PgAll 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 iffi(a)≤fi(b) And isfi(a)<fi(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 PgAll 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 PgAll the individuals in the system are sorted according to the target, and the sorted result is set as X1,X2,...,XN;
Step 5.1.2, non-dominated sorting:
(a) according to the order X1,X2,...,XNIn 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,…},FiIs the ith non-dominant layer, i ∈ 1, 2.., r, let k be 0;
(b) for any one individual XmM ∈ 1, 2.., N, individual X is first examinedmAnd a non-dominant layer FkDominant relationship of all individuals in (1), if the non-dominant layer FkIn the absence of any individual dominating individual XmThen subject XmIs set to k, i.e. the individual XmBelonging to the kth non-dominant layer Fk. Individual XmAnd a non-dominant layer FkWhen comparing individuals in (1), individual XmFirst and second non-dominant layer FkIs compared with the last individual in (1) and then with the non-dominant layer FiComparison of penultimate individuals, final and non-dominated layer FkThe first individual comparison, i.e., performing a reverse order comparison in the non-dominant layer;
(c) if the individual XmDominated layer FkAnd k < r;
(d) then add 1 to k and repeat steps b-c, otherwise, for the individual XmCreate a new rank r +1, Individual XmBelonging to the (r +1) th non-dominant layer Fr+1;
(e) After the non-dominant sorting is executed, the ranked non-dominant layer is marked as F1,F2,…,Fr;
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 vectorsAnd i is not equal to j,represents WiAnd WjThe included angle between them;
(b) then calculate the distance to each reference vector WiThe nearest T reference vectors, denoted by b (i) ═ i1,i2,...,iTDenotes approaching WiA set of T neighborhood reference vectors, each element in the set being
(c) Correspondingly, the closest sub-population SPiIs the corresponding reference vector WiT neighborhood reference vectorsThe constructed T sub-populations are marked as SPi1,SPi2,...,SPiT;
Step 5.2.2, two parent individuals are selected from the neighborhood population:
(a) individual one XiFor random slave sub-population SPiSelecting one individual with a non-dominant ranking;
(b) individual II XjSlave sub-population SPiOf any neighborhood population SPi1,SPi2,...,SPiTSelecting randomly;
step 5.2.3, performing genetic operations on the two parent individuals: for two selected individuals XiAnd XjPerforming 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 isThe probability of variation isK1And K2Are two constant coefficients, k representing the kth target, k ∈ 1, 2.., M,fk(Xi) Representing an individual XiOf the kth target value, fk(Xj) Representing an individual XjThe kth target value of (1);
step 6, updating the sub-population SP based on the individuals in the sub-population Q1,SP2,...,SPN:
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 Znad;
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,…,FrOn 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,…,FrBy the non-ranked ones of Q and non-dominated layer F1,F2,…,FrCompared 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 dividedgCombining with the population Q to obtain a combined population R ═ Pg∪ Q and updates the non-dominated layer F1,F2,…,FrWherein the size of the combo population R is 2N;
step 6.6, update sub-population SP1,SP2,…,SPN: adopting a step 4.2 individual allocation mechanism to allocate the individuals in the new population Q to the sub-population SPi,i∈{1,2,...,N}。
Step 7, selecting the execution environment of the combined population R and updating the next generation population Pg+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,...,SPNFirst, each sub-population SP is calculatediTarget vector F (X) of the middle individuali) To a reference vector WiEuclidean distance ofI.e. the target vector F (X)i) To a reference vector WiWherein F (X)i) I represents the target vector F (X)i) Is of a length, | Wi| | denotes a reference vector WiThe length of the die (c) is,<F(Xi),Wi>target vector F (X) representing an individuali) And a reference vector WiAngle between them sin<F(Xi),Wi>Indicating included angle<F(Xi),Wi>The sine value of (d);
step 7.1.2, calculate each sub-population SPiTarget vector F (X) of the middle individuali) To construct a reference vector WiDistance from the reference point ofWherein | | | F (X)i)-WiI represents a vector (F (X)i)-Wi) Is the target vector F (X)i) To the reference vector WiIf the sub-population SPiOne is located below the constructed hyperplane L, i.e. the individual dominates the ith reference point, at this timeIs negative;
step 7.1.3, in each sub-population SPiIn, synthesize two distancesAndselecting individuals according to the selection criteriaIndicated in each sub-population SPiTwo distances are selectedMinimum sum of pre-min (2, | SP)i|) two individuals. But each sub-population SP when performing step 6.6 individual assignmentsiMay have no individual, only one individual or a plurality of individuals, if the sub-population SPiNo individual in the population, no selection; if the sub-population SPiIf only one individual is present, the individual is selected; otherwise two distances are selectedAndthe 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 useRepresenting non-dominant layers after execution of a local environment selection policyIf 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 NAnd isThen the global environment selection policy is executed from the r-th non-dominant layerIn selectionEntering individual into next generation population;
step 7.2.2, calculate non-dominated layerThe Euclidean distance between the ith individual and the jth individual is equal toK and i ≠ j, orderRepresenting a non-dominant layerThe number of individuals;
step 7.2.3, calculate non-dominated layerAverage distance of all neighboring individuals inAnd performs global selection:
(a) if adjacent distanceGreater than average distanceGreater thanSelecting adjacent distancesMaximum frontEntering individual into next generation population;
(b) if adjacent distanceGreater than average distanceHas a number of individuals of K' andfirst select the adjacent distanceGreater than average distanceThe K' individuals enter the next generation population;
(c) then from adjacent distanceLess than the average distanceSelecting K' individuals from the individuals of (1) into a next generation population, whereinThe selection process is as follows:
1) for adjacent distanceLess than the average distanceThe 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 outThe distance between the ith individual and the (i + s) th individual after sorting, wherein the initial s is 2;
3) if it is notPerform s plus 1 and compare againAndup toThen selecting the (i + s) th individual Xi+sEntering a next generation population;
4) then the (i + s) th individual Xi+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 ″, whereinIndicating adjacent distanceGreater than average distanceδ 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 GmaxAnd 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 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,...,RPN};
Step 2.2, based on reference point RP1,RP2,...,RPNConstructing uniformly distributed reference vectors W in a target space1,W2,...,WN. The reference vector satisfies the following condition:
whereinRepresents a reference vector WiAnd component value of j-th dimension ofH represents that each target is divided into H equal parts, and the number of generated reference vectors is N, whereinIf 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 NgWhen 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 conditiong={X1,X2,...,XNIn which X isi={x1,x2,...,xDN, D denotes a decision variable XiG represents the algebra of the current population;
step 3.2, calculate population PgTarget vectors f (x) ═ f of all individuals in (c)1(X),f2(X),...,fM(X)),X∈PgAnd 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 ZnadSum worst point ZworstHowever, in the actual multi-objective optimization process, the ideal point Z*Worst point ZworstAnd extreme point ZnadCannot be predicted in advance, so based on the current population PgThe target vector of the medium body continuously updates the ideal point Z*Worst point ZworstAnd extreme point Znad:
(3) Extreme pointWhereinAnd extreme pointRepresenting the argument of the point correspondence of the minima, i.e.
Step 3.4, normalization of population PgTarget vectors of all individuals. Converting the target value range of each target to the interval [0,1 ] by normalization]. Normalized byIs carried out, wherein fi(X) denotes the actual target value, f' of the individual Xi(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 fi(X)=f′i(X), namely fi(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,...,WNThe population PgIs assigned to a sub-population SP ═ SPi,i∈1,2,...,N}:
Step 4.1, the reference vectors are associated with the sub-populations. Initializing N reference vectors W1,W2,...,WNCorresponding to N empty associated sub-populations SP1,SP2,...,SPNI.e. the reference vector WiAssociation sub-population SPi;
Step 4.2, assign individuals to each sub-population SPi:
(a) First, calculate the population PgTarget vector F (X) of each individuali) With all reference vectors WjAngle therebetween
(b) Comparison of θijIs largeSmall;
(c) if it is notThe ith individual XiTarget vector F (X)i) With reference vector W of jjAssociating, i.e. the ith individual XiTo the jth reference vector WjConstructed sub-population SPj;
(d) Repeating a-c until the population PgAll individuals X iniAre 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 PgAll 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 iffi(a)≤fi(b) And isfi(a)<fi(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 PgAll 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 PgAll the individuals in the system are sorted according to the target, and the sorted result is set as X1,X2,...,XN;
Step 5.1.2, non-dominated sorting:
(a) according to the order X1,X2,...,XNIn 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,…},FiIs the ith non-dominant layer, i ∈ 1, 2.., r, let k be 0;
(b) for any one individual XmM ∈ 1, 2.., N, individual X is first examinedmAnd a non-dominant layer FkDominant relationship of all individuals in (1), if the non-dominant layer FkIn the absence of any individual dominating individual XmThen subject XmIs set to k, i.e. the individual XmBelonging to the kth non-dominant layer Fk. Individual XmAnd a non-dominant layer FkWhen comparing individuals in (1), individual XmFirst and second non-dominant layer FkIs compared with the last individual in (1) and then with the non-dominant layer FiComparison of penultimate individuals, final and non-dominated layer FkThe first individual comparison, i.e., performing a reverse order comparison in the non-dominant layer;
(c) if the individual XmDominated layer FkAnd k < r;
(d) k is k +1 and steps b-c are repeated, otherwise the individual XmCreate a new rank r +1, Individual XmBelonging to the (r +1) th non-dominant layer Fr+1. Sorting the results X in step 5.1.1(c)1,X2,...,XNSequentially 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、…、FrIf 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,…,Fr;
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 vectorsAnd i is not equal to j,represents WiAnd WjThe included angle between them;
(b) then calculate the distance to each reference vector WiThe nearest T reference vectors, denoted by b (i) ═ i1,i2,...,iTDenotes approaching WiA set of T neighborhood reference vectors, each element in the set being
(c) Correspondingly, the closest sub-population SPiIs the corresponding reference vector WiT neighborhood reference vectorsThe constructed T sub-populations are marked as SPi1,SPi2,...,SPiT;
Step 5.2.2, two parent individuals are selected from the neighborhood population:
(a) individual one XiFor random slave sub-population SPiSelecting one individual with a non-dominant ranking;
(b) individual II XjSlave sub-population SPiOf any neighborhood population SPi1,SPi2,...,SPiTIs 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 XiAnd XjPerforming 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 isThe probability of variation isK1And K2Are two constant coefficients, k representing the kth target, k ∈ 1, 2.., M,representing an individual XiOf the kth target value, fk(Xj) Representing an individual XjThe kth target value of (1).
Step 6, updating the sub-population SP based on the individuals in the sub-population Q1,SP2,…,SPN:
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 Znad;
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,…,FrOn 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,…,FrBy the non-ranked ones of Q and non-dominated layer F1,F2,…,FrCompared 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 selectedgCombining with the population Q to obtain a combined population R ═ Pg∪ Q and updates the non-dominated layer F1,F2,…,FrWherein the size of the combo population R is 2N;
step 6.6, update sub-population SP1,SP2,...,SPN: adopting a step 4.2 individual distribution method to distribute the individuals in the new population Q to the sub-population SPi,i∈{1,2,...,N}。
Step 7, selecting the execution environment of the combined population R and updating the next generation population Pg=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,...,SPNFirst, each sub-population SP is calculatediTarget vector F (X) of the middle individuali) To a reference vector WiEuclidean distance ofI.e. the target vector F (X)i) To a reference vector WiWherein F (X)i) I represents the target vector F (X)i) Is of a length, | WiI denotes the reference directionQuantity WiThe length of the die (c) is,<F(Xi),Wi>target vector F (X) representing an individuali) And a reference vector WiAngle between them sin<F(Xi),Wi>Indicating included angle<F(Xi),Wi>The sine value of (d);
step 7.1.2, calculate each sub-population SPiTarget vector F (X) of the middle individuali) To construct a reference vector WiDistance from the reference point ofWherein | | | F (X)i)-WiI represents a vector (F (X)i)-Wi) Is the target vector F (X)i) To the reference vector WiIf the sub-population SPiOne is located below the constructed hyperplane L, i.e. the individual dominates the ith reference point, at this timeIs negative;
step 7.1.3, in each sub-population SPiIn, synthesize two distancesAndselecting individuals according to the selection criteriaIndicated in each sub-population SPiTwo distances are selectedMinimum sum of pre-min (2, | SP)i|) two individuals. But each sub-population SP when performing step 6.6 individual assignmentsiMay have no individual, only one individual or a plurality of individuals, if the sub-population SPiNo individual in the population, no selection; if the sub-population SPiOnly one of themSelecting the individual if the individual is selected; otherwise two distances are selectedAndthe 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 usingRepresenting non-dominant layers after execution of a local environment selection policyIf 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 NAnd isThen the global environment selection policy is executed from the r-th non-dominant layerIn selectionEntering individual into next generation population;
step 7.2.2, calculate non-dominated layerThe 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 ofK and i ≠ j, orderRepresenting a non-dominant layerThe number of individuals;
step 7.2.3, calculate non-dominated layerAverage distance of all neighboring individuals inAnd performs global selection.
(a) If adjacent distanceGreater than average distanceGreater thanSelecting adjacent distancesMaximum frontEntering individual into next generation population;
(b) if adjacent distanceGreater than average distanceHas a number of individuals of K' andfirst select the adjacent distanceGreater than average distanceThe K' individuals enter the next generation population;
(c) then from adjacent distanceLess than the average distanceSelecting K' individuals from the individuals of (1) into a next generation population, whereinThe selection process is as follows:
1) for adjacent distanceLess than the average distanceThe 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 outThe distance between the ith individual and the (i + s) th individual after sorting, wherein the initial s is 2;
3) if it is notPerforming s as s +1 and comparingAndup toThen selecting the (i + s) th individual Xi+sEntering a next generation population;
4) then the (i + s) th individual Xi+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 ″, whereIndicating adjacent distanceGreater than average distanceIs the minimum of all adjacent distances, δ is the parameter controlling the selection range.
Step 8, G is less than or equal to algebraic GmaxCircularly 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 GmaxThe 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,...,WN}:
Step 2.1, generating uniformly distributed reference points RP ═ RP on a unit hyperplane L by using a normal boundary intersection method1,RP2,...,RPN};
Step 2.2, based on reference point RP1,RP2,...,RPNConstructing uniformly distributed reference vectors W in a target space1,W2,...,WNThe reference vector satisfies the following condition:
whereinRepresents a reference vector WiA component value of the j-th dimension of (a), andi 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 Ng=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 conditiong={X1,X2,...,Xi...,XNIn which X isi={x1,x2,...,xDN, D denotes a decision variable XiG represents the generation of the current populationCounting;
step 3.2, calculate population PgTarget vectors f (x) ═ f of all individuals in (c)1(X),f2(X),...,fM(X)),X∈PgAnd 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 ZnadSum worst point ZworstHowever, in the actual multi-objective optimization process, the ideal point Z*Worst point ZworstAnd extreme point ZnadCannot be predicted in advance, based on the current population PgTarget vector calculation ideal point Z of the medium individual*Worst point ZworstAnd extreme point ZnadComprises the following steps:
(3) Extreme pointWhereinAnd extreme point Representing the argument of the point correspondence of the minima, i.e.
Step 3.4, normalization of population PgTarget vectors of all individuals in the group;
converting the target value range of each target to the interval [0,1 ] by normalization]Normalized byIs carried out, wherein fi(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 fi(X)=f′i(X), namely fi(X) represents the target vector after normalization;
step 4, based on the reference vector W1,W2,...,WNThe population PgIs assigned to a sub-population SP ═ SPi,i∈1,2,...,N}:
Step 4.1, reference vector association sub-population: initializing N reference vectors W1,W2,...,WNCorresponding to N empty associated sub-populations SP1,SP2,...,SPNI.e. the reference vector WiAssociation sub-population SPi;
Step 4.2, assign individuals to each sub-population SPi:
(a) First, calculate the population PgTarget vector F (X) of each individuali) With all reference vectors WjAngle therebetween
(b) Comparison of θijThe size of (d);
(c) if it is notThe ith individual XiTarget vector F (X)i) With the jth reference vector WjAssociating, i.e. the ith individual XiTo the jth reference vector WjConstructed sub-population SPj;
(d) Repeating steps a-c until the population PgAll individuals X iniAre 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 PgAll 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 iffi(a)≤fi(b) And isfi(a)<fi(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 PgAll 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 PgAll the individuals in the system are sorted according to the target, and the sorted result is set as X1,X2,…,XN;
Step 5.1.2, non-dominated sorting:
(a) according to the order X1,X2,...,XNSequentially selects individual execution non-dominant orderingIn advance, r is set to 0 non-dominant layers { F1,F2,…},FiIs the ith non-dominant layer, i ∈ 1, 2.., r, let k be 0;
(b) for any one individual XmM ∈ 1, 2.., N, individual X is first examinedmAnd a non-dominant layer FkDominant relationship of all individuals in (1), if the non-dominant layer FkIn the absence of any individual dominating individual XmThen subject XmIs set to k, i.e. the individual XmBelonging to the kth non-dominant layer Fk(ii) a Individual XmAnd a non-dominant layer FkWhen comparing individuals in (1), individual XmFirst and second non-dominant layer FkIs compared with the last individual in (1) and then with the non-dominant layer FiComparison of penultimate individuals, final and non-dominated layer FkThe first individual comparison, i.e., performing a reverse order comparison in the non-dominant layer;
(c) if the individual XmDominated layer FkAnd k < r;
(d) then add 1 to k and repeat steps b-c, otherwise, for the individual XmCreate a new rank r +1, Individual XmBelonging to the (r +1) th non-dominant layer Fr+1;
(e) After the non-dominant sorting is executed, the ranked non-dominant layer is marked as F1,F2,…,Fr;
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 vectorsAnd i is not equal to j,represents WiAnd WjThe included angle between them;
(b) then calculate the distance to each reference vector WiMost recent T referencesVector, using B (i) ═ i1,i2,...,iTDenotes approaching WiA set of T neighborhood reference vectors, each element in the set being
(c) Correspondingly, the closest sub-population SPiIs the corresponding reference vector WiT neighborhood reference vectorsThe constructed T sub-populations are marked as SPi1,SPi2,...,SPiT;
Step 5.2.2, two parent individuals are selected from the neighborhood population:
(a) individual one XiFor random slave sub-population SPiSelecting one individual with a non-dominant ranking;
(b) individual II XjSlave sub-population SPiOf any neighborhood population SPi1,SPi2,...,SPiTSelecting randomly;
step 5.2.3, performing genetic operations on the two parent individuals: for two selected individuals XiAnd XjPerforming 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 isThe probability of variation isK1And K2Are two constant coefficients, k representing the kth target, k ∈ 1, 2.., M,fk(Xi) Representing an individual XiOf the kth target value, fk(Xj) Representing an individual XjThe kth target value of (1);
step 6, updating the sub-population SP based on the individuals in the sub-population Q1,SP2,...,SPN:
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 Znad;
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,…,FrOn 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,…,FrBy the non-ranked ones of Q and non-dominated layer F1,F2,…,FrCompared 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 dividedgCombining with the population Q to obtain a combined population R ═ Pg∪ Q and updates the non-dominated layer F1,F2,…,FrWherein the size of the combo population R is 2N;
step 6.6, update sub-population SP1,SP2,...,SPN: adopting a step 4.2 individual allocation mechanism to allocate the individuals in the new population Q to the sub-population SPi,i∈{1,2,...,N};
Step 7, selecting the execution environment of the combined population R and updating the next generation population Pg+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,...,SPNFirst, each sub-population SP is calculatediTarget vector F (X) of the middle individuali) To a reference vector WiEuclidean distance ofI.e. the target vector F (X)i) To a reference vector WiWherein F (X)i) I represents the target vector F (X)i) Is of a length, | Wi| | denotes a reference vector WiThe length of the die (c) is,<F(Xi),Wi>target vector F (X) representing an individuali) And a reference vector WiAngle between them sin<F(Xi),Wi>Indicating included angle<F(Xi),Wi>The sine value of (d);
step 7.1.2, calculate each sub-population SPiTarget vector F (X) of the middle individuali) To construct a reference vector WiDistance from the reference point ofWherein | | | F (X)i)-WiI represents a vector (F (X)i)-Wi) Is the target vector F (X)i) To the reference vector WiIf the sub-population SPiOne is located below the constructed hyperplane L, i.e. the individual dominates the ith reference point, at this timeIs negative;
step 7.1.3, in each sub-population SPiIn, synthesize two distancesAndselecting individuals according to the selection criteria Indicated in each sub-population SPiTwo distances are selectedMinimum sum of pre-min (2, | SP)i|) two individuals, but when step 6.6 individual assignment is performed, each sub-population SPiMay have no individual, only one individual or a plurality of individuals, if the sub-population SPiNo individual in the population, no selection; if the sub-population SPiIf only one individual is present, the individual is selected; otherwise two distances are selectedAndthe 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 | Fi gI represents the non-dominant layer F after execution of the local environment selection policyi 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 NAnd isThen the global environment selection policy is executed from the r-th non-dominant layerIn selectionEntering individual into next generation population;
step 7.2.2, calculate non-dominated layerThe Euclidean distance between the ith individual and the jth individual is equal toAnd i ≠ j, let K ═ Fi gI denotes the non-dominant layerThe number of individuals;
step 7.2.3, calculate non-dominated layerAverage distance of all neighboring individuals inAnd performs global selection:
(a) if adjacent distanceGreater than average distanceGreater thanSelecting adjacent distancesMaximum frontEntering individual into next generation population;
(b) if adjacent distanceGreater than average distanceHas a number of individuals of K' andfirst select the adjacent distanceGreater than average distanceThe K' individuals enter the next generation population;
(c) then from adjacent distanceLess than the average distanceSelecting K' individuals from the individuals of (1) into a next generation population, whereinThe selection process is as follows:
1) for adjacent distanceLess than the average distanceThe 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 outThe distance between the ith individual and the (i + s) th individual after sorting, wherein the initial s is 2;
3) if it is notPerform s plus 1 and compare againAndup toThen selecting the (i + s) th individual Xi+sEntering a next generation population;
4) then the (i + s) th individual Xi+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 ″, whereinTo representAdjacent distanceGreater than average distanceδ 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 GmaxAnd terminating the calculation and outputting a Pareto solution set of the high-dimensional multi-objective optimization problem.
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