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MSPSO20210909V2.zip （38个子文件）

MSPSO20210909V2

CreateEmptyParticle.m 2KB

yueshuVoltageKrigingHJX.m 6KB

dacefit.m 9KB

lhsamp.m 592B

regpoly2.m 789B

RouletteWheelSelection.m 2KB

correxpg.m 1KB

DeleteFromRep.m 2KB

www.kalami.ir.url 110B

GetOccupiedCells.m 2KB

corrspline.m 2KB

gridsamp.m 1KB

dailiKrigingHJX.m 6KB

predictor.m 4KB

Dominates.m 2KB

mopso.m 9KB

correxp.m 1KB

corrgauss.m 1KB

GetNonDominatedParticles.m 2KB

MyCost4.m 597B

corrlin.m 1KB

Handling Multiple Objectives with Particle Swarm Optimization.pdf 1.5MB

regpoly0.m 377B

yueshuTorqueRippleKrigingHJX.m 6KB

MyCost3.m 2KB

MyCost1.m 2KB

CreateHypercubes.m 2KB

corrcubic.m 1KB

GetGridIndex.m 3KB

MyCost2.m 2KB

dsmerge.m 4KB

DetermineDomination.m 2KB

data1.mat 2KB

corrspherical.m 1KB

GetCosts.m 2KB

SelectLeader.m 2KB

Contents.m 1KB

regpoly1.m 385B

256 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 8, NO. 3, JUNE 2004

Handling Multiple Objectives With

Particle Swarm Optimization

Carlos A. Coello Coello, Member, IEEE, Gregorio Toscano Pulido, and Maximino Salazar Lechuga

Abstract—This paper presents an approach in which Pareto

dominance is incorporated into particle swarm optimization

(PSO) in order to allow this heuristic to handle problems with

several objective functions. Unlike other current proposals to

extend PSO to solve multiobjective optimization problems, our

algorithm uses a secondary (i.e., external) repository of particles

that is later used by other particles to guide their own flight. We

also incorporate a special mutation operator that enriches the

exploratory capabilities of our algorithm. The proposed approach

is validated using several test functions and metrics taken from the

standard literature on evolutionary multiobjective optimization.

Results indicate that the approach is highly competitive and that

can be considered a viable alternative to solve multiobjective

optimization problems.

Index Terms—Evolutionary multiobjective optimization,

multiobjective optimization, multiobjective particle swarm opti-

mization, particle swarm optimization.

I. INTRODUCTION

HE USE OF evolutionary algorithms for multiobjective

optimization (an area called “evolutionary multiobjective

optimization,” or EMO for short) has significantly grown in

the last few years, giving rise to a wide variety of algorithms

[7]. As any other research area, EMO currently presents cer-

tain trends. One of them is to improve the efficiency both of

the algorithms and of the data structures used to store nondom-

inated vectors. EMO researchers have produced some clever

techniques to maintain diversity (e.g., the adaptive grid used

by the Pareto Archive Evolutionary Strategy (PAES) [21]), new

algorithms that use very small populations (e.g., the microGA

[6]), and data structures that allow to handle unconstrained ex-

ternal archives (e.g., the dominated tree [12]).

Particle swarm optimization (PSO) is a relatively recent

heuristic inspired by the choreography of a bird flock. PSO has

been found to be successful in a wide variety of optimization

Manuscript received July 8, 2002; revised September 2, 2003. The work

of C. A. Coello Coello was supported in part by CONACyT under Project

34 201-A. G. T. Pulido acknowledges support from CONACyT through a

scholarship to pursue graduate studies at the Computer Science Section of the

Electrical Engineering Department, CINVESTAV-IPN, México. M. S. Lechuga

acknowledges support from CONACyT through a scholarship to pursue grad-

uate studies at the School of Computer Science, University of Birmingham,

Birmingham, U.K.

C. A. Coello Coello and G. T. Pulido are with CINVESTAV-IPN, Sección

de Computación, Departamento de Ing. Eléctrica, Sección de Computación,

México 07300, Mexico (e-mail: [email protected]vestav.mx; gtoscano@

computacion.cs.cinvestav.mx).

M. S. Lechuga is with the University of Birmingham, School of

Computer Science, Edgbaston, Birmingham B15 2TT, U.K. (e-mail:

[email protected]).

Digital Object Identifier 10.1109/TEVC.2004.826067

tasks [19], but until recently it had not been extended to deal

with multiple objectives.

PSO seems particularly suitable for multiobjective optimiza-

tion mainly because of the high speed of convergence that

the algorithm presents for single-objective optimization [19].

In this paper, we present a proposal, called “multiobjective

particle swarm optimization” (MOPSO), which allows the PSO

algorithm to be able to deal with multiobjective optimization

problems. Our current proposal is an improved version of

the algorithm reported in [5], in which we have added a

constraint-handling mechanism and a mutation operator that

considerably improves the exploratory capabilities of our

original algorithm.

MOPSO is validated using several standard test functions

reported in the specialized literature and compared against

three highly competitive EMO algorithms: the nondominated

sorting genetic algorithm-II [11] (NSGA-II), the PAES [21],

and the microgenetic algorithm for multiobjective optimization

(microGA) [6].

II. B

ASIC CONCEPTS

Definition 1 (Global Minimum): Given a function :

, , for the value is

called a global minimum if and only if

(1)

Then,

is the global minimum solution, is the objective func-

tion, and the set

is the feasible region , where rep-

resents the whole search space.

Definition 2 [General Multiobjective Optimization Problem

(MOP)]: Find the vector

which will

satisfy the

inequality constraints

(2)

the

equality constraints

(3)

and will optimize the vector function

(4)

where

is the vector of decision variables.

Definition 3 (Pareto Optimality): A point

is Pareto

optimal if for every

and either

(5)

or, there is at least one

such that

(6)

COELLO COELLO et al.: HANDLING MULTIPLE OBJECTIVES WITH PARTICLE SWARM OPTIMIZATION 257

In words, this definition says that is Pareto optimal if there

exists no feasible vector

which would decrease some criterion

without causing a simultaneous increase in at least one other cri-

terion. The phrase “Pareto optimal” is considered to mean with

respect to the entire decision variable space unless otherwise

specified.

Definition 4 (Pareto Dominance): A vector

is said to dominate (de-

noted by

) if and only if is partially less than , i.e.,

, : .

Definition 5 (Pareto Optimal Set): For a given MOP

the Pareto optimal set

is defined as

(7)

Definition 6 (Pareto Front): For a given MOP

and

Pareto optimal set

, the Pareto front is defined as

(8)

In the general case, it is impossible to find an analytical ex-

pression of the line or surface that contains these points. The

normal procedure to generate the Pareto front is to compute the

feasible points

and their corresponding . When there is

a sufficient number of these, it is then possible to determine the

nondominated points and to produce the Pareto front.

Pareto optimal solutions are also termed noninferior, admis-

sible,orefficient solutions [15]; their corresponding vectors are

termed nondominated.

III. R

ELATED

WORK

Kennedy and Eberhart [19] proposed an approach called

PSO, which was inspired by the choreography of a bird flock.

The approach can be seen as a distributed behavioral algorithm

that performs (in its more general version) multidimensional

search. In the simulation, the behavior of each individual is

affected by either the best local (i.e., within a certain neigh-

borhood) or the best global individual. The approach then

uses the concept of population and a measure of performance

similar to the fitness value used with evolutionary algorithms.

Also, the adjustments of individuals are analogous to the use

of a crossover operator. The approach also introduces the

use of flying potential solutions through hyperspace (used

to accelerate convergence), which can be seen as a mutation

operator. An interesting aspect of PSO is that it allows individ-

uals to benefit from their past experiences (note that in other

approaches such as the genetic algorithm, normally the current

population is the only “memory” used by the individuals). PSO

has been successfully used for both continuous nonlinear and

discrete binary single-objective optimization [19]. PSO seems

particularly suitable for multiobjective optimization mainly

because of the high speed of convergence that the algorithm

presents for single-objective optimization [19].

In fact, there have been several recent proposals to extend

PSO to handle multiobjectives. We will review the most impor-

tant of them.

• The algorithm of Moore and Chapman [24]: This al-

gorithm was presented in an unpublished document and it

is based on Pareto dominance. The authors emphasize the

importance of performing both an individual and a group

search (a cognitive component and a social component).

However, the authors did not adopt any scheme to main-

tain diversity.

• The swarm metaphor of Ray and Liew [28]: This algo-

rithm also uses Pareto dominance and combines concepts

of evolutionary techniques with the particle swarm. The

approach uses crowding to maintain diversity and a multi-

level sieve to handle constraints (for this, the authors adopt

the constraint and objective matrices proposed in some of

their previous research [27]).

• The algorithm of Parsopoulos and Vrahatis [26]: Un-

like the previous proposals, this algorithm adopts an ag-

gregating function (three types of approaches were imple-

mented: a conventional linear aggregating function, a dy-

namic aggregating function, and the bang-bang weighted

aggregation approach [18] in which the weights are varied

in such a way that concave portions of the Pareto front can

be generated).

• Dynamic neighborhood PSO proposed by Hu and

Eberhart [16]: In this algorithm, only one objective is

optimized at a time using a scheme similar to lexico-

graphic ordering [7]. Lexicographic ordering tends to be

useful only when few objective functions are used (two

or three), and it may be sensitive to the ordering of the

objectives. The idea of the dynamic neighborhood is, with

no doubt quite interesting and is novel in this context.

At the time in which this paper was originally prepared, none

of the existing proposals to extend PSO to solve multiobjective

optimization problems used a secondary population (the most

common notion of elitism in EMO). This may certainly limit

the performance of the algorithm, unless a very good diversity

maintainance approach is used (unfortunately, this is not the

case in most of the approaches available at that time). Also, none

of these techniques had been properly validated using test func-

tions and metrics normally adopted by EMO researchers.

Note however, that in more recent papers these ideas have

been already incorporated by other authors. The most represen-

tative proposals are the following (published after the submis-

sion of the original version of this paper).

• Fieldsend and Singh [13] proposed an approach in which

they use an unconstrained elite archive (in which a spe-

cial data structure called “dominated tree” is adopted) to

store the nondominated individuals found along the search

process. The archive interacts with the primary population

in order to define local guides. The approach is compared

(using four test functions and two metrics) against an algo-

rithm similar to PAES [21] and with a variation of our orig-

inal MOPSO [5]. Their approach also uses a “turbulence”

operator that is basically a mutation operator that acts on

the velocity value used by PSO. The approach seems to

have the same problems of our original MOPSO with mul-

tifrontal problems such as the fourth example included in

this paper. It is important to note that the new version of

258 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 8, NO. 3, JUNE 2004

our MOPSO provided in this paper does not have the prob-

lems of the original version in multifrontal problems.

• Hu

et al. [17] adopted a secondary population (called “ex-

tended memory”) and introduced some further improve-

ments to their dynamic neighborhood PSO approach [16].

Nevertheless, it is worth indicating that this approach com-

pletely fails in generating the true Pareto front of some

problems (see [17] for details). Hu et al. [17] also com-

pared their algorithm with respect to the strength Pareto

evolutionary algorithm (SPEA) [37] using the set coverage

metric [34].

• Mostaghim and Teich [25] proposed a sigma method in

which the best local guides for each particle are adopted

to improve the convergence and diversity of a PSO ap-

proach used for multiobjective optimization. They also use

a “turbulence” operator, but applied on decision variable

space. The idea of the sigma approach is similar to com-

promise programming [7]. The use of the sigma values in-

creases the selection pressure of PSO (which was already

high). This may cause premature convergence in some

cases (e.g., in multifrontal problems). In this approach,

the authors provide comparisons with SPEA2 [36] and the

dominated trees of Fieldsend and Singh [13] using four

test functions and the coverage metric.

• Li [23] proposed an approach in which the main mech-

anisms of the NSGA-II [11] are adopted in a PSO algo-

rithm. The proposed approach showed a very competitive

performance with respect to the NSGA-II (even outper-

forming it in some cases).

The main differences of our approach with respect to the other

proposals existing in the literature are:

• We adopt an external (or secondary) repository similar to

the adaptive grid of PAES [21] (see Section IV-B). None

of the other proposals use such a mechanism in the way

adopted in this paper.

• The mutation operator that we use acts both on the parti-

cles of the swarm, and on the range of each design variable

of the problem to be solved (see Section IV-C). This aims

not only to explore remote regions of the search space, but

also tries to ensure that the full range of each decision vari-

able is explored.

• We provide an extensive analysis of the impact of the pa-

rameters of our MOPSO on its performance. We also com-

pare our MOPSO with respect to three other algorithms

(which are representative of the state-of-the-art in evolu-

tionary multiobjective optimization), using three metrics.

IV. D

ESCRIPTION OF THE PROPOSED APPROACH

The analogy of PSO with evolutionary algorithms makes evi-

dent the notion that using a Pareto ranking scheme [14] could be

the straightforward way to extend the approach to handle mul-

tiobjective optimization problems. The historical record of best

solutions found by a particle (i.e., an individual) could be used to

store nondominated solutions generated in the past (this would

be similar to the notion of elitism used in evolutionary multiob-

jective optimization). The use of global attraction mechanisms

combined with a historical archive of previously found non-

dominated vectors would motivate convergence toward globally

nondominated solutions.

A. Main Algorithm

The algorithm of MOPSO is the following.

1) Initialize the population

(a) FOR

TO number of parti-

cles*/

(b) Initialize

2) Initialize the speed of each particle:

(a) FOR

(b)

3) Evaluate each of the particles in

4) Store the positions of the particles that represent nondom-

inated vectors in the repository

5) Generate hypercubes of the search space explored so far,

and locate the particles using these hypercubes as a co-

ordinate system where each particle’s coordinates are de-

fined according to the values of its objective functions.

6) Initialize the memory of each particle (this memory

serves as a guide to travel through the search space. This

memory is also stored in the repository):

(a) FOR

(b)

7) WHILE maximum number of cycles has not been reached

a) Compute the speed of each particle

using the fol-

lowing expression:

where (inertia weight) takes a value of 0.4;

and are random numbers in the range ;

is the best position that the particle

has had;

is a value that is taken from the

repository; the index

is selected in the following

way: those hypercubes containing more than one

particle are assigned a fitness equal to the result of

dividing any number

(we used in

our experiments) by the number of particles that

they contain. This aims to decrease the fitness of

those hypercubes that contain more particles and it

can be seen as a form of fitness sharing [10]. Then,

we apply roulette-wheel selection using these fit-

ness values to select the hypercube from which we

will take the corresponding particle. Once the hy-

percube has been selected, we select randomly a

particle within such hypercube.

is the cur-

rent value of the particle

b) Compute the new positions of the particles adding

the speed produced from the previous step

(9)

Each particle has a dimensionality that can vary depending on the problem

solved. When we say that we compute the speed of a particle, we refer to com-

puting the speed for each of its dimensions.

We will explain later on how do we define “better” in this context.

COELLO COELLO et al.: HANDLING MULTIPLE OBJECTIVES WITH PARTICLE SWARM OPTIMIZATION 259

Fig. 1. Possible cases for the archive controller.

c) Maintain the particles within the search space in

case they go beyond their boundaries (avoid gen-

erating solutions that do not lie on valid search

space). When a decision variable goes beyonds its

boundaries, then we do two things: 1) the deci-

sion variable takes the value of its corresponding

boundary (either the lower or the upper boundary)

and 2) its velocity is multiplied by (

1) so that it

searches in the opposite direction.

d) Evaluate each of the particles in

e) Update the contents of

together with the geo-

graphical representation of the particles within the

hypercubes. This update consists of inserting all the

currently nondominated locations into the reposi-

tory. Any dominated locations from the repository

are eliminated in the process. Since the size of the

repository is limited, whenever it gets full, we apply

a secondary criterion for retention: those particles

located in less populated areas of objective space

are given priority over those lying in highly popu-

lated regions.

f) When the current position of the particle is better

than the position contained in its memory, the par-

ticle’s position is updated using

(10)

The criterion to decide what position from memory

should be retained is simply to apply Pareto dom-

inance (i.e., if the current position is dominated

by the position in memory, then the position in

memory is kept; otherwise, the current position re-

places the one in memory; if neither of them is

dominated by the other, then we select one of them

randomly).

g) Increment the loop counter

8) END WHILE

B. External Repository

The main objective of the external repository (or archive) is

to keep a historical record of the nondominated vectors found

along the search process. The external repository consists of two

main parts: the archive controller and the grid.

We will proceed to discuss each of these two components in

more detail.

1) The Archive Controller: The function of the archive con-

troller is to decide whether a certain solution should be added

to the archive or not. The decision-making process is the fol-

lowing.

The nondominated vectors found at each iteration in the

primary population of our algorithm are compared (on a

one-per-one basis) with respect to the contents of the external

repository which, at the beginning of the search will be empty.

If the external archive is empty, then the current solution is

accepted (see case 1, in Fig. 1). If this new solution is domi-

nated by an individual within the external archive, then such

a solution is automatically discarded (see case 2, in Fig. 1).

Otherwise, if none of the elements contained in the external

population dominates the solution wishing to enter, then such a

solution is stored in the external archive. If there are solutions

in the archive that are dominated by the new element, then

such solutions are removed from the archive (see cases 3 and

4, in Fig. 1). Finally, if the external population has reached its

maximum allowable capacity, then the adaptive grid procedure

is invoked (see case 5, in Fig. 1).

2) The Grid: To produce well-distributed Pareto fronts, our

approach uses a variation of the adaptive grid proposed in [21].

The basic idea is to use an external archive to store all the solu-

tions that are nondominated with respect to the contents of the

archive. Into the archive, objective function space is divided into

regions as shown in Fig. 2. Note that if the individual inserted

into the external population lies outside the current bounds of

the grid, then the grid has to be recalculated and each individual

within it has to be relocated (see Fig. 3).

260 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTATION, VOL. 8, NO. 3, JUNE 2004

Fig. 2. Graphical representation of the insertion of a new element in the adaptive grid when the individual lies within the current boundaries of the grid.

Fig. 3. Graphical representation of the insertion of a new element in the adaptive grid when this lies outside the previous boundaries of the grid.

The adaptive grid is really a space formed by hypercubes.

Such hypercubes have as many components as objective

functions. Each hypercube can be interpreted as a geographical

region that contains an no number of individuals. The main

advantage of the adaptive grid is that its computational cost is

lower than niching (see [21] for a detailed complexity analysis).

The only exception would be if the grid had to be updated at

each generation. In such a case, the computational complexity

of the adaptive grid would be the same as niching [i.e.,

Strictly speaking, it is formed by hyperparallelepids when the ranges of the

objective functions are not scaled. If scaled, however, we are talking of hyper-

cubes, which is the assumption made in this paper.

The adaptive grid is used to distribute in a uniform way the

largest possible amount of hypercubes. In order to achieve this

goal, it is necessary to provide and obtain certain information

which is problem dependant (i.e., the number of grid subdivi-

sions).

C. Use of a Mutation Operator

This operator deserves a more detailed discussion. PSO

is known to have a very high convergence speed. However,

such convergence speed may be harmful in the context of

multiobjective optimization, because a PSO-based algorithm

may converge to a false Pareto front (i.e., the equivalent of a

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