a novel multi-objective estimation of ...novel multi-objective estimation of distribution algorithm...

International Journal of InnovativeComputing, Information and Control ICIC International c©2013 ISSN 1349-4198Volume 9, Number 6, June 2013 pp. 2543–2566

A NOVEL MULTI-OBJECTIVE ESTIMATION OF DISTRIBUTIONALGORITHM BASED ON SENSITIVITY OF OBJECTIVE FUNCTION

Hyunshik Seo and Chaewoo Lee

Department of Electrical and Computer EngineeringAjou University

San 5, Woncheon-dong, Youngtong-gu, Suwon 443-749, Koreamushs01; cwlee @ajou.ac.kr

Received March 2012; revised July 2012

Abstract. Most multi-objective optimization problems (MOPs) have a set of optimaltrade-off solutions known as the Pareto-optimal solutions since the objectives in MOPsare usually in conflict with one another. Recently proposed estimation of distributionalgorithms (EDAs) build a probability distribution model based on the probabilistic infor-mation about decision variables of solutions, and then produce new solutions from themodel. In the algorithms, the modeling technique enables the initial large search spaceto be reduced to small promising solution space during the search. However, the exist-ing EDAs might be inefficient at generating the promising solutions since they dependon the information extracted from the decision variables of current solutions expected toapproach the optimal solutions. For effective modeling of the promising solutions, wefirstly introduce new information about the relationship between decision variables andobjective functions; this information is called sensitivity of objective function. Secondly,we propose a multi-objective estimation of distribution algorithm based on the sensitivityof objective function (MOEDA-S). In the MOEDA-S, the sensitivity-based distributionmodeling adapts to the current search strategy such that the convergence-focused search atthe beginning part of the search is changed to a diversity-focused search at the latter partof the search. MOEDA-S is compared with two other leading multi-objective evolution-ary algorithms on a set of test instances. The simulation results show that MOEDA-Soutperforms the two compared algorithms in terms of both convergence and diversity per-formances of the solutions.Keywords: Multi-objective optimization problems, Multi-objective evolutionary algo-rithms, Estimation of distribution algorithms, Variables sensitivity-based modeling

1. Introduction. Many real-world problems, especially in the engineering area, includemultiple objectives conflicting with each other. Unlike in single-objective optimizationproblems, in multi-objective problems (MOPs), there is no unique solution which opti-mizes all the objectives at the same time. Many studies have focused on a reasonableform for optimal solution, and on efficient techniques to obtain the optimal solution [1, 2]in MOPs. This paper deals with an effective search technique for the following continuousMOP:

minimize F(x) = (f1(x), f1(x), . . . , fM(x))T (1)

subject to x ∈ Ω

where Ω is the decision variable space and x is the decision variable vector. F : Ω →RM changes the decision variable space into the M -dimensional objective space with Mobjectives.

Multiple objectives in MOP have a conflicting property that they cannot be optimizedat the same time. Pareto has proposed a useful approach for the MOP by defining

2543

2544 H. SEO AND C. LEE

the Pareto dominance and the Pareto optimality [3]. Let u,v ∈ RM be two vectors inobjective space; u is said to dominate v, which means u is a better solution than v, ifu 6= v and ui ≤ vi for all i ∈ 1, . . . ,M. x∗ is called Pareto optimal solution if there is noother x ∈ Ω that dominates x∗ ∈ Ω. The set of all the Pareto optimal solutions is calledthe Pareto set (PS), and the image of PS on the objective space is called the Pareto front(PF).Multi-objective evolutionary algorithms (MOEAs) aim at obtaining an approximation

to the PF; this is known as a complicated task to find many well-spread multiple optimalsolutions at a time [1, 4]. In such a situation, existing MOEAs focused on the followingtwo performance requirements:

• Convergence to PF: This requirement is related to how fast the solutions approachthe PF. If the solutions could have a high convergence speed, the final solutionsmight be close to the PF.

• Diversity of solutions: This requirement indicates how well distributed the finalsolutions are on search space. The set of final solutions with high diversity offersdecision makers a wide variety of solution options.

Most of the popular MOEAs such as NSGAII, SPEAII and PAES basically use thedominance-based fitness measure to guarantee the convergence performance in the finalsolutions. The diversity maintenance techniques such as niching and fitness sharing strat-egy [6], crowding distance [5], K-nearest-neighbor method [21], and ε-domination method[7, 22] have been also proposed to maintain well-spread solutions during the search. Somelocal search heuristics have combined with MOEAs to speed up the convergence [7, 8]or to sustain the diversity of solutions [9]. In addition, some efforts have been made toanalyze the change of the two performances on solutions during the search [10] and tobalance between them during the search [4, 12].In the existing MOEAs, genetic operators such as crossover and mutation have been

generally adopted to create now solutions. The operators directly recombine the selectedsolutions and change the solutions slightly without the consideration of the characteristicsof MOPs. If the characteristics of the given MOP such as variable linkage in solutions areobtained during the search, we can search for the optimal solutions efficiently by gener-ating new solutions based on the characteristics information. In [23], the performance ofthe conventional operators has been measured on some MOPs with variable linkage. Thesimulation results suggest that the operators might be inefficient at generating promisingsolutions in the MOPs, and the studies about effective reproduction of new solutions arereally crucial for finding optimal solutions.Recently, new stochastic optimization techniques named estimation of distribution al-

gorithms (EDAs) have been introduced for efficiently generating new solutions [11, 13, 14,15, 24, 25]. Instead of commonly used genetic operators such as crossover and mutation,EDAs search for optimal solutions by sampling new promising solutions based on a proba-bility distribution model. The model is built based on the probabilistic information aboutvariable linkage extracted from solutions such as dependence between the variables anddistribution patterns of them. In [11, 15], the conditional probability distribution modelbased on the dependence among the decision variables is considered to generate promis-ing solutions. In [24, 25], the Q-bit vector inspired from quantum computing is used torepresent a probability distribution of all decision variables in each solution that is 0 or1. New solutions are generated based on the probability distributions updated referringto the selected best group solutions. In [13, 14], a promising solution space is estimatedby properly extending the distribution of current solutions in the decision variable space.

NOVEL MULTI-OBJECTIVE ESTIMATION OF DISTRIBUTION ALGORITHM 2545

Basically, EDAs are based on the condition that the solutions naturally approach thePareto optimal solutions across generations, and the model built based on the solutionsis getting more and more characteristics of the optimal solutions [19]. Thus, the new so-lutions sampled from the model have a higher probability of being near-optimal solutionsacross generations. In other words, the reproduction of solutions from the probabilitydistribution model enables the initial large search space to be reduced to a small, promis-ing solution space. However, the existing EDAs are inefficient at generating promisingsolutions since they depend on the information about variable linkage extracted fromcurrent solutions expected to approach optimal solutions. Especially, in the beginningpart of the search, the search ability might be considerably low since the model could notrepresent the characteristic of the optimal solutions at the time. In this situation, if wecan obtain some information that can guide the search to the optimal solutions, such asthe good direction of variable-change and effectiveness of variables, the information willbe beneficial to generate the promising solutions persistently.

In this paper, we propose a new estimation of distribution algorithm named a novelestimation of distribution algorithm based on sensitivity of objective function (MOEDA-S) for approximating the Pareto-optimal solutions rapidly. The contributions of this paperinclude the following.

• For effective distribution modeling of the promising solutions, we introduce newinformation (never used in previous works) about the relationship between the ob-jective function and the decision variables, the sensitivity of objective function. Thesensitivity information is a metric that represents a level of sensitivity of objectivefunctions to the variation of decision variables. With this metric, we can evaluatethe relative importance of the variables for moving the candidate solutions towardsthe optimal solutions.

• Based on the sensitivity of objective function, we propose a multi-objective estima-tion of distribution algorithm with a novel distribution modeling technique. Thedistribution modeling is implemented to estimate the promising solution space byvarying the variables with high sensitivity of objective function.

• The proposed distribution modeling based on the sensitivity information is conductedconsidering the current search situation for rapid approximation of the optimal so-lutions. According to the current search strategy decided at every iteration, at thebeginning part of the search a convergence-focused search is implemented to esti-mate the promising solution space rapidly, and at the latter part of the search adiversity-focused search is implemented to search the diverse solutions.

The paper is organized as follows. Section 2 introduces the related works of EDAs andSection 3 presents the proposed MOEDA-S. Section 4 compares the MOEDA-S with twoother leading MOEAs, RM-MEDA and NSGAII, on a set of test instances, and Section 5concludes this paper.

2. Related Works. Estimation of distribution algorithms (EDAs) are new evolution-ary algorithms based on the model-based offspring generation method. EDAs employa reproduction operator based on a probability distribution model instead of crossoverand mutation operators used in the conventional MOEAs. Since the model is newlybuilt in every generation based on current solutions, it can gradually represent the exactcharacteristics of optimal solutions. Therefore, the core of EDAs lies in extracting usefulinformation from current solutions and building a model from the information to representcharacteristics of optimal solutions as accurately and quickly as possible.


Various modeling techniques have been proposed to generate near-optimal solutionseffectively. The dependence among decision variables has been used to build their proba-bility distribution model. The bayesian optimization algorithm (BOA) [15] uses a variabledependence-based model to sample new solutions in a single-objective problem with bi-nary variables. The variable dependence obtained from current solutions is representedwith a directed acyclic graph based on Bayesian networks. With the variable dependence-based graph model, independent decision variables are firstly set randomly, and then otherdependent decision variables are determined based on their conditional probability distri-butions depending on the pre-decided variables. As extended versions of BOA for MOP,multi-objective BOA (mBOA) [16], multi-objective hierarchical Bayesian optimization al-gorithm (mohBOA) [11], and multi-objective mixture-based iterated density estimationalgorithm (MIEDA) [17] have been proposed. In mBOA, the dominance-based selec-tion with crowding estimation technique proposed in NSGAII is applied to a new set ofsolutions sampled from a variable dependence-based probability model for maintainingdiversity of solutions. The mohBOA uses a clustering technique to improve diversityperformance in problems with a hierarchical dependence between decision variables. Thecomplex dependence is evaluated at each level of the relationship and integrated into aglobal structure to build a hierarchical dependence model. And then, the sampling ofdecision variables is implemented level by level based on the model. The MIEDA employsa mixture distribution model integrating multiple variable dependence-based probabilitymodels of clusters of solutions. The mixture distribution model is built as a weighted sumof the probability models at different positions in objective space, and various solutionsdistributed over the whole objective space can be generated from the integrated model.With the variable dependence-based modeling, the large search space could be graduallyreduced to a small, promising space. However, the task of getting variable dependencefrom the solutions is generally time consuming. As a result, the search ability is consider-ably low especially in the beginning part of the search since the model could not representthe variable dependence of the optimal solutions at the time.The distribution modeling of promising solutions has been considered. In regularity

model-based multi-objective EDA (RM-MEDA) [14], a probability distribution modelis built by enlarging the distribution of current solutions. The candidate solutions aregradually distributed with some pattern in decision variable space since the decision vari-ables have a link between one another. Therefore, if the distribution of decision variablecould be modeled properly, and if new solutions could be sampled around the currentdistribution model, search can be effectively focused on the promising space. Principlecomponent analysis (PCA) is employed to evaluate the distribution of solutions in re-duced dimensional space and to extend the distribution to a promising space which is notvery large. From these procedures, the distribution pattern of solutions is learned, andthe promising space is estimated effectively. Recently, the RM-MEDA has been improvedby combining it with an elitist strategy [20]. The distribution model is built not withthe entire population but part of the population, with better performance to estimate amore accurate model. The distribution modeling enables the initial large search spaceto be reduced to a small, promising space efficiently by estimating a new search spacewith a properly extended space from the distribution of current solutions. However, thismodeling technique is also inefficient at generating promising solutions since they dependon the information extracted from the decision variables of current solutions expected toapproach optimal solutions.Quantum-inspired evolutionary algorithms (QEAs) [24, 25] could be considered to be

EDA since they generate new solutions from the probability model in every generation.The QEAs maintain a number of the probability models, which is the same as the number


of solutions, called Q-bit vectors representing a probabilities of all decision variables ofsolutions as 0 or 1. The new solutions are generated based on the Q-bit vectors updatedreferring to the vectors of non-dominated solutions, in every generation. Therefore, theQ-bit vectors gradually converge on those representing optimal solutions. However, thesearch in QEAs cannot be assisted by the information about variable linkage since themodels in QEAs evolve individually without the consideration of the information aboutvariable linkage. Additionally, the convergence of the probability models to those for theoptimal solution requires a considerably long time.

To overcome the limitations mentioned above, new information guiding the search tothe optimal solutions is required for generating promising solutions. In the next section,we introduce a novel multi-objective estimation of distribution algorithm based on new in-formation, sensitivity of objective function, for persistently fast approximation of optimalsolutions.

3. Variables Sensitivity-Based Estimation of Distribution Algorithm.

3.1. Basic idea. In the proposed MOEDA-S, a distribution model of promising solutionspace is built based on the sensitivity of objective function to the variation of decisionvariables of solutions. The variables with high sensitivity are generally different in differentMOPs and at different moments even in a MOP. In such a situation, MOEDA-S makesnew solutions by varying mainly the variables with high sensitivity to search promisingsolutions effectively. The sensitivity can be obtained with a variation of values of objectivefunctions induced by that of decision variables. Here, the directions of variable changeare set with the principle vectors of variable change obtained by PCA. Figure 1 showsthe direction of evolution in objective space and the vector of variable change with highsensitivity of the evolution. At the beginning part in search, the population located farfrom PF tends to evolve toward PF as shown in Figure 1(a). At this time, it is efficientthat the variables are changed only to the direction with high sensitivity of the evolution(the variable changes might reduce the objective values). The vector of the variable

Figure 1. The direction of evolution in objective space and the vector ofvariable changes with high sensitivity of the evolution in decision variablespace


changes might indicate PS, the image of PF in decision variable space, as shown in Figure1(b). The omni-directional search can be called a convergence-focused search consideringthe magnitude and direction of the sensitivity information simultaneously.As the search progresses and as the population approaches PF, the evolution in objective

space occurs more toward the surroundings of PF than toward PF, as shown in Figure1(c), since the space for evolution toward PF is getting smaller. At this time, the variablesshould be changed to both directions that with high sensitivity of the evolution towardthe surroundings of PF. The vector of variable changes that causes the evolution mightindicate a surrounding of PS as shown in Figure 1(d). The bi-directional search canbe called a diversity-focused search considering only the magnitude of the sensitivity ofobjective function.To employ different search strategies in the beginning part and in the latter part of

the search, the search strategy should be changed from the convergence-focused searchto the diversity-focused search at the proper moment during search. In MOEDA-S, thechange of search strategy occurs when the volume of population is reduced to below afixed level. This is based on the fact that, as the search progresses, a large search spacemay be converged into a relatively small space near optimal solutions, and the volume ofpopulation decreases. From the next subsection, we address the procedure of MOEDA-S including the distribution modeling, the reproduction, and the search-strategy changetechnique.

3.2. Algorithm framework. The proposed MOEDA-S searches for optimal solutionswith a population of S solutions until a predetermined generation number is reached.MOEDA-S adopts the following widely used EDA framework.

Step 1) Initialization: Randomly generate an initial population for the first gener-ation, P 1 with S solutions.

Step 2) Modeling: Build the distribution model based on information extracted fromcurrent population of t-th generation, P t.

Step 3) Reproduction: Generate a set of new solutions Qt by sampling new S solu-tions from the model built in Step 2.

Step 4) Selection: Select S solutions from P t∪Qt and set P t+1 by replacing all the

solutions in P t with the selected solutions.Step 5) Termination condition: If the predetermined generation is reached, stopand return all the solutions. Otherwise, go to Step 2.

In the following subsection, we give details of modeling, reproduction, and selection.

3.3. Modeling. The distribution model of promising solutions is built based on statis-tical information extracted from current solutions to sampling new solutions. MOEDA-Semploys the sensitivity of objective function to model the promising space rapidly. Themodeling in MOEDA-S works as follows.

Step 2.1) Search-strategy change: Decide if search strategy should change fromthe convergence-focused search to the diversity-focused search or not at current gen-eration.

Step 2.2) Clustering: Divide the current population P t with K subpopultions (clus-ters), P t

1, Pt2, . . . , P

tK .

Step 2.3) PCA: Obtain the principle vectors of variable changes in each population.Step 2.4) Sensitivity-based modeling: Evaluate the sensitivity of objective func-tion on the principle vectors and build the distribution model based on the sensitivity.


Before modeling, MOEDA-S should decide the search strategy since the modeling isimplemented differently according to the search strategy. In general, solutions are dis-tributed in wide space far from the optimal solutions at the beginning part of search andconverge in a relatively small promising space as search progresses. Thus, the search-strategy change occurs when the volume of current population is reduced sufficientlycompared to that of the population at the first generation. In MOEDA-S for MOP withN -dimensional decision variable space, the search-strategy change occurs once at the firstgeneration that following condition is met.

N∏i=1

(maxx∈Pt

xi − minx∈Pt

xi)

≤ ε

N∏i=1

(maxx∈P1

xi − minx∈P1

xi)

(2)

where xi is the i-th variable of solution x ∈ Pt and a volume reduction ratio for search-strategy change, ε, is an algorithm parameter.

In clustering procedure, the population is divided into multiple subpopulations ac-cording to their locations in objective space, which helps to maintain the diversity inpopulation. The widely used k-means clustering method [18] is used to cluster P t into Ksubpopulations, P t

1, Pt2, . . . , P

tK . The number of subpopulation K is randomly chosen in

1, 2, . . . , Kmax and Kmax is an algorithm parameter.In each subpopulation, through the PCA, the principle vectors of variable changes are

obtained based on the distribution of solutions. The sensitivity of objective function isevaluated as variation of values of objective functions caused by variation of decisionvariables on the principle vectors, and then the distribution model is built based on thesensitivity. To obtain the sensitivity, at first, the sample mean xtk and sample covariancematrix Covt

k of the solutions in k-th population Ptk are calculated as follows:

xtk =1

|Ptk|∑x∈Pt

k

x (3)

Covtk =

1

|Ptk| − 1

∑x∈Pt

k

(x−xtk)(x− xtk)T

where |Ptk| is the cardinality of Pt

k. From the covariance matrix for Ptk, Covt

k, withN -dimensional decision variable space, the eigenvalues λtk,1 ≥ λtk,2 ≥ · · · ≥ λtk,N and thecorresponding eigenvectors V t

k,1, Vtk,2, · · · , V t

k,N can be obtained, which is the informationabout the new vectors listed in descending order by variation from the mean xtk. Theeigenvector V t

k,i is a the unit directional vector with i-th highest rank in terms of the

variance. The eigenvalue λtk,i is a variance value of the V tk,i. Suppose that Pt

k in N -dimensional decision variable space can be considered as a hypercube of reduced numberof dimension, nk. The nk is the smallest integer satisfying the following inequality

nk∑i=1

λtk,i ≥ θN∑i=1

λtk,i (4)

where 0 ≤ θ ≤ 1 is the variance consideration ratio for the PCA, which is an algorithmparameter. From (4), Pt

k can be expressed as a hypercube with at least 100θ% varianceby using the nk principle vectors. The directional vectors of the principle vectors areV tk,1, V

tk,2, · · · , V t

k,nkand their variances are λtk,1 ≥ λtk,2 ≥ · · · ≥ λtk,nk

, respectively.Based on the principle vectors obtained from PCA, the sensitivity of objective function

to decision variables is evaluated with its magnitude and direction separately to build dis-tribution model afterward. At the beginning part of search for convergence-focused search,both the magnitude and the direction of the sensitivity are used for omni-directional


Figure 2. The calculation of the effectiveness of a principle vector

search, while at the latter part of search for diversity-focused search, the magnitude ofthe sensitivity is employed for bi-directional search. The magnitude of the sensitivity foreach principle vector is obtained from normalization of effectiveness value for it. As shownin Figure 2, at first, the effectiveness for a principle vector v is evaluated as variation ofobjective values (d) caused by the variation of variables between the maximum (point a)and minimum values (point b) on the principle vector v.Let the coordination of the maximum and the minimum values on i-th principle vec-

tor be atk,i and btk,i respectively. The effectiveness for the i-th principle vector, Etk,i, is

calculated as follows:

Etk,i =

1∣∣dtk,i∣∣(

M∑j=1

(Fj

(atk,i)− Fj

(btk,i))2)1/2

(5)

where the coordinates of maximum value and minimum values on the i-th principle vectorare calculated in

atk,i = utk,iVtk,i +

nk∑n=1,n6=i

utk,n + ltk,n2

V tk,n (6)

btk,i = ltk,iVtk,i +

nk∑n=1,n6=i

utk,n + ltk,n2

V tk,n.

And the range of the projections of the points in Ptk onto the V t

k,i and their difference areobtained as follows:

ltk,i = minx∈Pt

k

xTV t

k,i

(7)

utk,i = maxx∈Pt

k

xTV t

k,i

dtk,i = utk,i − ltk,i, i = 1, . . . , nk.

The magnitude of the sensitivity of objective function for V tk,i is obtained from normal-

ization of its effectiveness as follows:

Stk,i =

Etk,i −minEt

k,jmaxEt

k,j −minEtk,j

, j = 1, . . . , nk (8)

where Etk = Et

k,1 , Etk,2, . . . , E

tk,nk

is the effectiveness set for principle vectors in Ptk.

The direction of the sensitivity represents whether the variable changes in direction ofeach principle vector lead solutions to the optimal solutions or to the opposite direction


of the optimal solutions. Let Dtk,i be an index to represent the direction of the sensitivity

for V tk,i. D

tk,i is obtained from

Dtk,i =

1, ifM∑i=1

(Fj

(atk,i)− Fj

(btk,i))

≥ 0

−1, otherwise.(9)

For effective search for PF in the minimization problem considered in this paper, whenthe summation of variations of values of objective functions caused by the variable changesto V t

k,i is positive (when Dtk,i is 1), it is sufficient that the variables should be changed to

the opposite direction of the principle vector, and vice versa.Based on the sensitivity of objective function for each principle vector, a distribution

model is built as an extended distribution model of distribution of each subpopulation.The distribution model for Pt

k, ζtk ∈ RN , can be described by

ζtk = Φtk + δtk (10)

where Φtk is the extended distribution model and δtk ∼ N(0, σt

kI) is a N -dimensionalzero-mean Gaussian noise vector. The variance considered in model for Pt

k, σtk, is set as

follows:

σtk =

1

N − nk

N∑i=nk+1

λtk,i (11)

At first, the distribution model for k-th subpopulation Ptk, ψ

tk, can be set based on the

principle vectors and their ranges obtained before as follows.

ψtk =

x ∈ RN

∣∣∣∣x =

nk∑i=1

αtk,iV

tk,i, l

tk,i ≤αt

k,i ≤ utk,i, i = 1, . . . , nk

(12)

The distribution model for sampling new solutions, Φtk, is built by extending ψt

k basedon the sensitivity for each principle vector. Figure 3 shows the change from ψt

k to Φtk.

At the beginning part of search for convergence-focused search, the model is extended to

Figure 3. The change from current from ψtk to Φt

k


only optimal solutions. And at the latter part of search for diversity-focused search, it isextended to both directions towards the surrounding of the population.The distribution model Φt

k is set as follows:

Φtk =

x ∈ RN

∣∣∣∣x =

nk∑i=1

γtk,iVtk,i, l

∗tk,i ≤γtk,i ≤ u∗tk,i, i = 1, . . . , nk

(13)

where l∗tk,i and u∗tk,i represent new extended range for variable variation that is computed

differently according to the current search strategy. Basically, the range of variable varia-tion on each principle vector is extended in proportion to the sensitivity for each principlevector. In addition, at the beginning part of search, the range extension occurs to only op-timal solutions (direction that objective values decreases) and at the latter part of searchit occurs to both directions toward the surrounding of the population. At the beginningpart of search, the new range of variable variation on i-th principle vector is set as

l∗tk,i =

ltk,i −(utk,i − ltk,i

)Stk,i

1+βnk∏i=1

(utk,i−ltk,i)(1+St

k,i)

1/nk

, if Dtk,i = 1

ltk,i, if Dtk,i = −1

,

u∗tk,i =

utk,i, if Dt

k,i = 1

utk,i +(utk,i − ltk,i

)Stk,i

1+βnk∏i=1


k,i)

1/nk

, if Dtk,i = −1.

(14)

And at the latter part of search,

l∗tk,i = ltk,i −utk,i−ltk,i

2Stk,i

1+βnk∏i=1


k,i)

1/nk

,

u∗tk,i = utk,i +utk,i−ltk,i

2Stk,i

1+βnk∏i=1


k,i)

1/nk

.

(15)

In (14) and (15), the extension ratio of the distribution of population for new searchspace β is an algorithm parameter. Φt

k is extended along each of the nk principle vectorssuch that the volume of Φt

k is 100β% larger than that of ψtk. In this paper, we set β = 1

such that the volume of new search space is 2 times larger than that of the space of currentpopulation.

3.4. Reproduction. New S solutions are generated based on our distribution model Φtk

in this step as follows.

Step 3.1) Model selection: Randomly select one subpopulation among K subpop-ulations.

Step 3.2) Reproduction: Randomly generate a new solution x from the selectedsubpopulation’s model Φt

k.Step 3.3) Boundary check: If the new solution x is outside the original ranges ofvariables, x is set randomly within the ranges.

One subpopulation model is firstly selected randomly to decide a model for samplingnew solution. A new solution x is uniformly generated from the selected subpopulationmodel Φt

k and δ′ that is generated from N(0, σtkI). If the generated solution x is beyond

the original boundaries of variables, the x is randomly set within the boundaries. Thereproduction procedure is repeated S times for generating S solutions.


3.5. Selection. For selection operator, we use the non-dominated solutions selection con-sidering crowding distance proposed in NSGAII [5] to set new population for next gener-ation. S solutions are selected among 2S solutions composed of existing S solutions fromprior generation and other S solutions newly generated in reproduction procedure. To dothis, firstly, groups of solutions with high rank in terms of Pareto dominance are selectedto fill the new population slots as possible, and then other solutions with larger crowdingdistance fill the remaining part of the new population. With the selection method, thepopulation can maintain good set of solutions in terms of both convergence and diversityperformances.

4. Performance Evaluation.

4.1. Comparing algorithm and parameter setting. The proposed MOEDA-S iscompared with two other leading algorithms, RM-MEDA and NSGAII, on a set of testproblems. RM-MEDA is an EDA that employs a probability distribution modeling ofpromising space and performs better than other EDAs for continuous MOPs with vari-able linkage [15]. NSGAII is a representative MOEA that employs non-dominated solutionselection with crowding distance estimation which is a widely used selection method. Forfair performance evaluation, three algorithms are started with same randomly generatedinitial population. The MOEDA-S is implemented with C language on a PC with 2.13GHz CPU, 1 GB RAM, Windows XP. The simulation codes for RM-MEDA and NS-GAII written by their authors are used in our experimental studies. The setting of eachalgorithm is summarized in Table 1.

Table 1. Experimental settings

Common parametersPopulation size: S = 100

Number of generations: 100

MOEDA-S

Extension ratio of search space: β = 1Variance consideration ratio for PCA: θ = 0.8

Volume reduction ratio for search-strategy change: ε = 0.3Maximum number of cluster: Kmax = 20

RM-MEDA Number of cluster: K = 10

NSGAII

Crossover parameter in SBX: nc = 20Crossover rate: Pc = 0.9

Parameter in polynomial mutation: nm = 20Mutation rate: Pm = 1/N (N : number of decision variable)

4.2. Test problems. We first compare MOEDA-S with two other algorithms on a setof problems (F1-F8) with different characteristics about variable linkage. We use testproblems used in RM-MEDA, which are modified problems from the widely used testproblems, ZDT1, ZDT2, ZDT6, DTLZ2, to get linear or nonlinear variable linkage. F1-F4 have a linear variable linkage, while F5-F8 have a nonlinear variable linkage. Secondly,we also evaluate the performance of MOEDA-S on the original problems with no variablelinkage. The number of decision variable is set 10 on F3, F7, and ZDT6, and 20 on othertest problems since we want to evaluate the performance clearly in given generations onthe three test problems, F3, F7, and ZDT6, that are relatively difficult to find optimalsolutions than other problems. All results in our simulation are averaged after 20 timessingle runs. All test problems are summarized in Table 2.


Table 2. Test problems

Instance Variables ] variables Objectives

F1 [0, 1]N 20

f1(x) = x1

f2(x) = g(x)[1−

√f1(x)/g(x)

]g(x) = 1 + 9

(n∑

i=2(xi − x1)

2

)/(n− 1)

F2 [0, 1]N 20

f1(x) = x1f2(x) = g(x)

[1− (f1(x)/g(x))

2]

g(x) = 1 + 9

(n∑

i=2(xi − x1)

2

)/(n− 1)

F3 [0, 1]N 10

f1(x) = 1− exp(−4x1) sin6(6πx1)

f2(x) = g(x)[1− (f1(x)/g(x))

2]

g(x) = 1 + 9

[n∑

i=2(xi − x1)

2/9

]0.25

F4 [0, 1]N 20

f1(x) = cos(π2x1)cos(π2x2)(1 + g(x))

f2(x) = cos(π2x1)sin(π2x2)(1 + g(x))

g(x) =n∑

i=3(xi − x1)

2

F5 [0, 1]N 20

f1(x) = x1

f2(x) = g(x)[1−

√f1(x)/g(x)

]g(x) = 1 + 9

(n∑

i=2(x2i − x1)

2

)/(n− 1)

F6 [0, 1]N 20

f1(x) =√x1

f2(x) = g(x)[1− (f1(x)/g(x))

2]

g(x) = 1 + 9

(n∑

i=2(x2i − x1)

2

)/(n− 1)

F7 [0, 1]N 10

f1(x) = 1− exp(−4x1) sin6(6πx1)

f2(x) = g(x)[1− (f1(x)/g(x))

2]

g(x) = 1 + 9

[n∑

i=2(x2i − x1)

2/9

]0.25

F8 [0, 1]N 20

f1(x) = cos(π2x1)cos(π2x2)(1 + g(x))

f2(x) = cos(π2x1)sin(π2x2)(1 + g(x))

g(x) =n∑

i=3(x2i − x1)

2

ZDT1 [0, 1]N 20

f1(x) = x1

f2(x) = g(x)[1−

√f1(x)/g(x)

]g(x) = 1 + 9

(n∑

i=2xi

)/(n− 1)

ZDT2 [0, 1]N 20

f1(x) = x1f2(x) = g(x)

[1− (f1(x)/g(x))

2]

g(x) = 1 + 9

(n∑

i=2xi

)/(n− 1)

ZDT6 [0, 1]N 10

f1(x) = 1− exp(−4x1) sin6(6πx1)

f2(x) = g(x)[1− (f1(x)/g(x))

2]

g(x) = 1 + 9

[n∑

i=2xi/9

]0.25

DTLZ2 [0, 1]N 20

f1(x) = cos(π2x1)cos(π2x2)(1 + g(x))

f2(x) = cos(π2x1)sin(π2x2)(1 + g(x))

g(x) =n∑

i=3(xi − 0.5)2


4.3. Performance metrics. We use two metrics, generational distance (GD) and di-versity metric (DM), to evaluate the convergence and diversity performance of solutionsrespectively in three algorithms.

4.3.1. Generational distance (GD). The generational distance (GD) is a widely used con-vergence metric, which is available since the PFs on the test problems are known before-hand. The uniformly distributed points are selected on PF as reference points. The GDis the average distance between the obtained solutions from algorithms and the nearestreference point on PF from the solutions. It is formulated as:

GD =ES

( S∑i=1

d2i

)1/2 (16)

where di is a distance between i-th solution and the nearest reference point from thesolution. With the GD, we can evaluate the convergence performance that is how theobtained solutions are nearer to the optimal solutions and how fast the solutions approachto the optimal solutions during search. In our simulation, 100 and 1000 points on PFare selected as reference points for test problems with two-objective and three-objective,respectively.

4.3.2. Diversity metric (DM). A new metric is used to assess diversity performance ofalgorithms. The existing diversity metrics [5, 10] can measure diversity only after at leastone solution reaches to PF or to the final solutions obtained finally through algorithm.Therefore, they can be said as diversity metrics considering convergence performance.

We propose a new diversity metric (DM) that evaluates the diversity performance inthe whole range of objective values of PF not only on PF during search. The ranges oneach dimension are divided into g grids in M-dimensional objective space and the DMassesses how uniformly solutions are distributed into the g grids. The proposed DM isdefined as

DM = EM

[g∑

i=1

n(h(i))

](17)

where h(i) =

1, if i-th grid has solution in population0, otherwise

where n(h) is a widely used neighboring scheme operator [10] to consider how evenly thesolutions are distributed on whole range of space. With the operator, evenly distributedsolutions have higher value of DM than those distributed unevenly though they are samein number. The neighboring scheme used in our simulation is shown in Table 3. In oursimulation, we divide the range of objective values of PF on each dimension into 100 grids.

4.4. Experimental results.

4.4.1. Test problems with linear variable linkage. We first evaluate MOEDA-S and twoother algorithms on 4 test problems with linear variable linkage, F1-F4. The test problemsF1-F3 have same line segment PS, x1 = x2 = . . . = xN , 0 ≤ xi ≤ 1, 1 ≤ i ≤ N and thetest function F4 has 2-D rectangle PS, x1 = x3 = . . . = xN , 0 ≤ xi ≤ 1, 1 ≤ i ≤ N .

Figure 4 shows the evolution of the average GD metric of nondominated solutions inthe current population in three algorithms on F1-F4. The GD of MOEDA-S approach to0 rapidly than that of other algorithms in all test problems, which means that MOEDA-Sfinds PF considerably faster than others. NSGAII has low convergence speed in these testproblems with variable linkage since it uses typical genetic operator with no consideration


Table 3. Neighboring scheme to evaluate n(h)

n(h(i)) h(i− 1) h(i) h(i+ 1)0.00 0 0 00.50 0 0 10.50 1 0 00.67 0 1 10.67 1 1 00.75 0 1 00.75 1 0 11.00 1 1 1

0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Number of Generations

GD

Met

ric

MOEDA−SRM−MEDANSGAII

(a) F1

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6


GD

Met

ric


(b) F2

0 20 40 60 80 1000

1

2

3

4

5

6

7


GD

Met

ric


(c) F3

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5


GD

Met

ric


(d) F4

Figure 4. The evolution of the average GD metric of nondominated solu-tions in the current population in three algorithms for F1-F4

of variable characteristics to find optimal solutions. As a result, NSGAII cannot find anyoptimal solutions within 100 generations in these test problems.Especially on F3 and F4, the three algorithms have a big difference of convergence

performance. In the test function of F3, basically, the density of candidate solutions is verylow near the PF. Therefore, F3 is considered as a difficult problem to find optimal solutionsrelated to other problems. In Figure 4(c) that shows performance on F3, MOEDA-Sobtains almost exact PF within 40 generations, while RM-MEDA achieves it after 60generations. This is because MOEDA-S leads the solutions to the PF effectively with a


0 20 40 60 80 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


DM

Met

ric


(a) F1

0 20 40 60 80 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


DM

Met

ric


(b) F2

0 20 40 60 80 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


DM

Met

ric


(c) F3

0 20 40 60 80 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


DM

Met

ric


(d) F4

Figure 5. The evolution of the DM metric of nondominated solutions inthe current populations in three algorithms for F1-F4

modeling based on sensitivity of objective function even at the beginning part of search.In case of F4 with three-objective, all algorithms acquire long time to the PF; however,MOEDA-S still has faster convergence speed to the PF than other algorithms.

Figure 5 shows the evolution of the DM metric of nondominated solutions in the currentpopulations in three algorithms on F1-F4. In Figure 5(c), the results for F3 show lowinitial DM than others since the solutions are located in limited area initially and thenchange the spread solutions later. In Figure 5(d), the situation is similar for F4 since thefeasible solution area of F4 is much larger than that of others. In terms of diversity per-formance, MOEDA-S outperforms other algorithms. As we confirm the GD performancefor F3, solutions in MOEDA-S reach to PF quickly; therefore, the DM of the solutionsstarts to increase early. In case of NSGAII, the DM stays at a low level since the solutionscannot reach to the PF enough.

Figure 6 gives the distribution of 10 sets of final non-dominated solutions for F1-F4from 10 single run. The MOEDA-S finds nearly same final solutions with the PF in alltest problems, while the final solutions found by RM-MEDA do not reach to the PF alittle and those obtained by NSGAII have big differences with the PF.

4.4.2. Test problems with nonlinear variable linkage. F5-F8 are the test problems withnonlinear variable linkage. F5-F7 have a same nonlinear PS, x1 = x2i , 0 ≤ x1 ≤ 1, 2 ≤ i ≤N , while F8 has 2-D bounded continuous surface PS, x1 = x2i , 3 ≤ i ≤ N, 0 ≤ x1, x2 ≤1 . Figure 7 shows the evolution of the average GD metric of nondominated solutions


(a) MOEDA-S for F1 (b) RM-MEDA for F1 (c) NSGAII for F1

(d) MOEDA-S for F2 (e) RM-MEDA for F2 (f) NSGAII for F2

(g) MOEDA-S for F3 (h) RM-MEDA for F3 (i) NSGAII for F3

(j) MOEDA-S for F4 (k) RM-MEDA for F4 (l) NSGAII for F4

Figure 6. The distribution of 10 sets of final non-dominated solutions forF1-F4 (some solutions found by NSGAII fall out of the figure range)

in the current populations in three algorithms for F5-F8. As we can see the results, thesolutions in MOEDA-S approach to the PF rapidly than those in other algorithms andreach exactly to PF approximately after 65 generations in F5 and F6. The reason thatNSGAII has quite good performance is that finding optimal solutions in some area on PFis very easy even in NSGAII, which is confirmed with distribution of final solutions shownin Figure 9.


0 20 40 60 80 1000

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


GD

Met

ric


(a) F5

0 20 40 60 80 1000

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2


GD

Met

ric


(b) F6

0 20 40 60 80 1000

1

2

3

4

5

6


GD

Met

ric


(c) F7

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9

10


GD

Met

ric


(d) F8

Figure 7. The evolution of the average GD metric of nondominated solu-tions in the current populations in three algorithms for F5-F8

Figure 8 shows the evolution of the average DM of nondominated solutions of thecurrent populations in each algorithm for F5-F8. In all test problems, MOEDA-S findsdiverse optimal solutions distributed evenly on the range of PF.

Figure 9 gives the distribution of 10 sets of final non-dominated solutions for F5-F8from 10 single runs in objective space. MOEDA-S finds almost exact PF for F5 and F6,while RM-MEDA miss considerable optimal solutions and NSGAII finds some part of thePF within given generations. In case of F7 and F8, MOEDA-S also cannot find solutionson PF; however, the final solutions from it are much nearer than those from the other twoalgorithms.

4.4.3. Test problems with no variable linkage. MOEDA-S is designed for solving an MOPwith variable linkage, in which a probability distribution model based on variable linkageis used to sample new solutions. To evaluate the performance of MOEDA-S in MOPwith no variable linkage, we compare MOEDA-S with other algorithms on a set of MOP,ZDT1, ZDT2, ZDT6, and DTLZ2, with no variable linkage.

Figure 10 shows the evolution of the average GD of nondominated solutions from thecurrent populations in three algorithms for ZDT1, ZDT2, ZDT6, and DTLZ2, respec-tively. Though the performance difference among three algorithms decreases dramatically,MOEDA-S has slightly better performance than other algorithms. The GD in RM-MEDAdecreases to low level quickly as search progresses since it can learn the distribution patternof variables and make new solutions based on the distribution. The proposed MOEDA-S


0 20 40 60 80 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


DM

Met

ric


(a) F5

0 20 40 60 80 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


DM

Met

ric


(b) F6

0 20 40 60 80 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


DM

Met

ric


(c) F7

0 20 40 60 80 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


DM

Met

ric


(d) F8

Figure 8. The evolution of the average DM metric of nondominated so-lutions in the current populations in three algorithms for F5-F8

has high convergence speed across generations, which may be because of its directionalsearch in the beginning part of search.Figure 11 shows the evolution of the average DM of nondominated solutions from the

current populations in three algorithm for ZDT1, ZDT2, ZDT6, and DTLZ2, respectively.The DM stays at low level since the solutions in three algorithms cannot approach to thePF enough. Compared with results in other algorithms, DM in MOEDA-S increasesquickly and reaches to high level within 100 generations, which means population foundby MOEDA-S approach to PF rapidly with diverse solutions.

4.4.4. Sensitivity of control parameters. We investigated the sensitivity of our algorithmrespect to the variation of the three control parameters used in MOEDA-S by evaluatingthe performance on test function of F3 according to the variation of each parameter. Ateach evaluation, the settings of the other parameters except observing parameter are thesame as in Table 1.Figures 12(a) and 12(b) show the average GD and DM metrics versus the variance

consideration ratio for PCA(θ). It is clear that MOEDA-S can approximate the PF wellwhen θ < 1 and the performance is very poor when θ = 1. It is because that when θ = 1,MOEDA-S assumes that the dimensionality of new search space is N and set the Gaussianvector σt

k to be zero in the modeling phase such that the search ability is significantlyreduced.


(a) MOEDA-S for F5 (b) RM-MEDA for F5 (c) NSGAII for F5

(d) MOEDA-S for F6 (e) RM-MEDA for F6 (f) NSGAII for F6

(g) MOEDA-S for F7 (h) RM-MEDA for F7 (i) NSGAII for F7

(j) MOEDA-S for F8 (k) RM-MEDA for F8 (l) NSGAII for F8

Figure 9. The distribution of 10 sets of final non-dominated solutions forF5-F8 (some solutions found by NSGAII fall out of the figure range)

Figures 12(c) and 12(d) show the performance versus the volume reduction ratio (ε).The convergence performance of final solutions is very high when ε < 0.4, while thediversity of final solutions is high when 0.15 ≤ ε ≤ 0.4. If ε is too small, the searchstrategy changes from the convergence-focused mode to the diversity-focused mode toolately such that the search for diverse solutions is limited. On the other hand, if ε is toohigh, the search strategy changes too early; as a result, the search ability toward the PFis reduced.


0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4


GD

Met

ric


(a) ZDT1

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5


GD

Met

ric


(b) ZDT2

0 20 40 60 80 1000

1

2

3

4

5

6

7

8

9


GD

Met

ric


(c) ZDT6

0 20 40 60 80 1000

0.5

1

1.5

2

2.5

3


GD

Met

ric


(d) DTLZ2

Figure 10. The evolution of the average GD of nondominated solutionsfrom the current populations in three algorithms for ZDT1, ZDT2, ZDT6,and DTLZ2, respectively

Figures 12(e) and 12(f) show the performance versus the maximum number of cluster(Kmax). MOEDA-S approximates the PF very well when Kmax ≥ 15. MOEDA-S hasdifficulty in estimating the PF when the number of cluster is too small since with fewclusters it could not built enough models to cover the widely distributed optimal solutions.From the observation of the sensitivity of the parameters, we confirmed similar perfor-

mance in reasonably large range of the four parameters. Therefore, we could concludethat the sensitivities of control parameters in MOEDA-S are not high and MOEDA-S isconsiderably robust respect to the variation of parameters.

4.4.5. Computational complexity. To evaluate computational complexity of MOEDA-S,we measured average CPU time spent to solve test problems. Table 4 shows the averageCPU time spent in three algorithms.

Table 4. The average CPU time spent (seconds)

F1 F2 F3 F4 F5 F6 F7 F8NSGAII 42.94 68.49 22.15 20.43 21.09 75.51 70.93 28.54

RM-MEDA 125.53 145.62 84.68 75.49 117.07 367.18 95.84 152.48MOEDA-S 130.68 151.14 88.18 81.15 122.54 380.62 102.19 169.75


0 20 40 60 80 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


DM

Met

ric


(a) ZDT1

0 20 40 60 80 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


DM

Met

ric


(b) ZDT2

0 20 40 60 80 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


DM

Met

ric


(c) ZDT6

0 20 40 60 80 1000.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1


DM

Met

ric


(d) DTLZ2

Figure 11. The evolution of the average DM of nondominated solutionsfrom the current populations in three algorithms for ZDT1, ZDT2, ZDT6,and DTLZ2, respectively

It is clear that MOEDA-S and RM-MEDA require much more CPU time than NSGAII.It might happen because the stages of modeling and reproduction have higher complexitythan general evolutionary operators, crossover and mutation. In addition, MOEDA-Srequires slightly more CPU time than RM-MEDA since it needs extra CPU time for k-means clustering and calculating of variables sensitivity at each generation. AlthoughMOEDA-S is more time consuming than other algorithms, it is still acceptable in manyapplications since it requires once the few minutes CPU time to obtain final solutions foreach test function.

5. Conclusion. Most multi-objective evolutionary algorithms aim at approximating thePF quickly and exactly. EDAs search optimal solutions on the PF by reproducing newsolutions based on the probability distribution model. Therefore, it is a critical issue thatuseful information is extracted from solutions and a probability distribution model builtwith the information reflects the characteristics of optimal solutions. We propose a novelestimation of distribution algorithm based on sensitivity of objective function (MOEDA-S)in which new information about the relationship between the objective functions and thedecision variables is employed for modeling of promising solution space. In the proposedMOEDA-S, according to the current search strategy decided, a convergence-focused searchis implemented to estimate the promising solution space rapidly at the beginning part ofthe search, and then a diversity-focused search is implemented to search diverse solutions


0.5 0.6 0.7 0.8 0.9 10

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

theta

GD

Met

ric

(a) GD metric versus θ

0.5 0.6 0.7 0.8 0.9 10.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

theta

DM

Met

ric

(b) DM metric versus θ

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

epsilon

GD

Met

ric

(c) GD metric versus ε

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.50.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

epsilon

DM

Met

ric

(d) DM metric versus ε

5 10 15 20 25 30 35 40 45 500

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

Kmax

GD

Met

ric

(e) GD metric versus Kmax

5 10 15 20 25 30 35 40 45 500.5

0.55

0.6

0.65

0.7

0.75

0.8

0.85

0.9

0.95

1

Kmax

DM

Met

ric

(f) DM metric versus Kmax

Figure 12. Average GD and DM metric of final nondominated solutionsversus four parameters in MOEDA-S for F3

at the latter part of the search. Our simulation results have indicated that MOEDA-S hasbetter performance in both terms of convergence and diversity on MOP with and withoutvariable linkage.

Acknowledgment. This research was supported by Basic Science Research Programthrough the National Research Foundation of Korea (NRF) funded by the Ministry ofEducation, Science and Technology (2012R1A1A2007605).


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