CN112270120A - Multi-objective optimization method based on hierarchical decomposition of tree structure - Google Patents

Multi-objective optimization method based on hierarchical decomposition of tree structure Download PDF

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CN112270120A
CN112270120A CN202011022305.4A CN202011022305A CN112270120A CN 112270120 A CN112270120 A CN 112270120A CN 202011022305 A CN202011022305 A CN 202011022305A CN 112270120 A CN112270120 A CN 112270120A
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辜方清
吴润佳
刘海林
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Abstract

The invention discloses a multi-objective optimization method based on hierarchical decomposition of a tree structure, which comprises the following steps: 1. initializing parameters and generating a group of uniformly distributed unit weight vectors; s2, constructing a tree and initializing a population; s3, evolutionary algorithm; in the proposed algorithm, the candidate solution only needs to be compared with the solution on the path from the root of the tree to the leaf node. Therefore, the computational complexity of processing the candidate single solution is only o (m logn), and the present invention approaches and refines the Pareto front by solving a few representative sub-problems. The strategy has the beneficial effects of low time complexity and high calculation efficiency for solving the large space optimization problem.

Description

Multi-objective optimization method based on hierarchical decomposition of tree structure
Technical Field
The invention relates to an optimization method, in particular to a multi-objective optimization method based on hierarchical decomposition of a tree structure.
Background
Over the past decades, researchers have proposed a variety of evolutionary multi-objective optimization algorithms. The evolutionary multi-objective optimization algorithms simultaneously evolve one population to approximate a Pareto frontier in one operation, and have achieved great success in practical engineering application. These algorithms can be roughly divided into three categories depending on the selection operator: the method comprises a Pareto domination based multi-objective optimization algorithm, a decomposition based multi-objective optimization algorithm and an index based multi-objective optimization algorithm.
Pareto dominated based algorithms, such as non-dominated ranking genetic algorithms and strong Pareto dominated multi-objective evolutionary algorithms, use Pareto dominated ranking as a primary selection mechanism and density based selection criteria (such as crowding distance) as a secondary selection mechanism. In recent years, many improved Pareto dominated based algorithms have been proposed for the challenge of specific problems. However, the algorithm complexity of Pareto dominant ranking calculation is high, and particularly when the population size increases, the calculation overhead will increase sharply. In addition, the algorithm based on Pareto domination sharply reduces the performance of the algorithm with the increase of the target number due to insufficient selection pressure when processing the super-multi-objective optimization problem.
Index-based evolutionary multi-objective optimization algorithms, such as S-metric-based evolutionary multi-objective algorithms, directly utilize the individual contribution to a performance index as a selection criterion to select offspring. These algorithms maintain the convergence and diversity of the candidate solution set well. However, the algorithm complexity of such algorithms is usually high. The running time complexity of the S-metric evolutionary multi-objective algorithm grows exponentially with the increase of the number of targets.
The decomposition-based multi-objective algorithm decomposes a multi-objective optimization problem into a number of simple optimization sub-problems through a set of weight vectors, but the performance is not very ideal in handling multi-objective optimization with irregular Pareto fronts.
The population scale of the evolutionary multi-objective optimization algorithm is a given constant. The size of the population size generally has a large impact on the performance of the evolutionary algorithm. Small population sizes may lead to premature convergence. And if the population scale is too large, a large amount of resources are wasted, and the searching efficiency of the algorithm is reduced.
Disclosure of Invention
The invention aims to solve the problems and provides a multi-objective optimization method based on hierarchical decomposition of a tree structure. The method comprises the steps of rapidly approaching a Pareto front edge through a small-scale population by a successive approximation algorithm mechanism to obtain a rough front edge interface, and gradually refining by introducing more sub-problems step by step. The strategy can effectively improve the efficiency of the algorithm, and particularly, the performance of the algorithm is obviously improved when a large space optimization problem is solved.
The purpose of the invention can be achieved by adopting the following technical scheme:
a multi-objective optimization method based on hierarchical decomposition of a tree structure comprises the following steps:
s1 initialization parameters
Generating a set of uniformly distributed unit weight vectors V ═ V1，v2，…，vn}；
S2, constructing a tree and initializing a population
According to the weight vector (v)cV) constructing a spanning tree T using an algorithm represented by the structure of the tree, each node of the tree containing a weight vector and being associated with the subproblems by means of its weight vector; the node of the first layer L2 is set as active, and the node without active descendant is recorded as a leaf node;
selecting an optimal individual from initial solutions by each subproblem of the active node, wherein the selected solutions form an initial overall T.X; population size N*| T · X | is the size of the solution, i.e. the number of active nodes;
uniformly and randomly generating N initial solutions in a decision space as an initial population P;
setting the number of active nodes L to 2, the improvement rate of minimum population convergence Delta to 0.1 and the central weight
Figure BDA0002701037190000021
Improved rate delta for initializing initial population convergence index 01, the current iteration time t is 1 and the external set A is P;
s3 evolutionary algorithm
In each generation, selecting a solution set Q through a mating selection strategy, and creating N offspring Y according to Q;
current one population improvement rate Δt-1< Delta and N*When the ratio is less than N, the reaction mixture is,
expanding the number of layers of active nodes in the tree, wherein the optimal solution P is the union of the descendant Y and T.X, the number L of the active nodes is L +1, and initializing delta againtThen, each active node sub-problem selects the best individual from solution set P, forming a new T · X, population size N*＝|T·X|；
Otherwise, updating the solution corresponding to the node in the tree T and the external set A thereof according to the population evolution algorithm, and calculating the current improvement rate deltat
The iteration number t is t +1
And (5) exiting the algorithm when the iteration times reach the maximum.
The specific content of step S2 is:
constructing a tree from the weight vectors
V＝{v1，v2，…，vN} (1)
Is a set of uniformly distributed unit weight vectors, wherein
Figure BDA0002701037190000031
Is the ith weight vector; constructing a tree structure based on the set of weight vectors, wherein each node of the tree comprises a weight vector, and the weight vector is a central vector v of all weight vectors distributed to the nodecThe center vector may not be in the set of weight vectors V, we first find the nearest center vector V in VcWeight vector v of
Figure BDA0002701037190000032
The node contains a weight vector V and removes V from a set of weight vectors V; determining a center vector v using different methods for root nodes, level two nodes, and other nodesc；
Weight vector v of each nodeiAll define the following sub-problems
g(x|vi)＝d1+θd2 (5)
Wherein d is1＝F(x)Tvi，d2＝||F(x)-d1viL. These subproblems are adaptively organized through a tree structure, where initially the node of the first L ═ 2 layers is set as the active node, and P ═ x is set as1，…，xNThe method comprises the steps that (1) an initial solution set is adopted, then each subproblem of each active node selects an optimal solution according to the value of a formula (5) to initialize a population, and an initial population is formed;
for the weight vector viX is the current best solution, yiFor the newly generated solution, P is the current node pointer; s is the weight vector v of formula (5) which is the same as F (x)iEquivalent points in direction, i.e.
s＝g(x|vi)vi (7)
Dividing the target space into three sub-regions I, II and III by an equivalence point s;
subregion I: if g (y)iI P.v) < g (P.x I P.v) indicates yiIs superior to x, comparing P.x with yiPerforming exchange processing to update the equivalence point P.s-g (P.x | P.v) P.v;
subregion III: if P.s < F (y)i) If the solution ratio x in the area is different, the operation is not performed;
subregion II: if P is a leaf node, yiRandomly replacing one solution in archive A; otherwise, yiIs transmitted to the nearest child node
Figure BDA0002701037190000033
Wherein<Tj.v，F(yi)>Representing the angle, moving the node pointer P to the child node Tj*；
When the convergence rate of the population is lower than a given value, the population scale is expanded; the yield of improvement of the population is defined as the total improvement of the individuals in the population:
Figure BDA0002701037190000041
Figure BDA0002701037190000042
and
Figure BDA0002701037190000043
which are the solutions of the same subproblem in the t-1 and t generation processes, respectively. When the convergence rate is lower than a given value, increasing the number of layers of active nodes in the tree; calculating the current improvement rate Delta by defining the population convergence improvement rate formula in the formula (6)t；
The specific content of step S3 is:
if straight line
Figure BDA0002701037190000044
fiThe variable is intersected with the Pareto front edge, the coordinate of the intersection point is the target vector of the optimal solution of the sub-optimization problem (5), and the included angle between the weight vector v and the target vector of the Pareto optimal solution of the corresponding sub-optimization problem is zero; the angle between the weight vector of the sub-optimization problem and the target vector of the solution is larger, and is farther from the optimal solution of the sub-optimization problem; if the line does not intersect the Pareto front, an algorithm for the solution life cycle is introduced:
for each solution x in the populationiWe remember the algebra generated from the individual to the current generation as the age of the individualiAnd calculating the included angle between the target vector and the corresponding weight vector, and recording as anglei. Selecting N solutions from a population as parents using tournament selection
Figure BDA0002701037190000045
In the selection of the championship, two individuals are randomly selected from the population, and the individuals with small ages are selected as father individuals to participate in hybridization variation to generate new individuals; when the ages of the two individuals are the same, selecting a solution with a larger angle value; for each individual in parent1
Figure BDA0002701037190000046
Individuals matched therewith for recombination
Figure BDA0002701037190000047
The selection of (A) is divided into the following two cases:
1) in the initial stage of the algorithm, when the size N of the population*When the ratio is less than or equal to N/2, introducing an external set A to improve the capability of the algorithm for keeping population diversity, and randomly selecting from the external set A
Figure BDA0002701037190000048
And
Figure BDA0002701037190000049
pairing and recombining to generate a new individual;
2) at the later stage of the algorithm, when the size N of the population*> N/2, recombination is performed based on two similar solutions,
Figure BDA00027010371900000410
is from
Figure BDA00027010371900000411
Randomly selecting pairs for recombination.
The central vector v is determined by different methods for the root node, the second layer node and other nodescThe specific contents are as follows:
the center vector of the root is set to
Figure BDA00027010371900000412
The center vector of the second layer node is
Figure BDA00027010371900000413
Extreme point in (1)
Figure BDA00027010371900000414
For other nodes, calculating a central vector by using an improved K-means clustering algorithm; each weight vector vjAssigning to the weighted nearest cluster;
Figure BDA0002701037190000051
λiis the percentage of the weight vector assigned to the ith cluster, initialized to 0.5,
Figure BDA0002701037190000052
the central vector of the ith cluster is the root node, the root node has M child nodes, and other nodes have at most 2 child nodes; the weight vector is assigned to the nearest center; i.e. for each weight vector
Figure BDA0002701037190000053
Calculate its nearest center:
Figure BDA0002701037190000054
the set of weight vectors that scores to assign to the ith cluster is Vi(ii) a A tree structure is obtained by recursively calling a tree construction algorithm flow chart.
The implementation of the invention has the following beneficial effects:
1. in the proposed algorithm, the candidate solution only needs to be compared with the solution on the path from the root of the tree to the leaf node. Therefore, the computation complexity of processing the candidate single solution is only o (mlogn), which has the advantage of low time complexity.
2. The invention provides a subproblem organization paradigm from coarse to fine. Thus, the algorithm herein approximates and progressively refines the Pareto front by solving a few representative sub-problems. The strategy has the advantage of high calculation efficiency for solving the large space optimization problem.
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In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the embodiments or the prior art will be briefly described below, it is obvious that the drawings in the following description are only some embodiments of the present invention, and for those skilled in the art, other drawings can be obtained according to the drawings without creative efforts.
FIG. 1 is a flow chart of the multi-objective optimization method based on hierarchical decomposition of tree structures.
FIG. 2 is a block diagram of a tree structure of the multi-objective optimization method based on hierarchical decomposition of a tree structure.
FIG. 3 is a flow chart of a set of uniformly distributed unit weight vectors of the multi-objective optimization method based on hierarchical decomposition of a tree structure.
Fig. 4 is a flow chart of a tree obtained by the tree construction method.
FIG. 5 is a region decomposition example diagram of the multi-objective optimization method based on hierarchical decomposition of tree structures according to the present invention.
Fig. 6 is a frontal interface surface obtained by the present invention when solving the 9 proposed large space optimization problems.
FIG. 7 is a front edge interface obtained by solving the ZDT series of test functions according to the present invention.
Detailed Description
The technical solutions in the embodiments of the present invention will be clearly and completely described below with reference to the drawings in the embodiments of the present invention, and it is obvious that the described embodiments are only a part of the embodiments of the present invention, and not all of the embodiments. All other embodiments, which can be derived by a person skilled in the art from the embodiments given herein without making any creative effort, shall fall within the protection scope of the present invention.
Examples
The embodiment relates to a multi-objective optimization method based on hierarchical decomposition of a tree structure, which comprises the following steps:
s1, initialization parameters:
generating a set of uniformly distributed unit weight vectors V ═ V1，v2，…，vn}；
S2, constructing a tree and initializing a population
According to the weight vector (v)cV) constructing a spanning tree T using an algorithm represented by the structure of the tree, each node of the tree containing a weight vector and being associated with the subproblems by means of its weight vector; the node of the first layer L2 is set as active, and the node without active descendant is recorded as a leaf node;
each subproblem of the active node selects the optimal individual from the initial solutions according to the aggregation function value defined in equation (5), these selected solutions forming an initial population T · X; population size N*| T · X | is the size of the solution, i.e. the number of active nodes;
uniformly and randomly generating N initial solutions in a decision space as an initial population P;
setting the number of active nodes L to 2, the improvement rate of minimum population convergence Delta to 0.1 and the central weight
Figure BDA0002701037190000061
Improved rate delta for initializing initial population convergence index 01, the current iteration time t is 1 and the external set A is P;
s3 evolutionary algorithm
Selecting a solution set Q through a mating selection strategy in each generation according to a mating selection algorithm, and creating N offspring Y according to Q;
current one population improvement rate Δt-1< Delta and N*When the ratio is less than N, the reaction mixture is,
expanding the number of layers of active nodes in the tree, wherein the optimal solution P is the union of the descendant Y and T.X, the number L of the active nodes is L +1, and initializing delta againtThen, the subproblem for each active node selects the optimal individual from the solution set P according to equation (5), forming a new T · X, population size N*＝|T·X|；
Otherwise, updating the solution corresponding to the node in the tree T and the external set A thereof according to the population evolution algorithm, and calculating the current improvement by defining the population convergence improvement rate formula in the formula (6)Rate deltat
The iteration number t is t +1
And (5) exiting the algorithm when the iteration times reach the maximum.
The above steps include two steps, as shown in fig. 2 to 4:
1. structure of tree
Constructing a tree from the weight vectors
V＝{v1，v2，…，vN} (1)
Is a set of uniformly distributed unit weight vectors, wherein
Figure BDA0002701037190000078
Is the ith weight vector; constructing a tree structure based on the set of weight vectors, wherein each node of the tree comprises a weight vector, and the weight vector is a central vector v of all weight vectors distributed to the nodecThe center vector may not be in the set of weight vectors V, we first find the nearest center vector V in VcWeight vector v of
Figure BDA0002701037190000071
The node contains a weight vector V and removes V from a set of weight vectors V; determining a center vector v using different methods for root nodes, level two nodes, and other nodescThe method comprises the following steps:
the center vector of the root is set to
Figure BDA0002701037190000072
The center vector of the second layer node is
Figure BDA0002701037190000073
Extreme point in (1)
Figure BDA0002701037190000074
For other nodes, calculating a central vector by using an improved K-means clustering algorithm; in particular, to balance each cluster, each weight vector v is appliedjAssigning to the weighted nearest cluster;
Figure BDA0002701037190000075
λiis the percentage of the weight vector assigned to the ith cluster, initialized to 0.5,
Figure BDA0002701037190000076
the central vector of the ith cluster is the root node, the root node has M child nodes, and other nodes have at most 2 child nodes; the weight vector is assigned to the nearest center; i.e. for each weight vector vjE.g. V, calculate its nearest center:
Figure BDA0002701037190000077
the set of weight vectors that scores to assign to the ith cluster is Vi(ii) a A tree structure is obtained by recursively calling a tree construction algorithm flow chart.
2. Population initialization and update based on tree structure
Weight vector v of each nodeiAll define the following sub-problems
g(x|vi)＝d1+θd2 (5)
Wherein d is1＝F(x)Tvi，d2＝||F(x)-d1viL. We can see that the weight vector of a child node is a refinement of the weight vector of its parent node. Thus, its corresponding child problem is a refinement of the child problem corresponding to its parent node. These subproblems are adaptively organized through a tree structure, where initially the node of the first L ═ 2 layers is set as the active node, and P ═ x is set as1，…，xNThe initial solution set is adopted, and then each subproblem of each active node is according to a formula(5) Selecting the optimal solution to initialize the population to form an initial population;
for the weight vector viX is the current best solution, yiFor the newly generated solution, P is the current node pointer; s is the weight vector v of formula (5) which is the same as F (x)iEquivalent points in direction, i.e.
s＝g(x|vi)vi (7)
As shown in fig. 5, the target space is divided into three sub-regions I, II and III by an equivalence point s;
subregion I: if g (y)iI P.v) < g (P.x I P.v) indicates yiIs superior to x, comparing P.x with yiPerforming exchange processing to update the equivalence point P.s-g (P.x | P.v) P.v;
subregion III: if P.s < F (y)i) If the solution ratio x in the area is different, the operation is not performed;
subregion II: if P is a leaf node, yiRandomly replacing one solution in archive A; otherwise, yiIs transmitted to the nearest child node
Figure BDA0002701037190000081
Wherein<Tj.v，F(yi)>Representing the angle, moving the node pointer P to the child node Tj*；
When the convergence rate of the population is lower than a given value, the population scale is expanded; the yield of improvement of the population is defined as the total improvement of the individuals in the population:
Figure BDA0002701037190000082
Figure BDA0002701037190000083
and
Figure BDA0002701037190000084
which are the solutions of the same subproblem in the t-1 and t generation processes, respectively. When the convergence rate is lower than a given value, increasing the number of layers of active nodes in the tree;
3. selection of individuals involved in hybrid variation
If straight line
Figure BDA0002701037190000085
fiThe variable is intersected with the Pareto front edge, the coordinate of the intersection point is the target vector of the optimal solution of the sub-optimization problem (5), and the included angle between the weight vector v and the target vector of the Pareto optimal solution of the corresponding sub-optimization problem is zero; the angle between the weight vector of the sub-optimization problem and the target vector of the solution is larger, and is farther from the optimal solution of the sub-optimization problem; therefore, we have reason to believe that this solution has a much greater capacity for improvement. However, this assumption will be invalid if the straight line does not intersect the Pareto front. We introduce a solution lifecycle to remedy this deficiency, i.e. if the straight line does not intersect the Pareto front, then a solution lifecycle algorithm is introduced. On the basis that the longer the life cycle of a solution is, the lower the probability of selecting the solution is, a selection strategy based on the matching of the life span and the hybridization variation individual of the included angle of the target vector and the corresponding weight vector is provided.
Specifically, for each solution x in the populationiWe remember the algebra generated from the individual to the current generation as the age of the individualiAnd calculating the included angle between the target vector and the corresponding weight vector, and recording as anglei. Selecting N solutions from a population as parents using tournament selection
Figure BDA0002701037190000091
In the selection of the championship, two individuals are randomly selected from the population, and the individuals with small ages are selected as father individuals to participate in hybridization variation to generate new individuals; when the ages of the two individuals are the same, selecting a solution with a larger angle value; to maintain diversity of individuals involved in recombination and to improve search efficiency, parent1 was used for each individual
Figure BDA0002701037190000092
Individuals matched therewith for recombination
Figure BDA0002701037190000093
The selection of (A) is divided into the following two cases:
1) in the initial stage of the algorithm, when the size N of the population*At ≦ N/2, the algorithm may suffer from loss of diversity and premature convergence if recombination is performed using only individuals in the population. Therefore, an external set A is introduced to improve the capability of the algorithm for keeping population diversity, and random selection is performed from the external set A
Figure BDA0002701037190000094
And
Figure BDA0002701037190000095
pairing and recombining to generate a new individual;
2) at the later stage of the algorithm, when the size N of the population*Recombination based on two similar solutions is advantageous for the search for better individuals > N/2. Therefore, the scheme provides a neighbor design based on a tree structure. For each active node TiIts neighbor is from TiTo TjAll active nodes T with side length less than given valuejIn the experiment of this embodiment, this value was set to 2. Neighborhoods of nodes of different levels have different ranges. A neighborhood of e.g. a root is distributed over the first octant because its neighborhood includes the extreme weight vector. While the neighbors of the leaf will focus on the smaller area. This may provide a balance between exploration and development.
Figure BDA0002701037190000096
Is from
Figure BDA0002701037190000097
Randomly selecting pairs for recombination.
Experimental data:
as shown in fig. 6 and 7, the proposed algorithm was compared with eight subsequent representative evolutionary multi-objective optimization algorithms:
1. NSGA-II: a fast and elitist multi-objective genetic algorithm NSGA-II (a fast and excellent multi-objective genetic algorithm: NSGA-II) is one of the most commonly used EMO algorithms to solve the multi-objective optimization problem. It uses Pareto-based dominance ordering as a primary selection mechanism and density-based selection criteria such as congestion distance as a secondary selection mechanism.
2. SMS-EMOA: SMS-EMOA Multi object selection based on dominated hypervolumes uses the individual's contribution in the hypervolume as a criterion for selecting offspring. Its purpose is to maximize the dominant hyper-volume of the resulting solution set in the optimization process. It preserves the convergence and diversity of the candidate solution set as much as possible.
3. IBEA: indicator-based selection in a multi-objective search (index-based selection in multi-objective search). IBEA uses indices to compare solutions and select next generation populations, which does not require additional diversity protection mechanisms. Binary index Is+The method is used for simulation experiments.
4. SRA 2: stochastic ranking algorithm for human-Objective optimization based on multiple index indicators (SRA) employs a Stochastic ranking technique to balance search bias of different indices. By storing well-converged solutions in conjunction with direction-based archiving, and maintaining diversity, the performance of the algorithm is further improved.
5. MOEA/D: a multi-objective evolution algorithm based on decomposition is taken as one of the most famous evolutionary multi-objective optimization decomposition algorithms, and the algorithm decomposes multi-objective optimization into a plurality of scalar subproblems through a group of weight vectors and optimizes the scalar subproblems at the same time. Since each sub-question is associated with a search direction, sufficient selection pressure can be provided for the Pareto front
6. M2M: the method decomposes a multi-objective optimization problem into a plurality of simple multi-objective optimization sub-problems and solves the sub-problems in a collaborative way. It prioritizes diversity. Therefore, the algorithm can well keep the population diversity and obtain a better effect on solving the imbalance optimization problem.
7. RVEA: reference vector-oriented evolution for human-object optimization (a multi-objective optimization evolution algorithm based on reference vectors) the method utilizes a set of reference vectors to decompose a multi-objective optimization problem into a set of single-objective subproblems. The angle penalty distance dynamically balances convergence and diversity according to the number of targets and the algebra.
8. Two _ Arch2: two _ Arch2: An improved Two-archive algorithm for management-objective optimization An improved Two-archive evolution multi-objective optimization algorithm Two _ Arch2 is proposed herein. In Two _ Arch2, Two profiles (convergence profile and diversity profile) are maintained during the evolutionary search and Two different selection principles (index-based and Pareto-based) are assigned to the Two profiles, which may balance convergence, diversity and complexity.
Table 1 below: the scheme provides average IGD values obtained by running 30 times of the REMOA and 8 most advanced algorithms on a ZDT series test function and an MF1-MF9 test function of a test problem. According to the rank sum test result, the algorithm provided by the scheme has good performance.
Figure BDA0002701037190000113
The significance of the proposed algorithm is superior to that of a comparison algorithm.
Figure BDA0002701037190000111
Table 2 below: the scheme provides the test functions of the ZDT series and the MF1 of the test problems of REMOA and 8 most advanced algorithmsMF9 test function runs 30 times resulting in an average HV value. According to the rank sum test result, the algorithm provided by the scheme has good performance.
Figure BDA0002701037190000114
The significance of the proposed algorithm is superior to that of a comparison algorithm.
Figure BDA0002701037190000112
While the invention has been described in connection with what is presently considered to be the most practical and preferred embodiment, it is to be understood that the invention is not to be limited to the disclosed embodiment, but on the contrary, is intended to cover various modifications and equivalent arrangements included within the spirit and scope of the appended claims.

Claims (4)

1. A multi-objective optimization method based on hierarchical decomposition of a tree structure is characterized by comprising the following steps:
s1 initialization parameters
Generating a set of uniformly distributed unit weight vectors V ═ V1，v2，…，vn}；
S2, constructing a tree and initializing a population
According to the weight vector (v)cV) constructing a spanning tree T using an algorithm represented by the structure of the tree, each node of the tree containing a weight vector and being associated with the subproblems by means of its weight vector; the node of the first layer L2 is set as active, and the node without active descendant is recorded as a leaf node;
selecting an optimal individual from initial solutions by each subproblem of the active node, wherein the selected solutions form an initial overall T.X; population size N*| T · X | is the size of the solution, i.e. the number of active nodes;
uniformly and randomly generating N initial solutions in a decision space as an initial population P;
setting the number of active nodes L to 2, the improvement rate of minimum population convergence Delta to 0.1 and the central weight
Figure FDA0002701037180000011
Improved rate delta for initializing initial population convergence index01, the current iteration time t is 1 and the external set A is P;
s3 evolutionary algorithm
In each generation, selecting a solution set Q through a mating selection strategy, and creating N offspring Y according to Q;
current one population improvement rate Δt-1< Delta and N*When the ratio is less than N, the reaction mixture is,
expanding the number of layers of active nodes in the tree, wherein the optimal solution P is the union of the descendant Y and T.X, the number L of the active nodes is L +1, and initializing delta againtThen, each active node sub-problem selects the best individual from solution set P, forming a new T · X, population size N*＝|T·X|；
Otherwise, updating the solution corresponding to the node in the tree T and the external set A thereof according to the population evolution algorithm, and calculating the current improvement rate deltat
The iteration number t is t +1
And (5) exiting the algorithm when the iteration times reach the maximum.
2. The multi-objective optimization method based on hierarchical decomposition of tree structure as claimed in claim 1, wherein the specific content of step S2 is:
constructing a tree from the weight vectors
V＝{v1，v2，…，vN} (1)
Is a set of uniformly distributed unit weight vectors, wherein
Figure FDA0002701037180000021
Is the ith weight vector; constructing a tree structure based on the set of weight vectors, wherein each node of the tree comprises a weight vector, and the weight vector is a central vector v of all weight vectors distributed to the nodecThe center vector may not be in the set of weight vectors V, we first find VTo the nearest central vector vcWeight vector v of
Figure FDA0002701037180000022
The node contains a weight vector V and removes V from a set of weight vectors V; determining a center vector v using different methods for root nodes, level two nodes, and other nodesc；
Weight vector v of each nodeiAll define the following sub-problems
g(x|vi)＝d1+θd2 (5)
Wherein d is1＝F(x)Tvi，d2＝||F(x)-d1viL. These subproblems are adaptively organized through a tree structure, where initially the node of the first L ═ 2 layers is set as the active node, and P ═ x is set as1，…，xNThe method comprises the steps that (1) an initial solution set is adopted, then each subproblem of each active node selects an optimal solution according to the value of a formula (5) to initialize a population, and an initial population is formed;
for the weight vector viX is the current best solution, yiFor the newly generated solution, P is the current node pointer; s is the weight vector v of formula (5) which is the same as F (x)iEquivalent points in direction, i.e.
s＝g(x|vi)vi (7)
Dividing the target space into three sub-regions I, II and III by an equivalence point s;
subregion I: if g (y)iI P.v) < g (P.x I P.v) indicates yiIs superior to x, comparing P.x with yiPerforming exchange processing to update the equivalence point P.s-g (P.x | P.v) P.v;
subregion III: if it is not
Figure FDA0002701037180000027
The solution ratio x in the area is poor, and no operation is performed；
Subregion II: if P is a leaf node, yiRandomly replacing one solution in archive A; otherwise, yiIs transmitted to the nearest child node
Figure FDA0002701037180000023
Wherein<Tj.v，F(yi)>Representing the angle, moving the node pointer P to the child node Tj*；
When the convergence rate of the population is lower than a given value, the population scale is expanded; the yield of improvement of the population is defined as the total improvement of the individuals in the population:
Figure FDA0002701037180000024
Figure FDA0002701037180000025
and
Figure FDA0002701037180000026
which are the solutions of the same subproblem in the t-1 and t generation processes, respectively. When the convergence rate is lower than a given value, increasing the number of layers of active nodes in the tree; calculating the current improvement rate Delta by defining the population convergence improvement rate formula in the formula (6)t。
3. The multi-objective optimization method based on hierarchical decomposition of tree structure as claimed in claim 2, wherein the specific content of step S3 is:
if straight line
Figure FDA0002701037180000031
fiIs a variable which is intersected with the Pareto front edge, the coordinate of the intersection point is a target vector of the optimal solution of the sub-optimization problem (5), and the weight vector v and the corresponding sub-optimization problem thereofThe included angle between the target vectors of the Pareto optimal solution is zero; the angle between the weight vector of the sub-optimization problem and the target vector of the solution is larger, and is farther from the optimal solution of the sub-optimization problem; if the line does not intersect the Pareto front, an algorithm for the solution life cycle is introduced:
for each solution x in the populationiWe remember the algebra generated from the individual to the current generation as the age of the individualiAnd calculating the included angle between the target vector and the corresponding weight vector, and recording as anglei. Selecting N solutions from a population as parents using tournament selection
Figure FDA0002701037180000032
In the selection of the championship, two individuals are randomly selected from the population, and the individuals with small ages are selected as father individuals to participate in hybridization variation to generate new individuals; when the ages of the two individuals are the same, selecting a solution with a larger angle value; for each individual in parent1
Figure FDA0002701037180000033
Individuals matched therewith for recombination
Figure FDA0002701037180000034
The selection of (A) is divided into the following two cases:
1) in the initial stage of the algorithm, when the size N of the population*When the ratio is less than or equal to N/2, introducing an external set A to improve the capability of the algorithm for keeping population diversity, and randomly selecting from the external set A
Figure FDA0002701037180000035
And
Figure FDA0002701037180000036
pairing and recombining to generate a new individual;
2) at the later stage of the algorithm, when the size N of the population*> N/2, recombination is performed based on two similar solutions,
Figure FDA0002701037180000037
is from
Figure FDA0002701037180000038
Randomly selecting pairs for recombination.
4. The multi-objective optimization method based on hierarchical decomposition of tree structure as claimed in claim 1, wherein the central vector v is determined by different methods for the root node, the second level node and other nodescThe specific contents are as follows:
the center vector of the root is set to
Figure FDA0002701037180000039
The center vector of the second layer node is
Figure FDA00027010371800000310
Extreme point in (1)
Figure FDA00027010371800000311
For other nodes, calculating a central vector by using an improved K-means clustering algorithm; each weight vector vjAssigning to the weighted nearest cluster;
Figure FDA00027010371800000312
λiis the percentage of the weight vector assigned to the ith cluster, initialized to 0.5,
Figure FDA0002701037180000041
the central vector of the ith cluster is the root node, the root node has M child nodes, and other nodes have at most 2 child nodes; the weight vector is assigned to the nearest center; i.e. for each weight vectorvjE.g. V, calculate its nearest center:
Figure FDA0002701037180000042
the set of weight vectors that scores to assign to the ith cluster is Vi(ii) a A tree structure is obtained by recursively calling a tree construction algorithm flow chart.
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