Preselection via Classiﬁcation: A Case Study on ... · 1 Preselection via Classiﬁcation: A Case Study on Evolutionary Multiobjective Optimization Jinyuan Zhang, Aimin Zhou, Ke

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Preselection via Classification: A Case Study onEvolutionary Multiobjective Optimization

Jinyuan Zhang, Aimin Zhou, Ke Tang, and Guixu Zhang

Abstract—In evolutionary algorithms, a preselection operatoraims to select the promising offspring solutions from a candidateoffspring set. It is usually based on the estimated or realobjective values of the candidate offspring solutions. In a sense,the preselection can be treated as a classification procedure,which classifies the candidate offspring solutions into promisingones and unpromising ones. Following this idea, we propose aclassification based preselection (CPS) strategy for evolutionarymultiobjective optimization. When applying classification basedpreselection, an evolutionary algorithm maintains two externalpopulations (training data set) that consist of some selected goodand bad solutions found so far; then it trains a classifier basedon the training data set in each generation. Finally it uses theclassifier to filter the unpromising candidate offspring solutionsand choose a promising one from the generated candidateoffspring set for each parent solution. In such cases, it is notnecessary to estimate or evaluate the objective values of thecandidate offspring solutions. The classification based preselec-tion is applied to three state-of-the-art multiobjective evolutionaryalgorithms (MOEAs) and is empirically studied on two sets oftest instances. The experimental results suggest that classificationbased preselection can successfully improve the performance ofthese MOEAs.

Index Terms—preselection, classification, multiobjective evolu-tionary algorithm

I. INTRODUCTION

In evolutionary algorithms (EAs), the preselection operatoris a component that has different meanings [1]. In this paper,we use the term “preselection” to denote the process thatselects the promising offspring solutions from a set of can-didate offspring solutions created by the offspring generationoperators before the environmental selection procedure.

A key issue in preselection is how to measure the qualityof the candidate offspring solutions. The surrogate model(metamodel) is one of the major techniques for this [2]–[5]. The surrogate model is a method to mimic the originaloptimization surface by finding an alternative mapping, whichis computationally cheaper, from the decision variable (modelinput) to the dependent variable (model output) through agiven training data set. Some popular surrogate models include

This work is supported by the National Natural Science Foundation ofChina under Grant No.61673180, and the Science and Technology Commis-sion of Shanghai Municipality under Grant No.14DZ2260800.

J. Zhang, A. Zhou, and G. Zhang are with the Shanghai Key Labo-ratory of Multidimensional Information Processing, Department of Com-puter Science and Technology, East China Normal University, 3663North Zhongshan Road, Shanghai, China. (email: [email protected],{amzhou,gxzhang}@cs.ecnu.edu.cn).

K. Tang is with the with the USTC-Birmingham Joint Research Institutein Intelligent Computation and Its Applications (UBRI), School of ComputerScience and Technology, University of Science and Technology of China,Hefei, Anhui, China. (email:[email protected]).

the Kriging [6], [7], Gaussian process [8]–[12], radial basisfunction [13], artificial neural networks [14]–[17], polynomialresponse surfaces [18], [19] and support vector machines [19],[20]. In the community of evolutionary computation, the sur-rogate model is usually used to replace the original objectivefunction especially when the objective function evaluation isexpensive [2], [4], [5], [21]–[29]. In the case of preselection,the surrogate model based preselection strategies have beenemployed to solve different optimization problems [7], [30]–[35]. To reduce the time complexity of the surrogate modelbuilding, recently we proposed using a “cheap surrogatemodel” [36] to estimate the offspring quality.

In EAs, the preselection can be naturally regarded as aclassification problem. More precisely, the preselection classi-fies the candidate offspring solutions into two categories: theselected promising ones, and the discarded unpromising ones.This indicates that what we need to know is whether a candi-date offspring solution is good or bad instead of how good it is.Following this idea, a classification method was proposed forexpensive optimization problem [23]. This work has also beenextended by using both classification and regression methodsin preselection [24]. A major difference between the regressionmodel based approach and the classification based approachis that the former measures the quality of the candidateoffspring solutions precisely while the latter roughly classifiesthe candidate offspring solutions into several categories. For acandidate offspring solution, an accurate quality measurementmight be useful, however in (pre)selection, a label has alreadyoffered enough information for making decisions.

It might be more suitable to use classification techniquesin multiobjective optimization. The reason is that the solu-tions in a multiobjective evolutionary algorithm (MOEA) areeither dominated or nondominated, which forms two classesnaturally. In [37], the classification algorithms are used tolearn Pareto dominance relations. As far as we know, weare the first ones to apply classification to the preselectionof MOEAs [38], [39]. Very recently, a similar idea has beenimplemented in [40].

In this paper, we extend our previous work [38], [39]and propose a general classification based preselection (CPS)scheme for evolutionary multiobjective optimization. The ma-jor procedure of CPS works as follows: in each generation, firsta training data set (external populations) is updated by recentlyfound solutions; secondly a classifier is built according to thetraining data set; thirdly for each solution, candidate offspringsolutions are generated and their labels are predicted by theclassifier. Finally for each solution, an offspring solution ischosen according to the predicted labels. The major differences

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between this paper and the previous work follow.• The data preparation strategy is improved: all the non-

dominated solutions found so far are used to update thenegative training set, and the training set contains morepoints. In [38], [39], only the solutions in the previousgeneration are used to update the negative training set.

• A general CPS strategy is proposed and applied to thethree main MOEA frameworks, i.e., the Pareto domi-nation based MOEA, the indicator based MOEA, andthe decomposition based MOEA. In [38], [39], CPS isapplied to one framework.

• A systematic empirical study, based on two sets oftest instances, has been performed to demonstrate theadvantages of CPS.

The rest of the paper is organized as follows. Section IIpresents the related work on evolutionary multiobjective opti-mization and briefly introduces the main MOEA frameworksused in the paper. Section III presents the classification basedpreselection in detail. The proposed CPS is systematicallystudied on some benchmark problems in Section IV. Finally,Section VI concludes this paper with some future work re-marks.

II. EVOLUTIONARY MULTIOBJECTIVE OPTIMIZATION

This paper considers the following continuous multiobjec-tive optimization problems (MOP)).

min F (x) = (f1(x), · · · , fm(x))T

s.t x ∈ Πni=1[ai, bi]

(1)

where x = (x1, · · · , xn)T ∈ Rn is a decision variable vector;Πn

i=1[ai, bi] ⊂ Rn defines the feasible region of the searchspace; ai < bi (i = 1, · · · , n) are the lower and upperboundaries of the search space respectively, fj : Rn → R(j =1, · · · ,m) is a continuous mapping, and F (x) is an objectivevector.

Due to the conflicting nature among the objectives in (1),there usually does not exist a single solution that can optimizeall the objectives at the same time. Instead, a set of tradeoffsolutions, named as Pareto optimal solutions [41], are ofinterest. In the decision space, the set of all the Pareto optimalsolutions is called the Pareto set (PS) and in the objectivespace, it is called the Pareto front (PF).

In practice, the target of different methods is to find anapproximation to the PF (PS) of (1). Among the methods,EAs have become a major method to deal with MOPs dueto their population based search property, which makes anMOEA be able to approximate the PF (PS) in a single run. Avariety of MOEAs have been proposed in last decades [42].Most of these algorithms can be classified into three cat-egories: the Pareto based MOEAs [43]–[46], the indica-tor based MOEAs [47]–[51], and the decomposition basedMOEAs [52]–[54]. In this paper, we demonstrate that CPS isable to improve the performance of the three kinds of MOEAs.Three algorithms- the regularity model based multiobjectiveestimation of distribution algorithm (RM-MEDA) [55], thehypervolume metric selection based EMOA (SMS-EMOA) [56],and the MOEA based on decomposition with multiple differen-tial evolution mutation operators (MOEA/D-MO) [57], from

the three MOEA categories respectively are used as the basicalgorithms. These algorithms are briefly introduced as follows.

A. RM-MEDA

Algorithm 1: RM-MEDA Framework

1 Initialize the population P = {x1, x2, · · · , xN};2 while not terminate do3 Build a probabilistic model for modeling the

distribution of the solutions in P ;4 Sample a set of offspring solutions Q from the

probabilistic model;5 Set P = P ∪Q;6 Partition P into fronts P 1, P 2, · · · ;7 Find the k-th front such that

k−1∑j=1

|P j | < N andk∑

j=1

|P j | ≥ N ;

8 while∑k

j=1 |P j | > N do9 Calculate the density of each solution in P k;

10 Find the solution with the worst density x∗ ∈ P k;11 Set P k = P k \ {x∗};12 end13 Set P = ∪kj=1P

j ;14 end15 return P .

The major contribution of the Pareto domination basedMOEAs is the environmental selection operator. It firstlypartitions the population into different clusters through thePareto domination relation. The solutions in the same clusterare nondominated with each other, and a solution in a worsecluster will be dominated by at least one solution in a bettercluster. Secondly, it assigns each solution in the same clustera density value. The extreme solutions and solutions in sparseareas are preferable. The sorting scheme based on both thePareto domination and density actually defines a full orderover the population.

RM-MEDA [55], shown in Algorithm 1, is a Pareto dom-ination based MOEA. It uses probabilistic models to guidethe offspring generation and utilizes a modified nondominatedsorting scheme [46] for environmental selection. RM-MEDAworks as follows: the population is initialized in Line 1,a probabilistic model is built in Line 3, a set of offspringsolutions are sampled from the probabilistic model in Line4, and the population is updated through the environmentalselection in Lines 5-13. Firstly, the offspring population andthe current population are merged in Line 5. The mergedpopulation is partitioned into different clusters in Line 6 wherea cluster with a low index value is preferable. A key cluster, ofwhich some solutions will be discarded, is found in Line 7. Thedensity values of the solutions in the key cluster are calculated,and the solutions with the worst density values are discardedone by one in Lines 8-12. This step is the major modificationwhere in the original version [46], the key cluster is sorted by

the density values and the worst ones are discarded directly.The crowding distance is used to estimate the density values.

B. SMS-EMOA

Algorithm 2: SMS-EMOA Framework

1 Initialize the population P = {x1, x2, · · · , xN};2 while not terminate do3 Generate an offspring solution y;4 Set P = P ∪ {y};5 Partition P into fronts P 1, P 2, · · · , P k according to

fast-nondominated-sort;6 Set x∗ = arg minx∈Pk{IH(P k)− IH(P k \ {x})};7 Set P = P \ {x∗};8 end9 return P .

For performance indicator based MOEAs, a performanceindicator is used to measure the population quality and select apromising population to the next generation. The hypervolumeindicator is widely used to do so. Some other indicators, forexample the R2 [56] and the fast hypervolume [49], have alsobeen utilized.

SMS-EMOA [48] is a typical algorithm in this category,and its framework is shown in Algorithm 2. The algorithmuses the steady state strategy that only one offspring solutionis generated in each generation. Similar to the environmentalselection in the Pareto domination based approaches, it firstlypartitions the combined population into different clusters inLine 5. It then finds the solution that has the minimumcontribution to the hypervolume (IH ) value of the mergedpopulation in Line 6. Finally, it removes this solution to keepa fixed population size in Line 7.

C. MOEA/D-MO

MOEA/D decomposes a MOP into a set of scalar-objectivesubproblems and solves them simultaneously. The optimalsolution of each subproblem will hopefully be a Pareto optimalsolution of the original MOP, and a set of well selectedsubproblems may produce a good approximation to the PS(PF). A key issue in MOEA/D is the subproblem definition.In this paper, we use the following Tchebycheff technique.

min g(x|λ, z∗) = max1≤j≤m

λj |fj(x)− z∗j | (2)

where λ = (λ1, · · · , λm)T is a weight vector with thesubproblem, and z∗ = (z∗1 , · · · , z∗m)T is a reference point,i.e., z∗j is the minimal value of fj in the search space. Forsimplicity, we use gi(x) to denote the i-th subproblem. Inmost cases, two subproblems with close weight vectors alsohave similar optimal solutions. Based on the distances betweenthe weight vectors, MOEA/D defines the neighborhood of asubproblem, which contains the subproblems with the nearestweight vectors. In MOEA/D, the offspring generation andsolution selection are based on the concept of neighborhood.

In MOEA/D, the i-th (i = 1, · · · , N ) subproblem maintainsthe following information:

• its weight vector λi and its objective function gi,• its current solution xi and the objective vector of xi, i.e.F i = F (xi), and

• the index set of its neighboring subproblems, Bi.

Algorithm 3: Framework of MOEA/D-MO

1 Initialize a set of subproblems (xi, F i, Bi, gi) for i = 1,· · · , N , initialize the reference point z∗ (j = 1, · · · , m)

z∗j = mini=1,··· ,N

fj(xi);

2 while not terminate do3 foreach i ∈ {1, · · · , N} do4 Set the mating pool as

π =

{Bi if rand() < pn{1, · · · , N} otherwise

5 Generate M offspring solutions {y1, · · · , yM} byparents from the mating pool;

6 foreach y ∈ {y1, · · · , yM} do7 for j=1:m do8

z∗j =

{fj(y) if fj(y) < z∗jz∗j otherwise

9 end10 Set counter c = 0;11 foreach j ∈ π do12 if gj(y) < gj(xj) and c < C then13 Replace xj by y;14 Set c = c+ 1;15 end16 end17 end18 end19 end20 return P .

In this paper, we use MOEA/D-MO [57], which modifiesthe offspring generation procedure of MOEA/D-DE [54] withmultiple differential evolution (DE) mutation operators, andthe algorithm framework is shown in Algorithm 3. In Line1, a set of N subproblems and the reference ideal point areinitialized. In Lines 4-5, the mating pool is set as the neighbor-hood solutions with probability pn and as the population withprobability 1 − pn, and M offspring solutions are generatedbased on the mating pool. The reference ideal point is thenupdated in Line 7-9. At most C neighboring solutions arereplaced by each of the newly generated offspring solution inLines 10-16. It should be noted that the replacement is basedon the subproblem objective (2) that is much different fromthe above two MOEA frameworks.

III. CLASSIFICATION BASED PRESELECTION

This section introduces the proposed CPS scheme for mul-tiobjective optimization. To implement CPS, three proceduresshould be employed: the training data set preparation, the

classification model building, and the choosing of promisingoffspring solutions. The details of these procedures follow.

A. Data Preparation

There are two issues to be considered for preparing a propertraining data set.

The first one is solution labeling. It is not suitable to directlyuse the objective values to label solutions in the case ofmultiobjective optimization. Instead, we can use the conceptof Pareto domination to label the solutions. The nondominatedsolutions can be labelled as ‘positive’ training points and thedominated ones can be labelled as ‘negative’ training points.

The second one is data set update. In this paper, we focuson binary classification, which usually requires to balance thenumber of the positive and negative sample points. To thisend, we use two external populations P+ and P−, denotingthe positive and negative data sets respectively, to store thetraining data points. Furthermore, we expect that P+ and P−have the same size and form a balanced training data set.The two external populations are updated in each generation.The reasons are (a) to reduce the time complexity in themodel building step, (b) to represent the current populationdistribution, and (c) to promote preselection efficiency.

Let Q = S(P,N) denote the nondominated sorting schemein [46], which selects the best N solutions from P and storesthe selected ones in Q. Q = NS(P ) is a procedure to choosethe nondominated solutions and store them in Q. The datapreparation works as follows:• In the initialization step: the two external populations are

set empty, i.e., P+ = P− = ∅.• In each step: let Q be the set of newly generated solutions

in each generation, Q+ and Q− contain the nondominatedand dominated solutions in Q respectively. P+ and P−are updated as

P+ = S (NS(P+ ∪Q+), 5N)

and

P− = S (P− ∪Q− ∪ P+ ∪Q+ \NS(P+ ∪Q+), 5N) .

Different to our previous work in [38], [39], this newapproach uses only nondominated solutions to update P+,and all the dominated solutions found so far to update P−.Furthermore, the external population sizes are expected atmost to be 5 times of the population size N . It should benoted that although it is expected that |P+| = |P−| = 5N , inearly states or on some complicated problems when it is hardto generate nondominated solutions, |P+| might be less than5N . The influence of the size of the training data set shall beempirically studied shortly in Section IV.

B. Classification Model

Let P+ = {x} and P− = {x} denote the positive andnegative training sets respectively where x is the feature vector,i.e., the decision variable vector in our case. +1 and −1 arethe labels of the feature vectors in P+ and P− respectively.

A classification procedure aims to find an approximated rela-tionship between a feature vector x and a label l ∈ {+1,−1}

l = C(x)

to replace the real relationship l = C(x) based on the giventraining set.

There exists a variety of classification methods [58]. Inthis paper, we focus on the following K-nearest neighbor(KNN) [59] model.

KNN(x) =

+1 ifK∑i=1

C(xi) ≥ 0

−1 otherwise

where K is an odd number, xi denotes the i-th closest featurevector, according to the Euclidean distance, in P+ ∪ P−, andC(xi) is the actual label of xi.

C. Offspring SelectionEach parent solution x generates M candidate offspring

solutions: Y = {y1, y2, · · · , yM}. In preselection, the qual-ities of these candidate solutions are estimated through theclassification model, and a promising one will be selected forthe real function evaluation.

In this paper, we randomly choose an offspring solutionwith a predicted label 1. It works as follows:

1) Set Y ∗ = {y ∈ Y |KNN(y) = 1},2) Reset it as Y ∗ = Y if Y ∗ is empty,3) Randomly choose a solution y∗ ∈ Y ∗ as the offspring

solution.It should be noted that the procedure may choose an

unpromising one when no candidate solutions are labeled as1.

D. CPS based MOEAFig. 1 illustrates a general flowchart of CPS based MOEA.To apply CPS in MOEAs, the following steps should be

added or modified in the original MOEAs.• The two external populations P+ and P− should be

set to empty in the initialization step, i.e., Line 1 inAlgorithms 1-3.

• In each step, the two external populations should befirstly updated as in Section III-A, i.e., below Line 2 inAlgorithms 1-3.

• Following the external population update, a classifier isconstructed as in Section III-B, i.e., before Line 3 inAlgorithms 1-3. It should be noted that in this paper,the KNN model is a nonparametric model and this stepcould be overlooked.

• The offspring generation step, i.e., Line 4 in Algorithm 1,and Line 3 in Algorithm 2 should be modified and aset of M candidate offspring solutions are generatedfor each parent by repeating the offspring generationprocedure for M times. A final offspring solution isthen chosen by the selection strategy in Section III-C.In Algorithm 3, the offspring generation in Line 5 shouldnot be changed. Then an offspring solution is chosen andonly this offspring solution is used to update the referenceidea point and to update the neighborhood in Lines 7-16.

Fig. 1: An illustration of the flowchart of CPS based MOEA

IV. EXPERIMENTAL STUDY

A. Experimental SettingsIn this section, we apply CPS into the variants of the three

major MOEA frameworks. The three chosen algorithms areRM-MEDA [55], SMS-EMOA [56], and MOEA/D-MO [57],respectively. And the three CPS based algorithms are denotedas RM-MEDA-CPS, SMS-EMOA-CPS, and MOEA/D-MO-CPS, respectively. These algorithms are applied to 10 testinstances, ZZJ1-ZZJ10, from [55] in the experiments.

The parameter settings are as follows:• The number of decision variables is n = 30 for all the

test instances.• The algorithms are executed 30 times independently on

each instance and stop after 20, 000 function evaluations(FEs) on ZZJ1, ZZJ2, ZZJ5, and ZZJ6, 40, 000 FEs onZZJ4 and ZZJ8, as well as 100, 000 FEs on ZZJ3, ZZJ7,ZZJ9, and ZZJ10.

• The population size is set as N = 100 on ZZJ1, ZZJ2,ZZJ5, ZZJ6, and 200 on ZZJ3, ZZJ4 and ZZJ7-ZZJ10respectively.

• In CPS, the number of nearest points used in KNN isK = 3.

The other parameter settings of algorithms are referred torelated work [55]–[57]. All of the algorithms are implementedin Matlab and are executed on the same computer.

B. Performance MetricsWe use the Inverted Generational Distance (IGD) met-

ric [60]–[62] and I−H metric [63] to assess the performance

of the algorithms in the experimental study.Let P ∗ be a set of Pareto optimal points to represent the

true PF, and P be the set of nondominated solutions found byan algorithm. The IGD and I−H metrics are briefly defined asfollows:

IGD(P ∗, P ) =

∑x∈P∗ d(x, P )

|P ∗|

where d(x, P ) denotes the minimum Euclidean distance be-tween x and any point in P , and |P ∗| denotes the cardinalityof P ∗.

I−H(P ∗, P ) = IH(P ∗, z∗)− IH(P, z∗)

where z∗ is a reference point, and IH(P, z∗) denotes thehypervolume of the space covered by the set P and thereference point z∗.

Both metrics can measure the diversity and convergence ofthe obtained set P . To have a small metric value, the obtainedset should be close to the PF and be well-distributed. In ourexperiments, 10, 000 evenly distributed points in PF are gener-ated as the P ∗. In the experiments, z∗ is set to (1.2, 1.2) and(1.2, 1.2, 1.2) for the bi-objective and tri-objective problemsrespectively.

In order to get statistically conclusions, the Wilcoxon’s ranksum test at a 5% significance level is employed to compare theIGD and I−H metric values obtained by different algorithms. Inthe table, ∼, +, and − denote that the results obtained by theCPS based version are similar to, better than, or worse thanthat obtained by the original version.

C. Comparison Study

1) RM-MEDA-CPS vs. RM-MEDA: Table I shows the sta-tistical results of the IGD and I−H metric values obtainedby RM-MEDA-CPS and RM-MEDA on ZZJ1-ZZJ10 over30 runs. The statistical test indicates that according to boththe two metrics, RM-MEDA-CPS outperforms RM-MEDAon 7 out of 10 instances, and on the other 3 instances, thetwo algorithms have similar performances. This results alsoindicates that given the same FEs, RM-MEDA-CPS works noworse than RM-MEDA. Since the only difference between thetwo algorithms is on the use of CPS, the experimental resultssuggest that CPS can successfully improve the performance ofRM-MEDA.

Fig. 2 plots the run time performance in terms of the IGDvalues obtained by the two algorithms on the 10 instances.The curves obtained by the two algorithms in Fig. 2 showthat on all test instances, RM-MEDA-CPS converges fasterthan RM-MEDA. Fig. 3 plots the mean FEs required to obtaindifferent levels of the IGD values. It suggests that to obtain thesame IGD values, RM-MEDA-CPS uses fewer computationalresources than RM-MEDA does. The experimental results inFigs. 2 and 3 suggest that CPS can help to speed up theconvergence of RM-MEDA on most of the instances.

In order to visualize the final obtained solutions, we takeZZJ1, ZZJ2 and ZZJ9 as examples and plot the final obtainedpopulations of the median runs according to the IGD valuesafter 10%, 20%, and 30% of the max FEs in Fig. 4. It is

TABLE I: The statistical results of IGD and I−H metric values obtained by RM-MEDA-CPS and RM-MEDA on ZZJ1-ZZJ10

instance metric RM-MEDA-CPS RM-MEDAmean std. min max mean std. min max

ZZJ1 IGD 4.19e-03(+) 8.61e-05 4.07e-03 4.42e-03 4.28e-03 1.11e-04 4.10e-03 4.49e-03I−H 5.73e-03(+) 2.74e-04 5.36e-03 6.45e-03 6.03e-03 3.72e-04 5.39e-03 6.76e-03

ZZJ2 IGD 4.13e-03(∼) 8.41e-05 3.97e-03 4.29e-03 4.15e-03 7.72e-05 3.98e-03 4.32e-03I−H 5.78e-03(∼) 4.02e-04 5.08e-03 6.97e-03 5.93e-03 3.72e-04 5.28e-03 6.77e-03








ZZJ10 IGD 1.34e+02(∼) 8.77e+00 1.03e+02 1.48e+02 1.32e+02 7.01e+00 1.18e+02 1.44e+02I−H 1.11e+00(∼) 2.26e-16 1.11e+00 1.11e+00 1.11e+00 2.26e-16 1.11e+00 1.11e+00

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(h) ZZJ8

10 5 1 0.5 0.1 0.050

2

4

6

8x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(i) ZZJ9

10 5 1 0.5 0.10

2

4

6

8

10x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(j) ZZJ10

Fig. 3: The mean FEs required by RM-MEDA-CPS and RM-MEDA to obtain different IGD values over 30 runs

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.21

0%

RM−MEDA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

10

%

RM−MEDAPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

20

%

RM−MEDA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

20

%

RM−MEDAPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

RM−MEDA−CPS

30%

RM−MEDA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

RM−MEDA

30

%

RM−MEDAPF

(a) ZZJ1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

10

%

RM−MEDA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

10

%

RM−MEDAPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

20

%

RM−MEDA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

20

%

RM−MEDAPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

RM−MEDA−CPS

30%

RM−MEDA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

RM−MEDA

30

%

RM−MEDAPF

(b) ZZJ2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

10

%

RM−MEDA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

10

%

RM−MEDAPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

20

%

RM−MEDA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

20

%

RM−MEDAPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

RM−MEDA−CPS

30

%

RM−MEDA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

RM−MEDA

30

%

RM−MEDAPF

(c) ZZJ9

Fig. 4: The median (according to the IGD metric values) approximations obtained by RM-MEDA-CPS and RM-MEDA after10%, 20%, 30% of the max FEs on (a) ZZJ1, (b) ZZJ2, and (c) ZZJ9

clear that RM-MEDA-CPS is able to achieve better resultsthan RM-MEDA with the same computational costs.

2) SMS-EMOA-CPS vs. SMS-EMOA: Table II presents thestatistical results of IGD and I−H metric values obtainedby SMS-EMOA-CPS and SMS-EMOA on ZZJ1-ZZJ10 over30 runs. The statistical test shows different results accord-ing to different performance metrics. According to the IGDmetric, SMS-EMOA-CPS works better than SMS-EMOA on3 instances, works worse than SMS-EMOA on 1 instance,and works similar to SMS-EMOA on the other 6 instances.According to the I−H metric, SMS-EMOA-CPS shows betterperformance on 5 instances, worse performance on 1 instance,and similar performance on the other 4 instances than SMS-EMOA. Even so, we can still say that SMS-EMOA-CPS isnot worse than SMS-EMOA on most of the given instancesaccording to the final obtained results.

Fig. 5 presents the run time performance in terms of the IGDvalues obtained by the two algorithms on the 10 instances. Itshows that on ZZJ5 and ZZJ8, SMS-EMOA-CPS convergesslower than SMS-EMOA, and on all the other instances,SMS-EMOA-CPS converges faster than or similar to SMS-EMOA. Fig. 6 plots the FEs required by the two algorithmsto obtain some levels of IGD values. This figure suggeststhat to obtain the same IGD values, SMS-EMOA-CPS takesfewer computational resources than SMS-EMOA does blue,especially in the early stages on most of the instances. Figs. 5and 6 suggest that CPS can improve the convergence speed ofSMS-EMOA on most of the instances.

In order to visualize the final obtained solutions, we drawthe final obtained populations of the median runs accordingto the IGD values after 10%, 20%, and 30% of the max FEsin Fig. 7 for ZZJ1, ZZJ2 and ZZJ9. The figure shows that theSMS-EMOA-CPS can achieve better results than SMS-EMOAwith the same computational costs on most of the instances.

3) MOEA/D-MO-CPS vs. MOEA/D-MO: Table 8 shows thestatistical results of IGD and I−H metric values obtained by

MOEA/D-MO-CPS and MOEA/D-MO on ZZJ1-ZZJ10 over30 runs. The statistical tests according to both the IGD and theI−H metric metrics are consistent with each other. It shows onZZJ1-ZZJ5, MOEA/D-MO-CPS outperforms MOEA/D-MO,and on ZZJ6-ZZJ10, the two algorithms performs similar witheach other. This suggests that with the given FEs, MOEA/D-MO-CPS works no worse than MOEA/D-MO. The reason isthat CPS helps MOEA/D-MO to obtain better results on someinstances.

Fig. 8 presents the run time performance in terms of the IGDvalues obtained by MOEA/D-MO-CPS and MOEA/D-MO onthe 10 instances. The curves obtained by the two algorithms inFig. 8 show that on most of the instances MOEA/D-MO-CPSconverges faster than MOEA/D-MO. Only on ZZJ6, ZZJ8, andZZJ10, MOEA/D-MO-CPS converges slower than MOEA/D-MO. Fig. 9 plots the FEs required by the two algorithmsto obtain some levels of IGD values on the 10 instances. Itsuggests that to obtain the same IGD values, MOEA/D-MO-CPS uses fewer computational resources than MOEA/D-MOon most of the instances.

To do a visual comparison, Fig. 10 plots the final obtainedpopulations of the median runs according to the IGD valuesafter 10%, 20%, and 30% of the max FEs for ZZJ1, ZZJ2and ZZJ9. The figure indicates that MOEA/D-MO-CPS canachieve better results than MOEA/D-MO with the same com-putational costs on most of the instances.

4) More Discussion: The experiments in the previous sec-tions clearly show that CPS can successfully improve theperformances of the three algorithms on most of the giventest instances according to both the quality of the final ob-tained solutions and the algorithm convergence. This sectioninvestigates why CPS works well.

We conduct the following experiment. For each parent,M = 3 candidate offspring solutions are generated and eval-uated, and their domination relationships are also calculated.CPS is used to select an offspring solution. The proportion of

TABLE II: The statistical results of IGD and I−H metric values obtained by SMS-EMOA-CPS and SMS-EMOA on ZZJ1-ZZJ10

instance metric SMS-EMOA-CPS SMS-EMOAmean std. min max mean std. min max

ZZJ1 IGD 3.67e-03(∼) 2.29e-05 3.62e-03 3.71e-03 3.67e-03 1.85e-05 3.64e-03 3.69e-03I−H 4.34e-03(+) 6.12e-05 4.26e-03 4.48e-03 4.37e-03 5.70e-05 4.28e-03 4.55e-03

ZZJ2 IGD 4.53e-03(∼) 1.85e-04 4.29e-03 5.22e-03 4.53e-03 1.26e-04 4.32e-03 4.76e-03I−H 4.40e-03(+) 6.38e-05 4.29e-03 4.57e-03 4.46e-03 7.73e-05 4.29e-03 4.62e-03






ZZJ8 IGD 4.29e-01(-) 6.93e-02 3.82e-01 6.22e-01 3.44e-01 1.13e-01 5.78e-02 3.88e-01I−H 3.13e-01(-) 7.06e-02 2.52e-01 4.10e-01 2.25e-01 7.20e-02 4.04e-02 2.55e-01



+/− / ∼ IGD 3/1/6+/− / ∼ I−

H 5/1/4

0 50 100 150 20010

−3

10−2

10−1

100

mea

n IG

D v

alue

FES(x102)

SMS−EMOA−CPSSMS−EMOA

(a) ZZJ1

0 50 100 150 20010

−3

10−2

10−1

100

mea

n IG

D v

alue

FES(x102)


(b) ZZJ2

0 100 200 300 400 500

100

mea

n IG

D v

alue

FES(x2x102)


(c) ZZJ3

0 50 100 150 20010

−2

10−1

100

101

mea

n IG

D v

alue

FES(x2x102)


(d) ZZJ4

0 50 100 150 20010

−3

10−2

10−1

100

mea

n IG

D v

alue

FES(x102)


(e) ZZJ5

0 50 100 150 200

10−0.6

10−0.4

10−0.2

mea

n IG

D v

alue

FES(x102)


(f) ZZJ6

0 100 200 300 400 500

100

mea

n IG

D v

alue

FES(x2x102)


(g) ZZJ7

0 50 100 150 200

10−0.4

10−0.2

100

100.2

mea

n IG

D v

alue

FES(x2x102)


(h) ZZJ8

0 100 200 300 400 50010

−2

10−1

100

101

mea

n IG

D v

alue

FES(x2x102)


(i) ZZJ9

0 100 200 300 400 50010

1

102

103

mea

n IG

D v

alue

FES(x2x102)


(j) ZZJ10

Fig. 5: The mean IGD values versus FEs obtained by SMS-EMOA-CPS and SMS-EMOA over 30 runs

10 5 1 0.5 0.1 0.050

2000

4000

6000

8000

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(a) ZZJ1

10 5 1 0.5 0.1 0.050

2000

4000

6000

8000

10000

12000

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(b) ZZJ2

10 5 10

2

4

6

8

10x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(c) ZZJ3

10 5 1 0.50

1000

2000

3000

4000

5000

6000

7000

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(d) ZZJ4

10 5 1 0.5 0.10

0.5

1

1.5

2x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(e) ZZJ5

10 5 10

0.5

1

1.5

2x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(f) ZZJ6

10 5 10

2

4

6

8

10x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(g) ZZJ7

10 5 10

1

2

3

4x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(h) ZZJ8

10 5 1 0.5 0.10

2

4

6

8

10x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(i) ZZJ9

10 5 10

2

4

6

8

10x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(j) ZZJ10

Fig. 6: The mean FEs required by SMS-EMOA-CPS and SMS-EMOA to obtain different IGD values over 30 runs

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOAPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOAPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOA−CPS

SMS−EMOA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOA

SMS−EMOAPF

(a) ZZJ1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOAPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOAPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOA−CPS

SMS−EMOA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOA

SMS−EMOAPF

(b) ZZJ2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOAPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOAPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOA−CPS

SMS−EMOA−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

SMS−EMOA

SMS−EMOAPF

(c) ZZJ9

Fig. 7: The median (according to the IGD metric values) approximations obtained by SMS-EMOA-CPS and SMS-EMOA after10%, 20%, 30% of the max FEs on (a) ZZJ1, (b) ZZJ2, and (c) ZZJ9

TABLE III: The statistical results of IGD and I−H metric values obtained by MOEA/D-MO-CPS and MOEA/D-MO on ZZJ1-ZZJ10

instance metric MOEA/D-MO-CPS MOEA/D-MOmean std. min max mean std. min max











+/− / ∼ IGD 5/0/5+/− / ∼ I−

H 5/0/5

offspring solutions, which selected by CPS are nondominatedones and dominated ones for each parent, are recorded foreach generation. The statistical results are shown in Fig. 11,Fig. 12 and Fig. 13.

From Fig. 11, Fig. 12, Fig. 13, we can see that for RM-MEDA, SMS-EMOA and MOEA/D-MO, 60.00%-100.00%offspring solutions chosen by CPS are nondominated ones.In most cases, the proportion is during 80.00%-100.00%. Andamong them, at the beginning of the generation, the proportionof nondominated solutions and dominated solutions are close.Along the increment of generation, the proportions of thenondominated solutions are increased while the dominatedones are decreased. This suggests that statistically, CPS has agood ability to obtain good solutions among the offsprings offor each parent. It shows that CPS can guide the search without

estimating the objective values of the candidate offspringsolutions. The reason might be that the classification modelcan successfully detect the boundary between the positive andnegative training sets, and thus to eliminate bad candidateoffspring solutions without function evaluations.

D. Sensitivity to Control Parameters

1) Sensitivity of the size of P+ and P−: This section studiesthe influence of the size of P+ and P−. |P+| = |P−| = 2N ,3N , 5N , 8N , and 10N are studied. The data preparationstrategy in our previous work [38], denoted as |P+|CEC

1, isalso compared here. RM-MEDA is used as the basic optimizerand all the other parameters are the same as in Section IV-A.

1|P+| = |P−| = 0.5N .

50 100 150 20010

−3

10−2

10−1

100

FES(x102)

mea

n IG

D v

alue

MOEA/D−MO−CPSMOEA/D−MO

(a) ZZJ1

50 100 150 20010

−3

10−2

10−1

100

FES(x102)

mea

n IG

D v

alue


(b) ZZJ2

100 200 300 400 500

100

FES(x2x102)

mea

n IG

D v

alue


(c) ZZJ3

50 100 150 20010

−2

10−1

100

101

FES(x2x102)

mea

n IG

D v

alue


(d) ZZJ4

50 100 150 20010

−3

10−2

10−1

100

FES(x102)

mea

n IG

D v

alue


(e) ZZJ5

50 100 150 200

10−0.9

10−0.7

10−0.5

10−0.3

FES(x102)

mea

n IG

D v

alue


(f) ZZJ6

100 200 300 400 500

100

FES(x2x102)

mea

n IG

D v

alue


(g) ZZJ7

50 100 150 20010

−0.9

10−0.6

10−0.3

100

FES(x2x102)

mea

n IG

D v

alue


(h) ZZJ8

100 200 300 400 50010

−2

10−1

100

101

FES(x2x102)

mea

n IG

D v

alue


(i) ZZJ9

100 200 300 400 50010

0

101

102

103

FES(x2x102)

mea

n IG

D v

alue


(j) ZZJ10

Fig. 8: The mean IGD values versus FEs obtained by MOEA/D-MO-CPS and MOEA/D-MO over 30 runs

10 5 1 0.5 0.1 0.050

5000

10000

15000

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(a) ZZJ1

10 5 1 0.5 0.1 0.050

2000

4000

6000

8000

10000

12000

14000

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(b) ZZJ2

10 5 10

2

4

6

8

10

12x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(c) ZZJ3

10 5 1 0.5 0.10

1

2

3

4x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(d) ZZJ4

10 5 1 0.5 0.1 0.050

0.5

1

1.5

2

2.5x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(e) ZZJ5

10 5 10

0.5

1

1.5

2

2.5x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(f) ZZJ6

10 5 10

2

4

6

8

10

12x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(g) ZZJ7

10 5 10

1

2

3

4

5x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(h) ZZJ8

10 5 1 0.5 0.10

2

4

6

8

10

12x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(i) ZZJ9

10 5 10

2

4

6

8

10

12x 10

4

IGD levels(x10−1)

mea

n fu

nctio

n ev

alua

tions


(j) ZZJ10

Fig. 9: The mean FEs required by MOEA/D-MO-CPS and MOEA/D-MO to obtain different IGD values over 30 runs

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

10

%

MOEA/D−MO−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

10

%

MOEA/D−MOPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

20

%

MOEA/D−MO−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

20

%

MOEA/D−MOPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

MOEA/D−MO−CPS

30%

MOEA/D−MO−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

MOEA/D−MO

30

%

MOEA/D−MOPF

(a) ZZJ1

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

10

%

MOEA/D−MO−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

10

%

MOEA/D−MOPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

20

%

MOEA/D−MO−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

20

%

MOEA/D−MOPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

MOEA/D−MO−CPS

30%

MOEA/D−MO−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

MOEA/D−MO

30

%

MOEA/D−MOPF

(b) ZZJ2

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

10

%

MOEA/D−MO−CPSPF

0 0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

1

1.2

10

%

MOEA/D−MOPF

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The mean FEs that required to achieve IGD = 5×10−2 havebeen recorded and are shown in Fig. 14.

f1 f2 f3 f4 f5 f6 f7 f8 f90

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The statistical results suggest that RM-MEDA-CPS with allthe size values perform similarly on ZZJ1 and ZZJ8; RM-MEDA-CPS with |P+| = |P−| = 5N works the best on ZZJ4and ZZJ6, and on ZZJ3, ZZJ5, ZZJ9, it works the secondbest. Our previous strategy |P+|CEC [38] performs best onZZJ3 and ZZJ7, but it also achieves the worst results onZZJ2, ZZJ4, ZZJ5, ZZJ6 and ZZJ9. These results showthat CPS with |P+| = |P−| = 5N performs the best in mostcases.

2) Sensitivity of the number of M : This section studies theinfluence of the number of candidate offspring solutions. RM-MEDA-CPS with M = 2, 3, 4, 5 is employed to the test suite.All the other parameters are the same as in Section IV-A. TheIGD values versus FEs have been recorded and are shown inFig. 15.

The statistical results suggest that RM-MEDA-CPS withM = 2 does not work well as RM-MEDA-CPS with othervalues. Furthermore, RM-MEDA-CPS with M = 3, 4, 5 workssimilarly on all of the instances. This results show that theCPS strategy is not sensitive to M . However, considering thecomputational cost required by offspring generation, M = 3might be a proper choice in practice.

V. EXPERIMENTAL RESULTS ON THE LZ TEST SUITE

In this section, we apply the three algorithms and theirCPS variants on the LZ test suite [54], of which the PSshave complicated shapes. The parameters are as follows. Thenumber of decision variables is n = 30 for all the testinstances. The algorithms are executed 30 times independentlyon each instance and stop after 150, 000 FEs on LZ1-LZ5and LZ7-LZ9, and 297, 500 FEs on LZ6. The population sizeis set as N = 300 on LZ1-LZ5 and LZ7-LZ9, and 595 onLZ6 respectively. All the other parameters are the same as inSection IV-A.

Table IV shows the mean and std. IGD and I−H metric val-ues obtained by RM-MEDA-CPS, RM-MEDA, SMS-EMOA-CPS, SMS-EMOA, MOEA/D-MO-CPS and MOEA/D-MO onLZ1− LZ9 after 30 runs.

The experimental results in Table IV suggests that CPS isable to improve the performances of the original algorithms

according to both the IGD and I−H metric values. It is clearthat the CPS based versions work no worse than the originalversions on all of the test instances. More precisely accordingto the IGD metric, (a) RM-MEDA-CPS performs better thanRM-MEDA on 5 instances, and on the other 4 instances,they work similarly; (b) SMS-EMOA-CPS achieves betterresults than SMS-EMOA does on 8 instances, and they achievesimilar results on the other instance; and (c) MOEA/D-MO-CPS outperforms MOEA/D-MO on 5 instances, and they getsimilar results on the other 4 instances. According to theI−H metric, (a) RM-MEDA-CPS wins on 7 instances, and onthe other 2 instances, they work similar; (b) SMS-EMOA-CPS performs better than SMS-EMOA on all 9 instances;and (c) MOEA/D-MO-CPS outperforms MOEA/D-MO on 4instances, and they get similar results on the other 4 instances.

VI. CONCLUSION

This paper proposes a classification based preselection(CPS) to improve the performance of multiobjective evolu-tionary algorithms (MOEAs). In CPS based MOEAs, somesolutions are chosen to form a training data set, and thena classifier is built based on the training data set in eachgeneration. Each parent solution generates a set of candidateoffspring solutions, and chooses a promising one as the realoffspring solution based on the classifier.

The CPS strategy is applied to three types of MOEAs. Thethree algorithms and their original versions are empiricallycompared on two test suites. The experimental results suggestthat the CPS can successfully improve the performance of theoriginal algorithms.

There is still some work that is worth further investigatingin the future. Firstly, the efficiency of CPS could be analyzed,secondly, more suitable data preparation strategies and moreclassification models should be tried, and thirdly, some strate-gies should be used to analyze CPS performance.

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0 2 4 6 8 1010

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10−2

10−1

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D v

alue

number of function evaluations(x2x103)

M=2M=3M=4M=5

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0 5 10

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Fig. 15: The IGD value versus FEs obtained by RM-MEDA with different values of M over 30 runs

TABLE IV: The statistical results of IGD and I−H metric values obtained by RM-MEDA-CPS and RM-MEDA, MOEA/D-MO-CPS and MOEA/D-MO, SMS-EMOA-CPS and SMS-EMOA on LZ1-LZ9

instance metric RM-MEDA-CPS RM-MEDA SMS-EMOA-CPS SMS-EMOA MOEA/D-MO-CPS MOEA/D-MOLZ1 IGD 1.54e-032.93e−05(+) 1.58e-033.23e−05 1.36e-032.04e−05(+) 1.45e-033.06e−05 1.35e-038.45e−06(+) 1.38e-031.46e−05

I−H 1.52e-037.34e−05(+) 1.63e-039.00e−05 1.13e-034.45e−05(+) 1.36e-035.89e−05 1.12e-032.61e−05(+) 1.20e-033.89e−05

LZ2 IGD 4.16e-023.09e−03(+) 4.66e-023.08e−03 3.52e-023.81e−03(+) 3.81e-023.72e−03 3.70e-035.51e−04(+) 4.62e-031.00e−03

I−H 7.66e-024.80e−03(+) 8.45e-025.63e−03 6.14e-027.01e−03(+) 6.75e-026.96e−03 9.00e-032.68e−03(+) 1.10e-022.76e−03


I−H 2.45e-024.39e−03(+) 2.61e-023.11e−03 4.32e-024.32e−03(+) 5.89e-024.10e−03 6.84e-032.21e−03(∼) 7.46e-032.34e−03


I−H 3.56e-027.95e−03(+) 4.19e-028.13e−03 4.43e-027.07e−03(+) 5.76e-023.73e−03 7.43e-033.02e−03(+) 9.94e-033.58e−03

LZ5 IGD 1.42e-022.13e−03(∼) 1.53e-023.06e−03 2.85e-023.96e−03(+) 3.51e-022.52e−03 7.23e-039.81e−04(∼) 7.72e-031.09e−03

I−H 2.44e-023.99e−03(∼) 2.62e-024.67e−03 4.11e-023.81e−03(+) 5.28e-023.23e−03 1.37e-022.80e−03(∼) 1.42e-022.94e−03

LZ6 IGD 1.64e-017.31e−02(+) 2.08e-016.38e−02 9.71e-026.67e−03(+) 1.00e-014.61e−03 5.84e-029.76e−03(∼) 5.96e-029.81e−03

I−H 3.35e-011.27e−01(+) 4.15e-019.47e−02 1.76e-015.86e−03(+) 2.07e-018.16e−03 1.26e-011.96e−02(∼) 1.30e-011.85e−02

LZ7 IGD 1.49e+003.09e−01(∼) 1.38e+004.46e−01 2.87e-011.61e−02(+) 3.17e-012.11e−02 1.42e-011.04e−01(∼) 1.17e-011.07e−01

I−H 1.08e+001.11e−01(∼) 1.04e+001.58e−01 4.68e-012.49e−02(+) 5.37e-013.35e−02 2.29e-011.84e−01(∼) 1.92e-011.77e−01

LZ8 IGD 1.61e-028.82e−03(∼) 2.23e-021.40e−02 1.40e-012.56e−02(+) 1.71e-011.76e−02 1.40e-021.26e−02(∼) 1.50e-021.30e−02

I−H 5.55e-022.47e−02(+) 7.12e-023.24e−02 2.23e-013.63e−02(+) 2.67e-012.59e−02 2.39e-022.41e−02(∼) 2.50e-022.33e−02

LZ9 IGD 4.40e-025.53e−03(∼) 4.47e-023.77e−03 3.89e-026.93e−03(∼) 4.18e-025.56e−03 4.79e-031.07e−03(+) 6.13e-031.42e−03

I−H 7.56e-027.67e−03(+) 8.06e-026.13e−03 6.60e-028.57e−03(+) 7.57e-028.07e−03 9.04e-032.42e−03(+) 1.16e-022.60e−03

+/− / ∼ IGD 5/0/4 8/0/1 5/0/4+/− / ∼ I−

H 7/0/2 9/0/0 4/0/5

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