I-MOPSO: A Suitable PSO Algorithm forMany-Objective Optimization

Andre Britto

Federal University of Parana

Curitiba, Parana, Brazil 81531-980

andrebc@inf.ufpr.br

Aurora Pozo

Federal University of Parana

Curitiba, Parana, Brazil 81531-980

aurora@inf.ufpr.br

Abstract—Multi-Objective Optimization Problems are prob-lems with more than one objective function. In the litera-ture, there are several Multi-Objective Evolutionary Algorithms(MOEAs) that deals with MOPs, including Multi-ObjectiveParticle Swarm Optimization (MOPSO). However, these al-gorithms scale poorly when the number of objective grows.Many-Objective Optimization researches methods to decreasethe negative effect of applying MOEAs into problems with morethan three objective functions. Here, it is proposed a new PSOalgorithm, called I-MOPSO, which explores specific aspects ofMOPSO to deal with Many-Objective Problems. This algorithmtakes advantage of an archiving method to introduce moreconvergence and from the strategy of the leader’s selection tointroduce diversity on the search. I-MOPSO is evaluated throughan empirical analysis aiming to observe how it works in Many-Objective scenarios in terms of convergence and diversity tothe Pareto front. The proposed algorithm is compared to otherMOEAs from the literature through the use of quality indicatorsand statistical tests.

Keywords-Multi-Objective Particle Swarm Optimization;Multi-Objective Optimization; Many-Objective Optimization

I. INTRODUCTION

Multi-Objective Particle Swarm Optimization (MOPSO) is a

population based Multi-Objective Meta-Heuristic that has been

used to solve several Multi Objective Optimization Problems

(MOP) [1]. MOPs involve the simultaneously optimization of

two or more conflicting objectives functions subject to certain

constraints. MOPSO algorithms are specifically designed to

provide robust and scalable solutions. In these algorithms, each

element, called particle, uses simple local rules to govern its

actions and by means of the interactions of the entire group,

the swarm achieves its objectives.

However, in spite of the good results of MOEAs, includ-

ing MOPSO algorithms [2], these algorithms scale poorly

when dealing with problems with more than 3 objective

functions [3], [4]. These problems are called Many-Objective

Optimization Problems (MaOPs). One of the main challenges

faced by MOEAs with many objectives is the deterioration

of the search ability. This deterioration occurs due to the

increase of the number of non-dominated solutions with the

number of objectives and, consequently, there is no pressure

towards the Pareto front. The Many-Objective Optimization is

the search for new techniques with the goal to overcome these

limitations [5].

In the literature, MaOPs have been tackled through different

approaches like: decomposition strategies, the proposal of

new preference relations, dimensionality reduction, among

others [3]. Our goal is to explore MOPSO in Many-Objective

Optimization, a topic few explored in the literature. However,

a different approach is taken, and here, specific features of

MOPSO are considered. A new algorithm is proposed, called

I-MOPSO (Ideal Point Guided MOPSO). This algorithm has

two main aspects: an archive method which introduces more

convergence on the search and the leader’s selection method

to deal with diversity. The archiving method uses the idea to

guide the solutions in the archive to a specific area of the ob-

jective space near the ideal point [6]. For the leader’s selection

it was chosen the NWSum method [7] which introduces more

diversity on the search and avoids the concentration of the

solutions on a small region of the Pareto Front.

I-MOPSO algorithm is evaluated through an empirical anal-

ysis. The algorithm is compared to other MOPSO algorithm

designed for Many-Objective Optimization, called CDAS-

SMPSO [8]. Furthermore, two MOEAs from the literature

are also compared: SMPSO [2] and NSGA-II [9]. In this

comparison, the algorithms solve the DTLZ2 many-objective

bechmarking problem [10]. Also, a set of quality indicators are

applied to investigate how these algorithms scale up in terms

of convergence and diversity in many objective scenarios,

they are: Generational Distance (GD), Inverse Generational

Distance (IGD), Spacing and also it is analysed the distribution

of the Tchebycheff distance over the ”knee” of the Pareto

front [11].

The rest of this paper is organized as follows: Section II

describes the main concepts of Many-Objective Optimization.

The Multi Objective Particle Swarm Optimization is presented

in Section III. After, the proposed algorithm is discussed

in Section IV. Section V presents the empirical analysis

performed to evaluate I-MOPSO. Finally, Section VI presents

the conclusions and future works.

II. MANY-OBJECTIVE OPTIMIZATION

A Multi-Objective problem (MOP) involves the simultane-

ous satisfaction of two or more objective functions. Further-

more, in such problems, the objectives to be optimized are

usually in conflict, which means that they do not have a single

best solution, but a set of solutions. To find this set of solutions

2012 Brazilian Symposium on Neural Networks

1522-4899/12 $26.00 © 2012 IEEE

DOI 10.1109/SBRN.2012.20

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it is used the Pareto Optimality Theory [12]. The general multi-

objective minimization problem, without constraints, can be

stated as (1).

Minimizef(x) = (f1(x), f2(x)..., fm(x)) (1)

subject to x ∈ Ω, where: x ∈ Ω is a feasible solution vector,

Ω is the feasible region of the problem, m is the number

of objectives and fi(x) is the i-th objective function of the

problem.

In this case, the purpose is to optimize m objective functions

simultaneously, with the goal to find a good trade-off of

solutions that represent the better compromise between the

objectives. So, given f(x) = (f1(x), f2(x)..., fm(x)) and

f(y) = (f1(y), f2(y)..., fy(x)), f(x) dominates f(y) , de-

noted by f(x) ≺ f(y), if and only if (minimization):

∀i ∈ {1, 2, ...,m} : fi(x) ≤ fi(y), and∃i ∈ {1, 2, ...,m} : fi(x) < fi(y)

f(x) is non-dominated if there is no f(y) that dominates f(x).

Also, if there is no solution y that dominates x, x is called

Pareto Optimal and f(x) is a non-dominated objective vector.

The set of all Pareto Optimal solutions is called Pareto Optimal

Set, denoted by P ∗, and the set of all non-dominated objective

vector is called Pareto Front, denoted by PF ∗.

In MOP, the MOEAs modify Evolutionary Algorithms by

incorporating a selection mechanism based on Pareto opti-

mality and adopting a diversity preservation mechanism that

avoids the convergence to a single solution [12]. However,

since in most applications the search for the Pareto optimal set

is NP-hard, the MOEAs focuses on finding an approximationPareto Front, as close as possible to the true Pareto Front.

Recently, research efforts have been oriented to investigate

the scalability of these algorithms with respect to the number

of objectives [3] [4]. Many-Objective Optimization is the area

that studies new techniques for problems that have more than

3 objectives, called Many-Objective Problems (MaOPs).

In the literature, some studies have showed that

MOEAs scale poorly in many-objective optimization prob-

lems [3] [8] [5] [13]. The main reason for this is the number

of non-dominated solutions that increases exponentially with

the number of objectives. As consequence: first, the search

ability is deteriorated because it is not possible to impose

preferences for selection purposes; Second, the number of so-

lutions required for approximating the entire Pareto front also

increases, and finally there exists difficulty of the visualization

of solutions. To avoid these problems, currently, these issues

has been tackled using mainly the adaptation of preference

relations that induce a finer order on the objective space, the

dimensionality reduction and decomposition strategies [3].

Our main goal, is to propose a PSO algorithm suitable for

Many-Objective Problems. The idea is to explore methods to

store the non-dominated solutions in the external archive and

the leader selection procedure. Until fairly recently most of

the research was concentrated on a small group of algorithms,

often Genetic Algorithms.

III. MULTI-OBJECTIVE PARTICLE SWARM OPTIMIZATION

Particle Swarm Optimization (PSO) is a cooperative

population-based heuristic inspired by the social behavior of

birds flocking to find food [1]. The set of possible solutions

is a set of particles, called a swarm, which moves in the

search space, in a cooperative search procedure. In PSO, a

set of solutions searches for optimal solutions by updating

generations. These movements are performed by the velocity

operator that is guided by a local and a social component. In

Multi-Objective Optimization, Multi-Objective Particle Swarm

Optimization (MOPSO), the Pareto dominance relation is

adopted to establish preferences among solutions to be consid-

ered as leaders. By exploring the Pareto dominance concepts,

each particle in the swarm could have different leaders, but

only one may be selected to update the velocity.

This set of leaders is stored in an external archive (or

repository) that contains the best non-dominated solutions

found so far. Normally, this archive is bounded and has a

maximum size. So, two important features of PSO are: the

method to archive the solutions in the repository and how

each particle will choose its leader (leader’s selection).

The basic steps of a MOPSO algorithm are: initialization of

the particles, computation of the velocity, position update and

update of leader’s archive.

Each particle pi, at a time step t, has a position x(t) ∈ Rn,

that represents a possible solution. The position of the particle,

at time t + 1, is obtained by adding its velocity, v(t) ∈ Rn,

to x(t), Equation 2. The velocity of a particle pi is based on

the best position already fetched by the particle, −→p best(t), and

the best position already fetched by the set of neighbors of pi,−→Rh(t), that is a leader from the repository, see Equation 3.

−→x (t+ 1) = −→x (t) +−→v (t+ 1) (2)

−→v (t+ 1) = � · −→v (t) + (C1 · φ1) · (−→p best(t)−−→x (t))

+(C2 · φ2) · (−→Rh(t)−−→x (t)) (3)

The variables φ1 and φ2, in (3), are coefficients that de-

termine the influence of the particle best position, randomly

obtained in each iteration. Constants C1 and C2 indicate how

much each component influences on velocity. The coefficient

� is the inertia of the particle, and controls how much the

previous velocity affects the current one. The local leader,−→p best(t), is the best position ever achieved by the particle. If

the new position and the current −→p best(t) are non-dominated,

the new value is chosen randomly between these two vectors.−→Rh is a particle from the repository, chosen as a guide of pi,obtained through a global neighborhood (totaly connected).

In the literature some works deal with MaOPs using PSO

algorithms. It can be highlighted the work presented at [8].

This work studied the influence of Control of Dominance Area

of Solutions [14] in different MOPSO algorithms. The study

167

showed that the technique improves the results of MOPSO for

problems with many objectives. It proposes a new algorithm,

called CDAS-SMPSO, that outperformed the SMPSO algo-

rithm in Many-Objective scenarios. In [15], a PSO algorithm

handles many-objectives using a Gradual Pareto dominance

relation to overcome the problem of finding non-dominated

solutions when the number of objectives grows and Mostaghim

and Schmeck [16] presented an overview of MOPSO with

many objectives, also two variants of MOPSO are proposed

based on the ranking of non-dominated solutions.

IV. I-MOPSO

It is known that Pareto based algorithms have several limita-

tions when dealing with MaOPs, but it is possible to introduce

new features into traditional MOEAs to avoid these problems.

As discussed in Section II, the Many-Objective Optimization

literature concentrates its work in tasks like the proposal

of new preference relation, dimensionality reduction, among

others. Here, our interest is to explore specific characteristic

of PSO algorithms willing to reduce the limitations observed

in Many-Objective Optimization.

A new Multi-Objective Particle Swarm Optimization al-

gorithm called, I-MOPSO (Ideal Point Guided MOPSO) is

proposed. This algorithm has to main features: the archiving

process which introduces more convergence and the leader’s

selection method which provides diversity to the search.

I-MOPSO has as basis the SMPSO algorithm [2]. It uses

the procedure that limits the velocity of each particle. The

velocity of the particle is limited by a constriction factor χ,

that varies based on the values of C1 and C2. Besides, the

SMPSO introduces a mechanism that links or constraints the

accumulated velocity of each variable j (in each particle).

Also, after the velocity of each particle has been updated

a mutation operation is applied. A polynomial mutation is

applied [9] in 15% of the population, randomly selected.

The proposed algorithm differs from SMPSO on the archiv-

ing method and the leader’s selection strategy. In I-MOPSO,

the archiving method introduces more convergence towards

the Pareto front. It is used the Ideal archiver, presented in [6]

For this, the archiving method guides the solutions in the

archive to a specific area of the objective space. So, in this

method the ideal point [12] is selected as guide. The ideal

point is a vector with the best value for every objective value,

obtained at each iteration between the points in the external

archive. In this approach, the distance to the ideal point defines

which solutions will remain in the archive. Here, when the

archive becomes full and a new solution tries to enter the

following procedure is executed: first the ideal point between

all solutions in the archive and the new solution is obtained;

second, the Euclidean distance from each point to the ideal

point is calculated; finally, it is removed the point with the

highest distance. The main idea of this archive is that guiding

the selection of the points in the archive to a region close to

the ideal point will increase the convergence of the search to

the Pareto front and will place the solutions in a good area of

the objective space.

However, this process that guides the solutions to a region

near the ideal point could introduce lack of diversity into the

PSO algorithm search. So, to avoid the concentration of the

generated approximation Front to a small region, it is chosen

a leader’s selection method which introduces diversity on the

search.

The NWSum method proposed in [7] is used. This method

consists in guiding the particle to the dimension where it closer

(dcloser). The select leader will be the particle in the repository

who is closer to dcloser. With this method, it is possible

to guide the particles closer to the axis of each dimension,

avoiding them to be only located near to the ideal point. This

method calculates weights for each objective values and give

more power for those objectives where the particle has good

values. It is defined by Equation 4, where xi represents the

position of the particle i and pi is a possible leader for xi

F =∑

j

fj(xi)∑k fk(xi)

fj(pi) (4)

The particle pi that generates the greatest weighted sum is

used for the update, aiming to push the particles towards the

axis it’s already close.

V. EMPIRICAL ANALYSIS

To evaluate the proposed algorithm, I-MOPSO, it was

compared with some state-of-art MOEA and some algorithms

specific designed for Many-Objective Problems. The MOEAs

chosen were the SMPSO algorithm [2], that have very good

results when compared to other MOPSO algorithms, and the

NSGA-II [9], that have very good results for MOP and is often

used as basis in MOP literature. Furthermore, the I-MOPSO

was compared with the CDAS-SMPSO algorithm [8], that was

designed for MaOPs.

Each algorithm executed 50000 fitness evaluation. For

SMPSO, CDAS-SMPSO and I-MOPSO the population was

limited to 250 particles. In each iteration ω varied randomly

in the interval [0, 0.8]. φ1 and φ2 varied randomly in [0, 1].C1 and C2 varied randomly over the interval [1.5, 2.5]. For the

CDAS-SMPSO, the parameter that control the dominance area,

Si, was set to 0.25, 0.30, 0.35, 0.40 and 0.45, that obtained

the best results in [8]. The archive as limited to 250 solutions.

NSGA-II was executed with a population of 250 individuals.

The algorithms were applied to the DTLZ2 many-objective

problem [10]. This problem can be scaled to any number of

objectives (m) and decision variables (n) and the global Pareto

front is known analytically. The DTLZ2 problem can be used

to investigate the ability of the algorithms to scale up their

performances in large numbers of objectives. In this analysis

the problem was scaled to 2, 3, 5, 10, 15 and 20 number of

objectives.

Here, our goal is to observe aspects like convergence

towards the Pareto Front and the diversity of the approxi-

mation of the Pareto front generated by each algorithm. To

measure the convergence it was used the Generational Distance

(GD) [12], that measures how far the approximated Pareto

168

Fig. 1: Mean of GD values for all 30 executions for all

algorithms

Front (PFapprox) generated by each archiver is from the true

Pareto front of the problem PFtrue. It is a minimization

measure. To observe if the (PFapprox) is well distributed

over the Pareto Front the Inverse Generational Distance (IGD)

was applied. IGD measures the minimum distance of each

point of PFtrue to the points of PFapprox. IGD allows us to

observe if PFapprox converges to the true Pareto front and

also if this set is well diversified. It is important to perform

a joint analysis of GD and IGD indicators because if only

GD is considered it is not possible to notice if the solutions

are distributed over the entire Pareto front. Finally, it was

used the Spacing quality indicator [12], that measures the

range variance between neighboring solutions in the front.

If the value of this metric is 0, all solutions are equally

distributed in the objective space. The Hypervolume metric

isn’t used here, because it has some limitations when applied

to Many-Objective Optimization, like to give higher values for

PFapprox near the edges in this context, as discussed in [11].

Besides the previous quality indicators, here it was also used

a methodology presented in [11]. Since, one of the problems

of the Many-Objective Optimization is the visualization of the

approximation set, one way to tackle this issue is the use of

histograms. So, seeking to observe where the approximation

set generated by each algorithm is located, it is made the

analysis of the distribution of Tchebycheff distance. This

methodology compares the Tchebycheff distance of each point

of the PFapprox to the ideal point (or the knee) of the Pareto

Front. The distributions of the Tchebycheff distance for all

solutions are presented in distribution charts, for all analyzed

objectives.

Every algorithm was executed thirty times. The quality

indicators are compared using the Friedman test at 5% sig-

nificance level. The test is applied to raw values of each

metric. The post-test of the Friedman test indicates if there

is any statistically difference between each analyzed data set,

to identify which data set has the best values it is used

some boxplot charts. The boxplot gives information about the

location, spread, skewness and tails of the data. Due to space

Fig. 2: Mean of IGD values for all 30 executions for all

algorithms

Fig. 3: Mean of Spacing values for all 30 executions for all

algorithms

limitations, the boxplots were omitted in this paper.

The results are presented in Figures 1 to 6 and Table I.

Figures 1 to 3 present the mean values of the GD, IGD and

Spacing for each algorithm. Every curve in each chart repre-

sents the GD, IGD and Spacing values evolution for different

objective numbers. Figures 4 to 6 present the distribution of

the Tchebycheff distance. Table I presents the summary of best

algorithm obtained by the Friedman test.

First, the GD is analyzed to observe if the algorithms

converged to the Pareto Front. At Table I, it can be observed

that only the algorithms designed for MaOPs obtained the best

results, according to the Friedman test. The CDAS-SMPSO

algorithm obtained the best results for 2, 3 and 5 objective

numbers, for different Si values. However, for high number

of objectives the proposed algorithm I-MOPSO obtained the

best results. This result is expected, since both algorithms

are specially designed to introduce more convergence to the

search, specially the I-MOPSO, that guide the search to a

region near the ideal point. In Figure 1, it can be observed that

the I-MOPSO had very good values of GD for all objective

values. In opposite way, the MOEAs from literature, SMPSO

169

TABLE I: Best archivers according to Friedman test for DTLZ2 problem

Obj Best algorithmsGD IGD Spacing

2 0.25, 0.30 and 0.35 NSGA-II 0.25, 0.30 and 0.353 0.25, 0.40 and 0.45 0.40, I-MOPSO, NSGA-II 0.25, 0.45 and SMPSO5 0.25, 0.35, 0.40 and I-MOPSO 0.35, 0.40 and I-MOPSO 0.25 and 0.45

10 I-MOPSO 0.35 and 0.40 0.25 and I-MOPSO15 I-MOPSO 0.30, 0.35 and 0.40 0.45 and I-MOPSO20 I-MOPSO 0.30, 0.35, 0.40 and I-MOPSO 0.45

and NSGA-II, suffer a huge deterioration when the number of

objective grows.

IGD is analyzed to observe the diversity properties of each

algorithm. Again, the algorithms with Many-Objective tech-

niques obtained the best results, specially when the number

of objective grows. The I-MOPSO obtained the best result,

along the CDAS-SMPSO, for 5 and 20 objective numbers.

The CDAS-SMPSO obtained the best diversity for almost all

objective numbers, with different Si values. The NSGA-II has

the best result only for two objective values, and the SMPSO

did not obtain any best result. Through Figure 2, again, it

can be observed that the I-MOPSO did not have a great

deterioration when the number of objective grows, but have

worse values for high number of objectives than for a low

number of objectives. It occurs, because the search is directed

to a region near the ideal point, favoring convergence instead

diversity. The results of the NSGA-II were omitted from the

chart, since it obtained very high values of IGD for a high

number of objectives.

Fig. 4: Tchebycheff distance distribution, 10 objective func-

tions

For the Spacing, the I-MOPSO algorithm generated a sim-

ilar value for all objectives and did not deteriorate when the

number of objective grows. It obtained the best values for 10and 15 objectives. The CDAS-SMPSO also had good results

obtaining the best value of Spacing for all objective values,

however, with different Si values. As presented in [8], often

the CDAS-SMPSO obtain good spacing values due to the

small size of the approximation Pareto set. Again, SMPSO

and NSGA-II had a high deterioration when the number of

objective grows. In Figure 3, it can be observed the good

Fig. 5: Tchebycheff distance distribution, 15 objective func-

tions

Fig. 6: Tchebycheff distance distribution, 20 objective func-

tions

results of I-MOPSO, specially for high number of objectives,

the good results of some CDAS-SMPSO configurations and

the deterioration of the NSGA-II and SMPSO.

Finally, the distributions of the Tchebycheff distance are

analyzed. In these charts, curves that have peaks near small

values of Tchebycheff distance concentrated their solutions

near the knee (ideal point). Here it is presented only the charts

for the higher objective numbers, 10, 15 and 20, however this

analysis relies for all objective numbers. For small number of

objectives (2, 3 and 5), almost all algorithms have a similar

distribution, often near the knee. However when the number

of objective grows, both NSGA-II and SMPSO tend to spread

170

their distribution in different regions of the Pareto front. Since

these algorithms can’t reach the Pareto Front, it is expected a

distribution with values far from the ideal point. I-MOPSO had

distributions near the knee of the Pareto front, often having

peaks near the origin of the chart. For CDAS-SMPSO, as

discussed in [8], one of the characteristic introduced by CDAS,

is to guide the search to a region near the knee. Therefore,

the distributions for the different configurations of the CDAS-

SMPSO were often located near the ideal point.

In summary, the I-MOPSO algorithm obtained good results

for many objective problems, in terms of convergence and

diversity. The algorithm has a good convergence towards the

Pareto front and generates the approximation Pareto front near

the knee, due to the archiving method. However, the algorithm

loss some diversity, since it tries to guide the search to the

ideal point. To avoid that the final solution concentrate in a

small region of the Pareto front, the leader selection method,

introduces more diversity on the search. The CDAS-SMPSO

had very good results, as presented in [8]. I-MOPSO had very

similar results but with the advantage of the using the original

Pareto dominance and not needing any additional configuration

parameters. Through the results, it can be observed that

different CDAS-SMPSO configurations obtained good values,

so it is not possible to obtain the best result in only one

execution of the CDAS-SMPSO. Finally, like observed in other

works in the literature [8] [9], the SMPSO and the NSGA-II

algorithms had a high deterioration on the search when the

number of objective grows.

VI. CONCLUSION

Multi-Objective Particle Swarm Optimization has demon-

strated to be very powerful, dealing with MOPs in a suitable

way and providing a set of good solutions for the problem

considering Pareto non-dominance concepts [8]. However,

just as other MOEAs, MOPSO algorithms suffer a great

deterioration in Many-Objective Problems.

This work presented a new MOPSO algorithm with the

goal to be suitable for problems with more than 3 objective

function, called I-MOPSO. The main idea was to explore con-

vergence and diversity through specific features of MOPSO.

Therefore, to introduce more convergence towards the Pareto

Front, I-MOPSO uses an archiving method that guide the

solutions to a region near the ideal point of the Pareto Front.

Besides, to avoid the solutions to be concentrate only in

a single point and introduce more diversity on the search,

the leader’s selection chosen for I-MOPSO was the NWSum

method.

I-MOPSO was evaluated through an empirical analysis, that

used the DTLZ2 many-objective problem, and was compared

to the CDAS-SMPSO, the SMPSO and NSGA-II. It was

concluded that the proposed algorithm could obtain good

results in a many-objective scenario. The algorithm presented

a good convergence and covered a region of the Pareto Front

near the ideal point. The results of I-MOPSO were very

similar to the CDAS-SMPSO algorithm, however the proposed

algorithm did not have any additional parameters and did use

the original Pareto dominance relation.

Future works include exploring other characteristics of

MOPSO algorithm, seeking to obtain a more diversified ap-

proximation of the Pareto front, without losing convergence.

Also, I-MOPSO will be analyzed in others bechmarking

problems, such as discontinuous Pareto fronts.

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