an efficient conical area evolutionary algorithm for bi-objective optimization

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IEICE TRANS.FUNDAMENTALS,VOL.E95–A,NO.8AUGUST2012

LETTER

An E?cient Conical Area Evolutionary Algorithm for Bi-objective Optimization

Weiqin YING?a),Member,Xing XU??,Yuxiang FENG?,and Yu WU???b),Nonmembers

SUMMARY A conical area evolutionary algorithm(CAEA)is pre-sented to further improve computational e?ciencies of evolutionary algo-rithms for bi-objective optimization.CAEA partitions the objective space into a number of conical subregions and then solves a scalar subproblem in each subregion that uses a conical area indicator as its scalar objective. The local Pareto optimality of the solution with the minimal conical area in each subregion is proved.Experimental results on bi-objective prob-lems have shown that CAEA o?ers a signi?cantly higher computational e?ciency than the multi-objective evolutionary algorithm based on decom-position(MOEA/D)while CAEA competes well with MOEA/D in terms of solution quality.

key words:bi-objective optimization,evolutionary algorithm,computa-tional complexity,conical area,local Pareto optimality

1.Introduction

Many engineering design and decision making problems in-volve simultaneous optimization of multiple,possibly con-?icting,objective functions[1].For instance,two objectives are usually considered at the same time in the deployment of wireless sensor networks[2]:maximizing the sensing cov-erage and maximizing the lifetime.Such design problems can generally be formulated as a multiobjective optimization problem(MOP)of the form:

minimize y=F(x)=(f1(x),f2(x),···,f m(x))T

sub ject to x∈Ω(1) whereΩis the decision space,x∈Ωis a decision vector,y is the objective vector,and R m is the objective space.If m=2, a MOP becomes a bi-objective optimization problem,which is the most elementary case of MOPs.

Since there exists no single solution which optimizes all the objectives simultaneously,the best trade-o?solutions can be de?ned in terms of Pareto optimality.Let u,v∈R m, u is said to dominate v(written as u?v)if and only if ?i∈{1,...,m}:u i≤v i∧?j∈{1,...,m}:u j

Manuscript received February27,2012.

?The authors are with the School of Software Engineering, South China University of Technology,Guangzhou510006,China.

??The author is with the School of Information Engineering, Jingdezhen Ceramic Institute,Jingdezhen333000,China.

???The author is with the School of Computer Science and Ed-ucational Software,Guangzhou University,Guangzhou510006, China.

a)E-mail:yingweiqin@https://www.360docs.net/doc/2f12353835.html,

b)E-mail:wuyu@https://www.360docs.net/doc/2f12353835.html,

DOI:10.1587/transfun.E95.A.1420Pareto optimal solutions in the decision space,which is also called Pareto Set(PS),is often mapped to a Pareto Frontier (PF)F(A?)={F(x?)|x?∈A?}in the objective space[3].

Evolutionary algorithms(EAs)particularly seem suit-able to solve MOPs because they simultaneously deal with a set of possible solutions in the form of a population[4]. This allows to?nd an approximation of the entire PF in a single run,instead of a series of separate runs as in the case of the traditional techniques.The majority of existing multi-objective EAs are based on Pareto dominance.The non-dominated sorting genetic algorithm II(NSGA-II)[5] is one of the most popular multi-objective EAs based on dominance.It sorts the individuals in the population by their dominance depths to keep the closeness to the PF.Be-sides,NSGA-II prefers the individuals with larger crowding distances to preserve the diversity of distribution along the PF.The main disadvantage of multi-objective EAs based on dominance is the absence of e?cient algorithms to check for non-dominance in a population[4].Therefore,any EA based on dominance exhibits a serious degradation in per-formance as the size N of its population increases.

In traditional methods,a MOP can be decomposed into a number of scalar objective optimization subproblems. Zhang et al.borrow this idea from traditional methods to build a more e?cient EA,the multi-objective evolution-ary algorithm based on decomposition(MOEA/D)[6],[7]. MOEA/D explicitly decomposes a MOP into N scalar op-timization subproblems with the same decision spaceΩ.It solves these subproblems simultaneously by evolving a pop-ulation of solutions.At each generation,the population is composed of the N best solutions found so far for N sub-problems.An o?spring for each subproblem needs to update T best solutions found so far for its T neighboring subprob-lems.Since T is smaller than N,MOEA/D usually has the lower computational complexity than NSGA-II does.

In this letter,we propose a conical area evolutionary algorithm(CAEA)to further improve the computational ef-?ciencies of EAs for bi-objective optimization by introduc-ing a conical partition strategy and a conical area indicator. In contrast with MOEA/D,CAEA not only decomposes the MOP into N scalar subproblems,but also assigns each sub-problem an exclusive decision subsetΩ(k).Moreover,each subproblem uses the conical area as its scalar objective to ?nd a local nondominated solution in its own decision sub-set.Since each o?spring only needs to update the individual for the subproblem over the subset it belongs to,CAEA of-fers a lower computational complexity than MOEA/D does.

Copyright c 2012The Institute of Electronics,Information and Communication Engineers

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2.Conical Partition and Conical Area Indicator

2.1Conical Partition Strategy

To partition the decision space,an utopia point and a nadir point are ?rst required in the objective space.Given a set of solutions A ?Ω,the utopia point and the nadir point over

A are respectively F (A )=(f 1

(A ),f 2(A ))and F (A )=(f 1

(A ),f 2(A ))where f i (A )=min x ∈A

f i (x )and f i (A )=max x ∈A

f i (x ),i =1,2.For convenience,we transform an objec-tive vector y to y by y =y ?F (A ).After this transformation,the utopia point is at the origin (0,0).

De?nition 1(Observation vector).The observation vector for any transformed point y =(y 1,y 2)is V (y )=(v 1(y ),v 2(y ))

where v i (y )=y

i y 1

+y 2

,i =1,2.

Notice that the direction from the utopia point (0,0)to V (y )is consistent with the one from the utopia point to y .Furthermore,the observation vector has these features:v i (y )≥0,and v 1(y )+v 2(y )=1.Given a prescribed number of divisions N ,a series of uniformly distributed observa-tion vectors V (k )can be de?ned as V (k )=(k N ?1,1?k

N ?1),k =0...N ?1.Each vector V (k )

can induce a resulting coni-cal subregion where the observation vector of any point is

closer to V (k )

than the other vectors,as shown in Fig.1(a).De?nition 2(Conical subregion).For the central observa-tion vectors V (k )=(k N ?1,1?k

N ?1),k =0...N ?1,the region C ={y =(y 1,y 2)|y 1≥0∧y 2≥0}can be divided into N

conical subregions:C (k )={y ∈C |k ?0.5N ?1≤v 1(y )

N ?1},k =0...N ?1where k is called the index of the subregion C (k )

.Given a set A ,A (k )

={x ∈A |F (x )?F

(A )∈C (k )

}is called

the k -th decision subset of A .If x ∈A and F (x )?F (A )∈C (k )

,

we say that x lies in the conical subregion C (k )

.It can be inferred by Def.2that the index of the subregion in which a

solution x ∈A lies is k = (N ?1)(f 1(x )?f 1(A ))

f 1(x )?f 1(A )+f 2(x )?f 2(A )+12 where · denotes the bottom integral function.

2.2Conical Area Indicator and Local Pareto Optimality Among a variety of quality indicators,the hypervolume in-dicator,which measures the volume of the dominated por-tion of the objective space,is of exceptional interest as it possesses the highly desirable feature of strict Pareto com-pliance.However,It is computationally expensive.In this letter,we borrow the feature of hypervolume to present an e ?cient conical area indicator for bi-objective cases.De?nition 3(Conical area).Let y ∈C (k )

,0≤k ≤N ?1,y r denote the reference point which should be an approxi-mate in?nity point dominated by all feasible solutions,then

the conical area for y is the area of the portion C

(y )

=Fig.1(a)Conical subregions and (b)Conical areas in the transformed objective space.

{y ∈C (k )

|?(y ?y )∧y ?y r },written as S (y ).

In fact,S (y )represents the portion undominated by

y in the conical subregion C (k )

in which y lies.Let a =

k ?0.5N ?k ?0.5and b =k +0.5

N ?k ?1.5

,y =(y 1,y 2)∈C (k ),if 1≤k ≤N ?2,the portion C (y )is a concave quadrilateral OL (1)y U (2)as shown in Fig.1(b)where O =(0,0),L (1)=(b y 2,y

2)is the

intersection between the line y 2=y

2and the lower bound-ary of C (k ),and U (2)=(y 1,1a y

1)is the intersection between y 1=y

1and the upper boundary of C (k )

.Therefore,the

conical area S (y )is the sum of the areas of two triangles OL (1)U (1)and y U (2)U (1)where U (1)

=(a y 2,y 2)is the in-tersection between y 2=y 2and the upper boundary of C (k )

:

S (y )=0.5(b ?a )(y 2)2+0.51a

(y 1?a y 2)2

=0.51a y 12

+0.5b y 22?y 1y 2.

(2)

However,the reference point y r is required to calcu-late S (y )if y ∈C (0)or y ∈C (N ?1)

.The nadir point is used

to de?ne (103(f 1(A )?f 1(A )),103(f 2

(A )?f

2(A )))as the reference point y r .If y ∈C (0)

,the portion C (y )is a con-cave pentagon OL (1)y T (1)T (2)as shown in Fig.1(b)where T (1)=(y 1,y r 2)is the intersection between y 1=y

1and y 2=y r 2,and T (2)

=(0,y r 2)is the intersection between

y 2=y r

2and y 1=0.Thus,S (y )is the sum of areas of

a triangle OL (1)U (1)

and a rectangle y T (1)T (2)U (1)where U (1)

=(0,y

2)is the intersection between y 2=y 2and y 1=0:

S (y )=0.5b (y 2)2+(y r 2?y 2)y

1.Similarly,if y ∈C (N ?1),the conical area S (y )=0.51a (y 1)2

+(y r 1?y 1)y 2.

Lemma 1.Let y

=(y

1,y

2)∈C (k )and y =(y 1,y 2)∈C (k )

,

0≤k ≤N ?1,if y ?y ,then S (y )

).

Proof.First,the cases 1≤k ≤K ?2are taken into consider-ation.In these cases,It can be reached from Equation 2:S (y )?S (y )=0.51a (y 12

?y 12)+0.5b (y 22?y 22)?(y 1y 2?y 1y

2)

=0.5(y 1?y 1)((1a y 1?y 2)+(1a y

1?y 2))+0.5(y 2?y 2)((b y

2?y 1)+(b y 2?y

1)).

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Since y

∈C (k )

,then k ?0.5N ?1≤y

1y 1+y 2

N ?1by Def.2.It follows that a y 2≤y 1

1?y 2≥0and b y 2?y

1>0.

Similarly,we have 1a y 1?y 2≥0and b y 2?y 1>0by y ∈C (k )

.Besides,it can be inferred from y ?y

that y

1?y

1≥0and

y 2?y

2≥0.Therefore S (y )?S (y )≥0.

Then the method of proof by contradiction is used to show S (y )?S (y ) 0.Assume that S (y

)?S (y )=0.Thus (y 1?y 1)((1a y 1?y 2)+(1a y

1?y 2))=0and (y 2?

y 2)((b y 2?y 1)+(b y 2?y

1))=0.Because y ?y ,

?i ∈{1,2}:y i ≤y i ∧?j ∈{1,2}:y i

i can be reached.With-out loss of generality,suppose that y 1

1>0

and (y 1?y 1)((1a y 1?y 2)+(1a y 1?y

2))=0,we have (1a y

1?y 2)+(1a y 1?y

2)=0.Because 1a y 1?y 2≥0and 1a y

1?y 2≥0,

then 1a y 1?y 2=0and 1a y

1?y

2=0.Besides,it can be inferred that y 2?y 2=0from (y 2?y 2)((b y 2?y 1)+(b y 2?y

1))=0,b y 2?y 1>0and b y 2?y 1>0.By y 2?y 2=0,1a y 1?y 2=0and 1

a y 1?y 2=0,we have y 1=y 1∧y 2=y 2,which contradicts y ?y

.So the assumption S (y )?S (y

)=0must be false.

Since S (y )?S (y )≥0and S (y )?S (y ) 0,S (y )

)in the cases 1≤k ≤N ?2.Similarly,we also have S (y )

)in the cases k =

0and k =N ?1.Therefore S (y )

)in all cases. Theorem 1.Given a set A ∈Ω,let A (k )={x ∈A |F (x )?

F (A )∈C (k )

},0≤k ≤N ?1,if x ?∈A (k )and ?x ∈A (k ):S (F (x ?)?F (A ))≤S (F (x )?F (A )),then ??x ∈A (k ):x ?x ?.

Proof.The proof is by contradiction.First,assuming that there exists x ∈A (k )and x ?x ?,which means ?i ∈{1,2}:f i (x )≤f i (x ?)∧?j ∈{1,2}:f j (x )

and

F (x )?F (A )∈C (k )

.By F (x )?F (A )?F (x ?)?F (A )and Lemma 1,we have S (F (x ?)?F (A ))>S (F (x )?F (A ))which contradicts ?x ∈A (k ):S (F (x ?)?F (A ))≤S (F (x )?F (A )).Therefore the assumption must be false and the op-posite ??x ∈A (k ):x ?x ?is proven.

Theorem 1tells us that the conical area indicator pos-sesses the feature of local Pareto optimality,which means

the solution with the minimal conical area must be non-dominated in the decision subset.3.

Conical Area Evolutionary Algorithm

MOEA /D decomposes a MOP into N scalar optimization

subproblems with the same decision space Ωin the form:

minimize g moead ,k (x )=max 1≤i ≤2

{v (k )i |f i (x )?f

i (Ω)|}

sub ject to x ∈Ω.(3)

Thus an o ?spring for each subproblem needs to update T individuals for its T neighboring subproblems.In contrast

with MOEA /D,the proposed algorithm,CAEA,not only decomposes the MOP into N scalar subproblems,but also assigns each subproblem an exclusive decision subset Ω(k ).Moreover,the k -th scalar subproblem g caea ,k ,0≤k ≤N ?1,uses the conical area as its scalar objective in the form:

minimize g caea ,k (x )=S (F (x )?F (Ω))sub ject to x ∈Ω(k ).

(4)

The k -th individual in the population of CAEA is just the best solution x k found so far for the subproblem g caea ,k .Only one o ?spring x is reproduced at each generation of CAEA.Since the solutions with smaller conical areas should be pre-ferred in the neighboring conical subregions,CAEA uses an area-based tournament selection AreaTour ().AreaTour ()selects the one with the larger di ?erence between the aver-age conical areas of its two nearest neighbors and its own conical area from two random individuals,as shown below.AreaTour ()

Output:k -the index of the selected solution in P .1)Randomly select two indexes a ,b from 0...N ?1.2)Set a 1=a ?1and a 2=a +1.If a =0then a 1=2else if a =N ?1then a 2=N ?3.Then b 1and b 2are set similarly in terms of b .

3)If S (F (x a 1)?F

)+S (F (x a 2)?F )

2

?S (F (x a )?F )>S (F (x b 1)?F )+S (F (x b 2)?F )

2

?S (F (x b )?F )then k =a

else k =b .

Due to the fact that each o ?spring x lies in the conical subregion of only one subproblem,CAEA only needs to up-date the individual associated with this subproblem in terms of conical areas.The conical partition keeps the diversity of distribution and the minimization of conical area as the scalar objective maintains the local Pareto optimality of in-dividuals in the population.The proposed algorithm works as follows.

Step 1Initialization:

Step 1.1Generate N initial solutions

A ={z 0,...,z N ?1}by randomly sampling from Ω.

Step 1.2Initialize F =(f 1,f 2)and F =(f 1,f

2)

by setting f i =min 0≤j ≤N ?1

f i (z j ),f i =max 0≤j ≤N ?1

f i (z j

).

Step 1.3Create a population P ={x 0,...,x N ?1

}where x 0=...=x N ?1=null .Set A =A .For each k =0...N ?1:if A (k ) ?,then x k =arg min z ∈A (k )

S (F (z )?F

)and A

=A

?x k

.For each k =0...N ?1:if x k =null ,then

x k =arg min z ∈A

d (V (F (z )?F

),V (k ))and A =A ?x k .Set nE =N and L = N

10

.Step 2Evolution:

Step 2.1Selection:Set iL =nE mod L .If iL <2,then p 1=iL ×(N ?1)else p 1=AreaTour ().Set p 2=AreaTour ().

Step 2.2Reproduction:Produce an o ?spring x from

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x p 1and x p 2using genetic operators.Set nE =nE +1.Step 2.3Update of F and F :For each i =1,2,if f i (x )f i ,then set f i =f i (x ).

Step 2.4Update of x k :Set k 1= (N ?1)(f 1

(x )?f 1

)

f 1(x )?f 1

+f 2(x )?f

2

+12

and k 2=

(N ?1)(f 1(x k 1)?f

1)

f 1(x k 1)?f 1+f 2(x k 1)?f 2+

1

2 .

If k 2 k 1∨

S (F (x )?F )

),then set x k 1=x .Step 3Stoping Criterion:If the stopping criterion is not satis?ed,then go to Step 2.Step 4Post-processing:Remove the dominated solutions

from the ?nal population P to obtain the approximate PS P .Then output A and the approximate PF F (P ).where · denotes the top integral function,d (·,·)represents the Euclidean distance between two vectors in Step 1.3and mod in Step 2.1denotes the modulus operator.Since the local non-dominated solution with the minimal conical area in a subregion may be dominated by a solution in another subregion occasionally,the post-processing in Step 4is nec-essary to obtain a ?nal set P of global non-dominated so-lutions to approximate the PS.Since it is trivial and time-consuming to ?nd the exact utopian point of the whole de-cision space Ω,a set Ω of all solutions discovered so far by CAEA are used as a substitute for Ωin fact.

The main operation in NSGA-II is the fast non-dominated sorting whose computational complexity is O (N 2)at each generation for bi-objective cases.For an o ?-spring of each subproblem,the major computational costs of MOEA /D are to update the individuals of its T neighboring subproblems.Thus MOEA /D has the computational com-plexity of O (NT )for N o ?springs at each generation.The major computational costs of CAEA at each generation are the update operation in Step 2.4.Since each o ?spring needs to update the individual of only one subproblems in Step 2.4,the computational complexity of CAEA is O (1)for an o ?spring and O (N )for N o ?springs.Therefore,if NSGA-II,MOEA /D and CAEA use the same population size and the same stopping criterion,the ratio among their computational complexities is O (N 2):O (NT ):O (N )=O (N ):O (T ):O (1).Since usually T

Experimental Results and Analysis

In this section,the proposed algorithm CAEA is com-pared with MOEA /D and NSGA-II on six widely used bi-objective test problems:ZDT 1(convex),ZDT 2(non-convex),ZDT 3(nonconvex,disconnected),ZDT 4(convex,multimodal),ZDT 6(nonconvex,nonuniformly spaced)[8],and LZ 2(complicated PS shape)[9].In our experiments,ZDT 1?3use 30decision variables while ZDT 4,6and LZ 2use 10decision variables.All three algorithms are implemented in C ++and run on an Intel Core i72.70GHz PC with 4GB RAM.For the sake of fairness,the population size N =

100

Fig.2Evolution of the average IGD-metric values with the number of function evaluations in three algorithms for six test instances.

and the stopping criterion is to reach 25000function eval-uations for all the test instances in all three algorithms in our experiments.The simulated binary crossover (SBX)and polynomial mutation are used in three algorithms to gener-ate a new o ?spring.The distribution indexes in both SBX and the polynomial mutation are set to be 20.The crossover rate is 1.00,while the mutation rate is 1/n ,where n is the number of decision variables.The size T of the neighbor-hood of each weight vector in MOEA /D is set to 20,which is the same as in [6].To assess the overall performance of three algorithms,a total of 30statistically independent runs of each algorithm have been performed for each test instance.

A performance metric,the inverted generational dis-tance (IGD)[10],is adopted to assess the overall perfor-mance of the algorithms in our experimental studies.Mea-suring how far the elements in the PF are from those in the discovered frontier,IGD is de?ned as IGD (P ?,

B )=1|P ?|Σv ∈P ?

d (v,B ),wher

e P ?is a set o

f uniformly distributed points alon

g the PF,B is the discovered frontier,and d (v,B )is the minimum Euclidean distance between v and the el-ements in B .A low value of IGD (P ?,B )indicates that B must be very close to the PF and cover nearly all the exten-sion of the PF.Since the PFs of all six test cases are pro-

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2012

Fig.3Plot of the ?nal population and the nondominated frontier found by CAEA in the run with the lowest IGD-metric value.Table 1Average CPU time (in seconds)spent by three algorithms.ZDT1ZDT2ZDT3ZDT4ZDT6LZ2CAEA 0.7960.8180.7960.4450.4100.372MOEA /D 2.178 2.190 2.370 2.036 2.004 1.882NSGA-II

35.54

30.30

35.32

27.01

25.48

27.85

vided [8],[9],1000uniformly distributed points along the PF have been choosen to constitute P ?for each case in our experiments.

Figure 2presents the evolution of the average IGD-metric values of the population with the growth of the num-ber of function evaluations in three algorithms for each test instance.Since the IGD-metric values are usually very small,we use the logarithmic scale for the axes of the IGD-metric value in Fig.2to clearly reveal the di ?erences.It is evident from Fig.2that CAEA obtains obviously lower IGD-metric values than MOEA /D and NSGA-II for all the test instances except ZDT 3,which suggests that CAEA is more e ?ective than MOEA /D and NSGA-II to ?nd the ap-proximate frontier with both a better closeness and a better distribution for all the test instances except ZDT 3.In the only case of ZDT 3,CAEA is slightly worse than NSGA-II in terms of IGD-metric although CAEA still outperforms MOEA /D.It could be attributed to the fact that the PF of ZDT 3consists of ?ve disconnected parts,which leads to that there are a fewer number of nondominated solutions in the Pareto frontier obtained by CAEA than NSGA-II since some individuals in the ?nal populations found by CAEA might be dominated by others and be eliminated during the post-processing as shown in Fig.3.

Table 1shows the average CPU time in seconds spent by each algorithm for each test problem.Table 1clearly indicates that CAEA among three algorithms needs least CPU time for all test problems.On average,CAEA requires respectively about 35.8%of the time MOEA /D needs and 2.38%of that NSGA-II does for the previous three test cases with 30decision variables.Moreover,there is a larger im-provement in CPU time for the last three cases with only 10decision variables.As we can see,CAEA runs respectively about ?ve times as fast as MOEA /D and more than 60times as fast as NSGA-II for these three cases.This observation is consistent with our analysis of the computational complexi-ties of CAEA,MOEA /D and NSGA-II in Sect.3.

As a whole,we can claim that CAEA is computa-

tionally much more e ?cient than MOEA /D and NSGA-II while CAEA performs similarly to and even outperforms MOEA /D and NSGA-II in terms of the solution quality.5.

Conclusions

In this letter,the algorithm CAEA is proposed to further im-prove computational e ?ciencies of EAs for bi-objective op-timization.In CAEA,the objective space of a bi-objective problem is partitioned into a number of conical subregions.Then the bi-objective problem in each subregion is con-verted to a scalar subproblem that adopts the conical area indicator as its scalar objective.We have also proved the lo-cal Pareto optimality of the solution with the minimal con-ical area in each subregion.The experimental results in-dicate that CAEA o ?ers the signi?cantly higher computa-tional e ?ciency than both MOEA /D and NSGA-II while CAEA competes well with MOEA /D and NSGA-II in terms of solution quality.Our future research will focus on ex-panding the study to the cases of three and more objectives.Acknowledgments

This work is supported by Natural Science Foundation of Guangdong Province,China (No.S2011040002472),Doctoral Fund of Ministry of Education of China (No.20110172120035),the Fundamental Research Funds for the Central Universities (No.2011ZM0107),and the Youth Science Foundation of Jiangxi Provincial Department of Education (No.GJJ12514).

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