CN106997486A - A kind of higher-dimension multi-objective optimization algorithm split based on space - Google Patents

Abstract

The present invention provides a kind of higher-dimension multi-objective optimization algorithm split based on space.Higher-dimension multi-objective optimization question is mapped as the local two dimensional optimization problem of vector correlation using the reference vector for dividing equally distribution by the algorithm, and maintain a sub- population for each reference vector, then space segmentation is carried out centered on reference vector, by index of angle, whole solution space is divided into some sub-spaces.When carrying out environmental selection, to subspace from closely to far selecting successively during the best individual of convergence index is put into follow-on sub- population, guiding population more rapidly, more uniformly to Pareto leading surface approaches.The method of the present invention is not limited by the dimension of multi-objective optimization question, can effectively handle the insoluble challenge of traditional multi-objective optimization algorithm.

Description

A kind of higher-dimension multi-objective optimization algorithm split based on space

Technical field

The present invention relates to intelligence computation field, more particularly, to a kind of higher-dimension multiple-objection optimization split based on space Algorithm.

Background technology

Many problems of real world are generally made up of multiple targets, and solution multi-objective problem is generally highly difficult, because mesh Often collided with each other between mark.The lifting of one target capabilities frequently can lead to the decline of another target capabilities, make institute Have target at the same be all optimal it is often impossible, therefore, for multi-objective problem, it is necessary to which what is found is one group equal Weighing apparatus solution, that is, Pareto optimal solution, make each target be optimal as far as possible, this process is multiple-objection optimization.In recent years Come, multi-objective Evolutionary Algorithm has been achieved for development at full speed, has extremely be widely applied in many aspects.But calculated Most of method is all processing low-dimensional target problem, such as 2 dimensions or 3-dimensional, and it is partially the higher-dimension problem for handling 4 to 9 dimensions, place also to have Manage considerably less more than more than 10 dimensions.However as the increase of target dimension, the effect of optimization of traditional multi-objective Evolutionary Algorithm Substantially reduce.In practical application, the target number of many problems is more than 3, i.e. higher-dimension multi-objective optimization question.Therefore, higher-dimension is more Objective optimisation problems are one of the Research Challenges in current Evolutionary multiobjective optimization field.

Currently proposed higher-dimension multi-objective Evolutionary Algorithm is broadly divided into following a few classes：

1) still using the sort method dominated based on Pareto.But algorithm combines other technologies to reduce search space Or reduction target dimension combines preference information to reduce Pareto front porch area, using reduction target dimension such as in search procedure Degree simplifies problem etc.；

2) sort method dominated using loose Pareto.This kind of method, can be right by relaxing Pareto dominance relation Many non-dominant individuals are compared and selected, such as winning relation, and k- is optimal, the concept such as α-domination；

3) using the sort method of non-Pareto.This kind of method using new interpretational criteria population at individual is compared with Sequence, such as based on finger calibration method, the method based on decomposition.

The content of the invention

The present invention provides a kind of higher-dimension multi-objective optimization algorithm split based on space, and the algorithm can quickly promote population Leading surface is approached, and good diversity can be obtained.

In order to reach above-mentioned technique effect, technical scheme is as follows：

A kind of higher-dimension multi-objective optimization algorithm split based on space, is comprised the following steps：

S1：Reference vector number N, subspace scale n, maximum assessment number of times MFE are set, and maintains N number of sub- population P= {P₁,P₂,...,P_N}；

S2：Generate equally distributed reference vector set V={ V₁,V₂,...,V_N, random initializtion population Q；

S3：To each reference vector V_i(i=1,2 ..., N), with V_iCentered on, by index of angle by whole space It is divided into n sub-spaces

S4：To each sub- population P_i(i=1,2 ..., N), sets n set D={ D₁,D₂,...,D_nAnd subspaceCorrespond, make P_i=P_i∪ Q, if individualX is then added into D_j(j=1,2 ..., n). Then D is made_jIn individual sorted from small to large by convergence index, retain the best individual of convergence, that is, obtain follow-on Sub- population P_i；

S5：If function evaluation number of times is more than MFE, merge sub- population Q=P₁∪P₂∪...∪P_N, select N number of body conduct Final population output；Otherwise S6 is gone to；

S6：Select several individuals to carry out cross and variation operation as father and mother in P and produce new progeny population Q, return S3。

Wherein, higher-dimension multi-objective optimization question can be expressed as：

Min F (x)=(f₁(x),f₂(x),…f_M(x))

s.t.x∈Ω

Further, the calculating process of the hollow segmentation of step S3 is as follows：

Further, the calculation formula of convergence index is as follows in the step S4：

Further, the calculating process of genetic operator is as follows in the step S6：

Wherein, η_cFor cross parameter；

The calculating process of mutation operator is as follows：

Wherein, η_mFor Mutation parameter, u is the random number between [0,1].

Compared with prior art, the beneficial effect of technical solution of the present invention is：

The essential requirement that the present invention is solved for higher-dimension multi-objective optimization question, i.e. convergence and distributivity, using uniform The local two dimensional optimization problem that higher-dimension multi-objective optimization question is converted into being more easily handled by the reference vector of distribution, then with angle Solution space is divided into some subspaces for index, simple target two-dimensional optimization being converted into subspace carries out genetic algorithm Iteration, guiding population more rapidly, more uniformly to Pareto leading surface approach.The method of the present invention is not asked by multiple-objection optimization The dimension limitation of topic, can effectively handle the insoluble challenge of traditional multi-objective optimization algorithm.

Brief description of the drawings

Fig. 1 is the inventive method flow chart.

Embodiment

Accompanying drawing being given for example only property explanation, it is impossible to be interpreted as the limitation to this patent；

To those skilled in the art, it is to be appreciated that some known features and its explanation, which may be omitted, in accompanying drawing 's.

Technical scheme is described further with reference to the accompanying drawings and examples.

Embodiment 1

As shown in figure 1, a kind of higher-dimension multiple-objection optimization evolution algorithm split based on space, is comprised the following steps：

S1：Reference vector number N, subspace scale n, maximum assessment number of times MFE are set, and maintains N number of sub- population P= {P₁,P₂,...,P_N}；

S2：Generate equally distributed reference vector set V={ V₁,V₂,...,V_N, random initializtion population Q；

S3：To each reference vector V_i(i=1,2 ..., N), with V_iCentered on, by index of angle by whole space It is divided into n sub-spaces

S4：To each sub- population P_i(i=1,2 ..., N), sets n set D={ D₁,D₂,...,D_nAnd subspaceCorrespond, make P_i=P_i∪ Q, if individualX is then added into D_j(j=1,2 ..., n). Then D is made_jIn individual sorted from small to large by convergence index, retain the best individual of convergence, that is, obtain follow-on Sub- population P_i；

S5：If function evaluation number of times is more than MFE, merge sub- population Q=P₁∪P₂∪...∪P_N, select N number of body conduct Final population output；Otherwise S6 is gone to；

S6：Select several individuals to carry out cross and variation operation as father and mother in P and produce new progeny population Q, return Step 3.

In the present embodiment, choose four and classical illustrate the present invention's without constraining non-linear higher-dimension multiple-objection optimization example Implementation steps, example description is as shown in table 1：

The test function information table of the embodiment of table 1

Specific solution procedure is as follows：

Step 1：Target dimension is set to M=3, and 5,8,10, corresponding reference vector number is N=91,210,156,275, Interval scale is set to n=30, and maximum assesses number of times and is set to MFE=250*M*N, and cross parameter is set to η_c=30, variation Parameter setting η_m=20；

Step 2：Generate equally distributed reference vector set V={ V₁,V₂,...,V_N, random initializtion population Q；

Step 3：To each reference vector V_i(i=1,2 ..., N), with V_iCentered on, will be whole empty by index of angle Between be divided into n sub-spacesSpace segmentation formula is as follows：”

Step 4：To each sub- population P_i(i=1,2 ..., N), sets n set D={ D₁,D₂,...,D_nAnd son SpaceCorrespond, make P_i=P_i∪ Q, if individualX is then added into D_j(j=1,2 ..., n).Then D is made_jIn individual sorted from small to large by convergence index, retain the best individual of convergence, that is, obtain the next generation Sub- population P_i, convergence index is calculated as follows：

Step 5：If function evaluation number of times is more than MFE, merge sub- population Q=P₁∪P₂∪...∪P_N, select individual Exported as final population；Otherwise step 6 is gone to；

Step 6：Select several individuals to carry out cross and variation operation as father and mother in P and produce new progeny population Q, return Step 3 is returned, crossover operator and mutation operator are as follows：

Crossover operator is calculated as follows：

Mutation operator is calculated as follows：

In the present embodiment, in order to illustrate advantage of the present invention to classical multi-objective optimization algorithm, classic algorithm is chosen MOEA/D is as comparison algorithm, and every kind of algorithm reruns 30 times to each example, calculates the HV indexs of each run result, takes it Median is as evaluation criterion, and optimum results are as shown in table 2：

The optimum results of table 2

It is can be seen that by the test result of the present embodiment relative to classical multi-objective optimization algorithm, the present invention is for height The solution of dimension multi-objective optimization question has significant advantage, it was demonstrated that the feasibility and superiority of the present invention.

The same or analogous part of same or analogous label correspondence；

Diagram the being given for example only property explanation of position relationship described in accompanying drawing, it is impossible to be interpreted as the limitation to this patent；

Obviously, the above embodiment of the present invention is only intended to clearly illustrate example of the present invention, and is not pair The restriction of embodiments of the present invention.For those of ordinary skill in the field, may be used also on the basis of the above description To make other changes in different forms.There is no necessity and possibility to exhaust all the enbodiments.It is all this Any modifications, equivalent substitutions and improvements made within the spirit and principle of invention etc., should be included in the claims in the present invention Protection domain within.

Claims (4)

1. a kind of higher-dimension multi-objective optimization algorithm split based on space, it is characterised in that comprise the following steps：

S1：Reference vector number N, subspace scale n, maximum assessment number of times MFE are set, and maintains N number of sub- population P={ P₁, P₂,...,P_N}；

S2：Generate equally distributed reference vector set V={ V₁,V₂,...,V_N, random initializtion population Q；

S3：To each reference vector V_i(i=1,2 ..., N), with V_iCentered on, by index of angle whole space split Into n sub-spaces

S4：To each sub- population P_i(i=1,2 ..., N), sets n set D={ D₁,D₂,...,D_nAnd subspaceCorrespond, make P_i=P_i∪ Q, if individualX is then added into D_j(j=1,2 ..., n). Then D is made_jIn individual sorted from small to large by convergence index, retain the best individual of convergence, that is, obtain follow-on Sub- population P_i；

S5：If function evaluation number of times is more than MFE, merge sub- population Q=P₁∪P₂∪...∪P_N, individual is selected as most Whole population output；Otherwise S6 is gone to；

S6：Select several individuals to carry out cross and variation operation as father and mother in P and produce new progeny population Q, return to S3.

2. the higher-dimension multi-objective optimization algorithm according to claim 1 split based on space, it is characterised in that the step The calculating process of hollow segmentation of S3 is as follows：

$g_{angle}(x,V_i)=1-\frac{f\·V_i}{\left|f\right|\·\left|V_i\right|}$

$S_j^i=\left\{\begin{array}{c}\left\{x\right|0\&le;g_{angle}(x,V_i)<\frac{1}{2^n}\},j=1\\ \left\{x\right|\underset{k=n-j+2}{\overset{n}{\mathrm{\&Sigma;}}}\frac{1}{2^k}\&le;g_{angle}(x,V_i)<\underset{k=n-j+1}{\overset{n}{\mathrm{\&Sigma;}}}\frac{1}{2^k}\},1<j<n\\ \left\{x\right|\underset{k=2}{\overset{n}{\mathrm{\&Sigma;}}}\frac{1}{2^k}\&le;g_{angle}(x,V_i)\&le;1\},j=n\end{array}\right..$

3. the higher-dimension multi-objective optimization algorithm according to claim 2 split based on space, it is characterised in that the step Convergence index is calculated as follows in S4：

$h\left(x\right)=\underset{i=1}{\overset{M}{\&Sigma;}}{f}_i\left(x\right).$

4. the higher-dimension multi-objective optimization algorithm according to claim 3 split based on space, it is characterised in that the step The calculating process of genetic operator is as follows in S6：

$X_1^{new}=\frac{x_1+x_2}{2}+\mathrm{\&beta;}\&CenterDot;\frac{x_1-x_2}{2}$

$X_2^{new}=\frac{x_1+x_2}{2}-\mathrm{\&beta;}\&CenterDot;\frac{x_1-x_2}{2}$

$\mathrm{\&beta;}=\left\{\begin{array}{cc}{\left(2u\right)}^{\frac{1}{{\mathrm{\&eta;}}_c+1}}& 0\&le;u\&le;0.5\\ {\left(\frac{1}{2(1-u)}\right)}^{\frac{1}{{\mathrm{\&eta;}}_c+1}}& 0.5<u\&le;1\end{array}\right.$

Wherein, η_cFor cross parameter；

The calculating process of mutation operator is as follows：

$X_{new}=\left\{\begin{array}{cc}x+\mathrm{\&beta;}(x-x_{min})& 0\&le;u\&le;0.5\\ x+\mathrm{\&beta;}(x_{max}-x)& 0.5\&le;u\&le;1\end{array}\right.$

$\mathrm{\&beta;}=\left\{\begin{array}{cc}{\left(2u\right)}^{\frac{1}{{\mathrm{\&eta;}}_m+1}}-1& 0\&le;u\&le;0.5\\ 1-{\left(2\right(1-u\left)\right)}^{\frac{1}{{\mathrm{\&eta;}}_m+1}}& 0.5<u\&le;1\end{array}\right.$

Wherein, η_mFor Mutation parameter, u is the random number between [0,1].