Adaptive differential evolution algorithm

Adaptive di?erential evolution algorithm for multiobjective optimization problems

Weiyi Qian *,Ajun li

Department of Mathematics,Bohai University,Jinzhou,Liaoning 121000,PR China

Abstract

In this paper,a new adaptive di?erential evolution algorithm (ADEA)is proposed for multiobjective optimization problems.In ADEA,the variable parameter F based on the number of the current Pareto-front and the diversity of the current solutions is given for adjusting search size in every generation to ?nd Pareto solutions in mutation operator,and the select operator combines the advantages of DE with the mechanisms of Pareto-based ranking and crowding dis-tance sorting.ADEA is implemented on ?ve classical multiobjective problems,the results illustrate that ADEA e?ciently achieves two goals of multiobjective optimization problems:?nd the solutions converge to the true Pareto-front and uni-form spread along the front.

ó2008Elsevier Inc.All rights reserved.

Keywords:Multiobjective optimization problems;Di?erential evolution algorithm;Adaptive;Select operator

1.Introduction

Multiobjective optimization problems (MOPs)which consist of several competing and incommensurable objective functions are frequently encountered in real-world problem such as scienti?c and engineering appli-cations.Consequently,there are two goals in multiobjective optimization:(i)to discover solutions as close to the Pareto-front as possible,(ii)to ?nd solutions as diverse as possible in the obtained nondominated front.In recent years,many optimization techniques have been proposed in some literatures to solve MOPs.Some of the most attractive algorithms are evolution algorithms (EAs)such as NSGAII [1],MODE [2],PAES [3].In contrast to traditional gradient-based techniques,EAs use a set of potential solutions to detect feasible region.So several solutions of a multiobjective problem can be obtained in a single run.The properties enable EAs converge fast to the true Pareto-front (the concept will be explained in the following section).In MOPs,a large number of optimal solutions exist,and each correspond to a di?erent trade-o?among the objective functions.Di?erential evolution algorithm (DE)is designed for minimizing functions of real variable.It is extremely robust in locating the global minimum.DE is a simple yet powerful evolutionary optimization algorithm that

0096-3003/$-see front matter ó2008Elsevier Inc.All rights reserved.doi:10.1016/j.amc.2007.12.052

*

Corresponding author.

E-mail address:qianweiyi@https://www.360docs.net/doc/b82941930.html, (W.Qian).

Available online at https://www.360docs.net/doc/b82941930.html,

Applied Mathematics and Computation 201(2008)

431–440

432W.Qian,A.li/Applied Mathematics and Computation201(2008)431–440

has been successfully used in solving single-objective problems by Price and Storn[4].After that,it was used to handle MOPs.Abbass was the?rst to apply DE to MOPs in the so-called Pareto di?erential evolution(PDE) algorithm[5,6].Madavan achieved good results with the Pareto di?erential evolution approach(PDEA)[7]. Xue introduced multiobjective di?erential evolution[2].Tea Robi^c proposed di?erential evolution for multi-objective optimization(DEMO)[8].

In this paper,we propose a new way of extending DE to be suitable for solving MOPs.A novel approach, called adaptive di?erential evolution algorithm(ADEA)which incorporated a new select operator and an adaptive parameter,is introduced to search the global solutions.In each generation,the select operator emphasize Elitist to promote the research towards the Pareto-front.And the adaptive parameter F was used to adjust step size for the need of algorithm.From the simulate results on?ve test problems,we?nd the speed of converging to the true Pareto-front of ADEA is more fast and the diversity is better than most of other optimization algorithms for multiobjective problems.

The rest of the paper is organized as follows:Section2provides the concept of Pareto-front and Pareto-ranking,DE scheme which was used as a background for ADEA.In Section3,a method called ADEA is dis-cussed in detail.Section4outlines the applied test problems and performance measures.Experimental results showing the e?ectiveness of our approach and the further comparison and discussion of the results are also provided in this section.Section5concludes the paper.

2.Background

2.1.Pareto-ranking

Two or more objectives in a multiobjective problem are usually con?ict and compete,they cannot be min-imized concurrently.So one solution often cannot be said to be better or worse than another according their function values.There exists not a single optimal solution,but a set optimal solutions-called Pareto-front.

Starting from a set of initial solutions,multiobjective evolutionary algorithms use an iteratively improving optimization techniques to?nd the optimal solutions.In every iterative progress,EAs favor population-based Pareto dominance as a measure of?tness.That makes for exploring the potential promising areas of search space and obtain more the?nal approximation of the Pareto-optimal front.The Pareto-optimal front is de?ned as follows:

For any two decision vectors a and b,

a1bea dominates bTif feaT6febTK feaT?febT;

a#bea weakly dominates bTif feaT6febT;

a$bea is incomparable to bTif feaTi febTK febTi feaT;

where relations 2f?;<;6;i g are for every j2f1;...;n g if f jeaT f jebT.

A decision vector v is called Pareto-optimal solution when it is not dominate by any other decision vector ~v2X in all objectives.The set of all Pareto optimal solutions is called the Pareto-optimal front.We rest Par-eto-ranking–a concept that has been prevalent since Goldberg’s early work[2]and features in a host of tech-niques to?nd Pareto-optimal front.The computation of Pareto-ranking is very complex before Deb et al.[8] proposed a fast ranking approach which requires at most oeMN2Tcomputations in NSGAII.This approach can sort a population of size N according to the level of non-domination fast.

2.2.Di?erential evolution

DE is an e?cient evolutionary optimization algorithm.It has been successfully applied on a plethora of applications.DE is a population set of solution vectors which are successively updated by addition,subtrac-tion and component swapping,until the populations converge,hopefully to optimum.An initial set S consists of N points with corresponding function values is generated randomly in X.Mutation and crossover are used to generate new points according S.A selection progress is then used to drive these points to the vicinity of the global minimizer.That is,the?nal minimizers will be obtained through irritating the initial populations.

Now we will describe the progress of DE in detail.

N points are sampled in X as initial set S?f x1;x2;...;x N g.Take N)n,n being the dimension of the func-tion f(x).In each iteration,N competitions between target points and trial points which are generated through mutation and crossover operators are held to determine the members of S for the next generation.In mutation phrase,DE randomly selects three distinct points x(1),x(2),x(3)from the current set S.The weighted di?erent of any points is then added to the third points which can be mathematically described as: ^x i?x pe1TtFex pe2Tàx pe3TT;e1T

where F61is a selecting factor.Then a trial point y

i as a competition to target point x i will be found from its

parents x i and^x i using the following crossover rules:

y j i?

^x j

i

if R j6C R or j?I i;

x j i if R j>C R and j?I i;

(

e2T

where I i is a randomly chosen integer in the set I,i.e.I i?f1;2;...;n g;the superscript j represents the j th com-ponent of respective vectors:R j2e0;1T;draw randomly for each j.The entity C R is a constant.Notice that for C R?1,the trial vector y i is equal to the mutated vector,that is,only the mutation operation is used to repro-

duction.The goal of the crossover operation is to get the trial vector y

i .The selection mechanism decides

which point(x i and y

i )is accept.All the solutions in the population have the same chance of being selected

as parents without dependence on their?tness value.The child produced after the mutation and crossover operations is evaluated.Then,the performance of the child vector and its parents is compared and the better one is selected.If the parent is still better,it is retained in the population.

The di?erential evolution algorithm(DE):

Step1.Determine the initial set S?f x1;x2;...;x N g

where the points x i;i?1;...;N are sampled randomly in X.Evaluate f(x)at each x i2S.Take N)n, n being the dimension of the f(x).Set generation counter k=0.

Step2.While stopping criterion not meet,do

Step2.1.Generate points to replace points in S.For each x i2S,determine y i by the following two reproduction operations.

Mutation:Randomly select three points from S except x i,?nd the second parent^x i by the muta-

tion rule(1).

Crossover:Calculate the trial vector y i corresponding to x i from x i and^x i using the crossover rule

(2).

Step2.2.Replace points in S.Evaluate the trial vector y

i ,if y

i

is better than its parent x i;y

i

replaces

x i,otherwise y

i

is discarded.Set k=k+1.Go to step2.

3.Adaptive di?erential evolution algorithm

When applying DE to MOPs,we face many di?culties.How to replace the parent with trial solutions and preserve a uniform spread of nondominated solutions are two main challenge for DE algorithm.The selection operator of original DE is simple but it is not?t for the MOPs.The selection is easy in single-objective opti-mization,but the selection is not so straight forward in MOPs.It exists that the trial solutions and the parent solutions are incomparable.According to the original DE,if the trial solution is not dominance the parent,the trial is discarded.This method lead to miss the optimal solutions that have been found.Meanwhile,the sen-sitivity of the parameter F was not study,authors just chose their values randomly and mainly between0.5and 0.9.The e?uence of F to the new points was ignored.

To overcome the above mentioned drawbacks,we proposed a adaptive parameter F and a new selection method.We do not compare trial solution with its parent when a trial point is generated,but keep it in the trial set.When all the members in the population set generate trial solutions,we get a new population size of two popsize with trial set and parent set.Then truncate it to the size of popsize for the next step of the algorithm.The truncation consists of sorting the individuals with nondominated sorting and evaluating the W.Qian,A.li/Applied Mathematics and Computation201(2008)431–440433

individuals of the same front with the crowding distance metric.The truncation procedure continue until only the best popsize individuals are get.The truncation is similar to what in NSGAII,but ADEA introduces a di?erent crowding distance metric.It is outlined in Fig.1.We use diagonal of the largest cuboid which enclos-ing the point x i without including any other point in the population as density estimation of the population.Fig.1outlines the crowing distance computation procedure of the solutions in an nondominated set I.In DE,the generation of new points is mainly dependent on mutation operator.In mutation,parameter F plays an important role.F has in?uence on the speed of converge and the diversity of the solutions.F decides the search https://www.360docs.net/doc/b82941930.html,ually,optimization algorithms favor global search at the early stage for exploring feasible domain and ?nding the global optimization solutions and local search at the latter stage for accelerating con-verge.Based on above strategy,we de?nes a parameter F as follows:

F ?max P k j ?1

P m j i ?1j d ij à d j j td f P j Q j á d td f ;1à2j P j j Q j ;l min !

where d ij is the crowding distance of the i th solution in the j th Pareto level; d

j is the average value of crowd-ing distance of the solutions in the j th Pareto level; d

is the average value of crowding distance of the solu-tions in every iteration;j P j is the number of the nondominated solutions;j Q j is the number of the population set;the parameter d f represent the Euclidean distance between two boundary solutions in Q ;l min is a lower bound for F .

ADEA is implemented as follows:

Initially,a random parent population S is created.The truncation procedure is introduced for calculating F in the ?rst iteration and all the solutions in S are retained.In the other iteration,truncation procedure used to select the parent set (for next generation)and calculating F .F is decided by the number of the current Pareto-front and the diversity of the current solutions.It can adjust its size with the need of the algorithm.The gen-eration of new points is as the same as DE (according mutation rule and crossover rule).Then a combined population S ?S [Q (including new points)is formed.The population S will be of size 2N .Since all previous and current population members are included in S ,the elitism is insured.The ADEA procedure is shown in Fig.2.4.Experimental results 4.1.Performance measures

To validate our approach,we used the methodology normally adopted in the evolutionary multiobjective optimization literature.Because we wanted to compare ADEA to other MOEAs on their published results,we use three metrics that have been used in these studies.They represent both quantitative comparisons and qual-

Fig.1.Crowding distance metric in ADEA.

434W.Qian,A.li /Applied Mathematics and Computation 201(2008)431–440

itative comparisons with MOEAs that are respective of the state-of-the-art:the Nondominated Sorting Genetic Algorithm II (NSGAII)the Strength Pareto Evolutionary Algorithm (SPEA),the Pareto Archived Evolution Strategy (PAES),the Pareto-frontier Di?erential Evolution Approach (PDEA),the Multiobjective Optimization Evolution Algorithm (MOEA).(1)Convergence metric ?.

This metric is de?ned as:

??P n

i ?1d i

n

;

where n is the number of nondominated vector found by the algorithm being analyzed and d i is the Euclidean distance (measured in the objective space)between the obtained nondominated front Q and the nearest member in the true Pareto front P .It measures the distance between the Q and P .

(2)Generational distance (GD).

This metric is similar to ?.It is de?ned as:

GD ???????????????P n

i ?1d i

p n ;

where d i is the Euclidean distance (measured in the objective space)between the obtained nondominated front Q and the nearest member in the true Pareto-front P .It should be clear that a value of GD =0indicates that all the elements generated are in the true Pareto-front of the problem.(3)Diversity metric D .

This metric measures the distant of spread achieved among the nondominated solutions.It is de?ned as:

D ?d f td l tP n

i ?1ed i à d

Td f td l ten à1T d

where d i is the Euclidean distance (measured in the objective space)between consecutive solutions in the

obtained nondominated front Q and d

is the average of these distances.The parameters d f and d l rep-resent the Euclidean distance between the extreme solutions of the true Pareto-front P and the boundary solutions of the obtained front Q .

4.2.Test problems and results

The test problems for evaluating the performance of our methods are chosen based on signi?cant past stud-ies in multiobjective evolutionary algorithms.We chose ?ve problems from benchmark domains commonly used in past multiobjective GA research (in the literature [1,7,2,10]).

For every test problem,number of population points (NP)=100,number of iterations (NG)=200:

Fig.2.Outline of ADEA.

W.Qian,A.li /Applied Mathematics and Computation 201(2008)431–440435

Test problem1(ZDT1):Convex Pareto-front:

f1exT?x1;

f2exT?gexTe1à

???????????????x1=gexTp

T;

gexT?1t

9

nà1

X n

i?2

x i

with n=30and x i2?0;1 .

Test problem2(ZDT2):Nonconvex Pareto-front f1exT?x1;

f2exT?gexTe1àex1=gexTT2T;

gexT?1t

9

nà1

X n

i?2

x i

with n=30and x i2?0;1 .

Test problem3(ZDT3):Discontinuous Pareto-front:

f1exT?x1;

f2exT?gexTe1à

???????????????

x1=gexT

p

à

x1

gexT

sine10p x1TT;

gexT?1t

9

nà1

X n

i?2

x i

with n=30and x i2?0;1 .

Test problem4(ZDT4):Many local Pareto-fronts:

f1exT?x1;

f2exT?gexTe1à

???????????????x1=gexTp

T;

gexT?1t10enà1Tt

X n

i?2ex2

i

à10cose4p x iTT

with n=10and x12?0;1 x i2?à5;5 for i?2; (9)

Test problem5(ZDT6):Low density of solutions near Pareto-fronts: f1exT?1àexpà4x1sine6p x1T6;

f2exT?gexTe1àef1exT=gexTT2T;

gexT?1t

9

nà1

X n

i?2

x i

with n=10and x i2?0;1 .

The results of ADEA is shown in Figs.3–7,the behavior of ADEA is compared to NSGAII on a number of test problems.To match the settings of the algorithms used for comparison,the population size was set100 and the algorithm was run for200generations.As can be seen,ADEA is able to generate uniform distribution of solutions and the results is better than NSGAII on both quality and quantity on zdt1,2,3,4,6.

https://www.360docs.net/doc/b82941930.html,parison and discussion

Results on?ve test functions,in relation to the convergence metric C and diversity metric D,are presented in Table1–5,the mean and variance of the values are averaged over10runs.The results for ADEA and other ?ve algorithms which are taken from the literature(see[9]for the results and parameter settings of both ver-sions of NSGAII,SPEA and PAES,[7]for PDEA,and[2]for MODE)are provided.

436W.Qian,A.li/Applied Mathematics and Computation201(2008)431–440

W.Qian,A.li/Applied Mathematics and Computation201(2008)431–440437

Results for PDEA in[6]were evaluated with generational distance instead of the convergence metric.

As can be seen,ADEA is able to converge better in all problems except in ZDT1and ZDT2,where NSGA-II(binary-coded)and SPEA found better convergence.The challenge for MOEAs in the?rst three test prob-lems(ZDT1,ZDT2and ZDT3)lies in the high-dimensionality of these problems.Many MOEAs have achieved very good results on these problems in both goals of multiobjective optimization(convergence to the true Pareto-front and uniform spread of solutions along the front).The results for the problems ZDT1 and ZDT2(see Tables1and2)show that ADEA achieves good results,which are comparable to the results of the algorithms NSGA-II(real-coded),PAES and MODE.On ZDE3(see Table3),ADEA outperform all other algorithms used in comparison.

438W.Qian,A.li/Applied Mathematics and Computation201(2008)431–440

Table1

Statistics of results on test problems ZDT1

Algorithm Convergence metric Diversity metric NSGA-II(real-coded)0.033482±0.0047500.390307±0.001876 NSGA-II(binary-coded)0.000894±0.0000000.463292±0.041622 SPEA0.001799±0.0000010.784525±0.004440 PAES0.082085±0.008679 1.229794±0.000742 PDEA N/A0.298567±0.000742 MODE0.005800±0.000000N/A

SDE0.002741±0.0003850.382890±0.001435 Table2

Statistics of results on test problems ZDT2

Algorithm Convergence metric Diversity metric NSGA-II(real-coded)0.072391±0.0316890.430776±0.004721 NSGA-II(binary-coded)0.000824±0.0000000.435112±0.024607 SPEA0.001339±0.0000000.755184±0.004521 PAES0.126276±0.036877 1.165942±0.007682 PDEA N/A0.317958±0.001389 MODE0.005500±0.000000N/A

SDE0.002203±0.0002970.345780±0.003900

W.Qian,A.li/Applied Mathematics and Computation201(2008)431–440439 Table3

Statistics of results on test problems ZDT3

Algorithm Convergence metric Diversity metric NSGA-II(real-coded)0.114500±0.0049400.738540±0.019706 NSGA-II(binary-coded)0.043411±0.0000420.575606±0.005078 SPEA0.047517±0.0000470.672938±0.003587 PAES0.023872±0.0000100.789920±0.001653 PDEA N/A0.623812±0.000225 MODE0.021560±0.000000N/A

SDE0.002741±0.0001200.525770±0.043030 Table4

Statistics of results on test problems ZDT4

Algorithm Convergence metric Diversity metric NSGA-II(real-coded)0.513053±0.1184600.702612±0.064648 NSGA-II(binary-coded) 3.227636±7.3076300.479475±0.009841 SPEA7.340299±6.5725160.798463±0.014616 PAES0.854816±0.5272380.870458±0.101399 PDEA N/A0.840852±0.035741 MODE0.638950±0.500200N/A

SDE0.100100±0.4462000.436300±0.110000 Table5

Statistics of results on test problems ZDT6

Algorithm Convergence metric Diversity metric NSGA-II(real-coded)0.296564±0.0131350.668025±0.009923 NSGA-II(binary-coded)7.806798±0.0016670.644477±0.035042 SPEA0.221138±0.0004490.849389±0.002713 PAES0.085469±0.006664 1.153052±0.003916 PDEA N/A0.473074±0.021721 MODE0.026230±0.000861N/A

SDE0.000624±0.0000600.361100±0.036100 ZDT4is a hard optimization problem with manye219Tlocal Pareto-fronts that tend to mislead the optimi-zation algorithm.In Table4,we can see that all algorithms have di?culties in converging to the true Pareto-front.But ADEA get very good result.

With the?rst test problem ZDT6,there are two major di?culties.The?rst one is thin density of solutions towards the Pareto-front and the second one nonuniform spread of solutions along the front.On this problem, ADEA outperform all other algorithms.

In the?rst three and the last problems,crossover rate is0.7,ZDT4is0.3.For the diversity metric,ADEA outperform any other proposed algorithms.

5.Conclusion

ADEA is a new DE implementation dealing with multiple objectives.The biggest di?erence between ADEA and other MOEAs is that ADEA introduced a new de?ned self-parameter and a new select operator.We tested the approach on?ve benchmark problems and it was found that our approach is competitive to most other approaches.We also experimented with di?erent crossover rate on these problems to?nd their best solu-tions.The crossover rate is found to be sensitive on problem4to the solutions.In the near future,we also plan to evaluate ADEA on additional test problems.

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