a novel elitist multiobjective optimization algorithm

__________________ *Corresponding author. E-mail address: [email protected]

A novel elitist multiobjective optimization algorithm:

multiobjective extremal optimization

Min-Rong Chen*, Yong-Zai Lu

Department of Automation, Shanghai Jiao Tong University,

Dongchuan Road 800, Class A0403291, 200240 Shanghai, China

Abstract: Recently, a general-purpose local-search heuristic method called Extremal

Optimization (EO) has been successfully applied to some NP-hard combinatorial

optimization problems. This paper presents an investigation on EO with its application

in multiobjective optimization and proposes a new novel elitist (1+ λ ) multiobjective

algorithm, called Multiobjective Extremal Optimization (MOEO). In order to extend

EO to solve the multiobjective optimization problems, the Pareto dominance strategy is

introduced to the fitness assignment of the proposed approach. We also present a new

hybrid mutation operator that enhances the exploratory capabilities of our algorithm.

The proposed approach is validated using five popular benchmark functions. The

simulation results indicate that the proposed approach is highly competitive with the

state-of-the-art multiobjective evolutionary algorithms. Thus MOEO can be considered

a good alternative to solve multiobjective optimization problems.

Keywords: Multiple objective programming; Metaheuristics; Extremal optimization;

Self-organized criticality

1. Introduction Most real-world engineering optimization problems are multiobjective in nature, since

they normally have several (possibly conflicting) objectives that must be satisfied at the

same time. Instead of aiming at finding a single solution, the multiobjective optimization

methods try to produce a set of good trade-off solutions from which the decision maker

may select one.

The Operations Research (OR) community has developed a lot of mathematical

programming techniques to solve multiobejctive optimization problems (MOPs) since the

1950s [1, 2]. However, mathematical programming techniques have some limitations

when dealing with MOPs [3]. For example, many of them may not work when the Pareto

front is concave or disconnected. Some require differentiability of the objective functions

and the constraints. In addition, most of them only generate a single solution from each

run. During the past two decades, a considerable amount of multiobjective evolutionary

algorithms (MOEAs) have been presented to solve MOPs [3-7]. Evolutionary algorithms

seem particularly suitable to solve MOPs, because they can deal with a set of possible

solutions (or so-called population) simultaneously. This allows us to find several

members of the Pareto-optimal set in a single run of the algorithm [3]. There is also no

requirement for differentiability of the objective functions and the constraints. Moreover,

evolutionary algorithms are susceptible to the shape of the Pareto front and can easily

deal with discontinuous or concave Pareto fronts.

Recently, a general-purpose local-search heuristic algorithm named Extremal

Optimization (EO) was proposed by Boettcher and Percus [8]. EO is based on the Bak-

Sneppen (BS) model [9], which shows the emergence of self-organized criticality (SOC)

[10] in ecosystems. The evolution in this model is driven by a process where the weakest

species in the population, together with its nearest neighbors, is always forced to mutate.

The dynamics of this extremal process exhibits the characteristics of SOC, such as

punctuated equilibrium [9]. EO opens the door to applying non-equilibrium process,

while the simulated annealing (SA) applies equilibrium statistical mechanics. In contrast

to genetic algorithm (GA) which operates on an entire “gene-pool” of huge number of

possible solutions, EO successively eliminates those worst components in the sub-

optimal solutions. Its large fluctuations provide significant hill-climbing ability, which

enables EO to perform well particularly at the phase transitions. EO has been

successfully applied to some NP-hard combinatorial optimization problems such as graph

bi-partitioning [8], graph coloring [11], spin glasses [12], MAXSAT [13], production

scheduling [14,15], function optimization [16] and dynamic combinatorial problems [17].

So far there have been some papers studying on the multiobjective optimization using

extremal dynamics. Ahmed and Elettreby [18,19] introduced random version of BS model.

They also generalized the single objective BS model to a multiobjective one by weighted

sum aggregation method. The method is easy to implement but its most serious drawback

is that it cannot generate proper members of the Pareto-optimal set when the Pareto front

is concave regardless of the weights used [20]. Galski et al. [21] applied the Generalized

Extremal Optimization (GEO) algorithm [22] to design a spacecraft thermal control

system. The design procedure was tackled as a multiobjective optimization problem and

they also resorted to the weighted sum aggregation method to solve the problem. In order

to extend GEO to solve the MOPs effectively, Galski et al. [23] further present a revised

multiobjective version of the GEO algorithm, called M-GEO. The M-GEO algorithm

does not use the weighted sum aggregation method. Instead, the Pareto dominance

concept was introduced to M-GEO in order to find out the approximate Pareto front, and

at the same time the approximate Pareto front was stored and updated each run. Since the fitness

assignment in the M-GEO is not based on the Pareto dominance strategy, M-GEO belongs to the

non-Pareto approach [23]. M-GEO was successfully applied to the inverse design of a

remote sensing satellite constellation [23]. In this work, we develop a novel elitist multiobjective optimization method, called

Multiobjective Extremal Optimization (MOEO). Our approach does not use the weighted

sum aggregation method to solve MOPs. Instead, we adopt the fitness assignment method

based on the Pareto dominance strategy. Thus, MOEO is a Pareto-based approach. It is

interesting to note that, similar to the (1+ λ ) Pareto Archived Evolution Strategy (PAES)

[24], MOEO is also a single-parent λ -offspring multiobjective optimization algorithm.

Furthermore, we propose a new hybrid mutation operator that enhances the exploratory

capabilities of our algorithm. Our approach has been validated using five benchmark

functions reported in the specialized literature and compared with four competitive

MOEAs: the Nondominated Sorting Genetic Algorithm-II (NSGA-II) [25], the Pareto

Archived Evolution Strategy (PAES) [24], the Strength Pareto Evolutionary Algorithm

(SPEA) [26] and the Strength Pareto Evolutionary Algorithm2 (SPEA2) [27]. The

simulation results demonstrate that the proposed approach is highly competitive with the state-

of-the-art MOEAs. Hence, MOEO may be a good alternative to solve the multiobjective

optimization problems.

This paper is organized as follows. In Section 2, we give the problem formulation of

multiobjective optimization problem. Section 3 describes four state-of-the-art elitist multiobjective

evolutionary algorithms. The extremal optimization algorithm is introduced in Section 4. In

Section 5, we propose the MOEO algorithm and describe its main components in detail.

In Section 6, the proposed approach is validated using five popular benchmark functions.

In addition, the simulation results are compared with those of four state-of-the-art

multiobjective evolutionary algorithms. Finally, Section 7 concludes the paper with an

outlook on future work.

2. Problem formulation Without loss of generality, the MOPs are mathematically defined as follows:

Find x which minimizes 1 2( ) ( ( ), ( ), , ( ))kF x f x f x f x=

subject to :

( ) 0 1,2, ,

( ) 0, 1, 2, ,i

j

g x i m

h x j p

≥ =

= =

， (1)

where 1 2( , , , )Tnx x x x= ∈Ω is a vector of decision variables, each decision variable is bounded

by lower and upper limits , 1, ,l l ll x u l n≤ ≤ = , k is the number of objectives, m is the number

of inequality constraints and p is the number of equality constraints. The following four

concepts are of importance [28]:

Definition 1. Pareto dominance: A vector 1( , , )ku u u= is said to dominate another

vector 1( , , )kv v v= (denoted by u v≺ ) if and only if u is partially less than v , i.e., 1, , : ( 1, , : ).i i j ji k u v j k u v∀ ∈ ≤ ∧ ∃ ∈ <

Definition 2. Pareto optimality: A solution x∈Ω is said to be Pareto optimal with respect to Ω if and only if there is no 'x ∈Ω for which ' ' '

1( ) ( ( ), , ( ))kv F x f x f x= =

dominates 1( ) ( ( ), , ( ))ku F x f x f x= = . The phrase “Pareto optimal” is taken to mean with

respect to the entire decision variable space unless otherwise specified. Definition 3. Pareto-optimal set: The Pareto optimal set SP is defined as the set of all

Pareto optimal solutions, i.e., ' ' | : ( ) ( )SP x x F x F x= ∈Ω ¬∃ ∈Ω ≺ . Definition 4. Pareto-optimal front: The Pareto-optimal front FP is defined as the set of

all objective functions values corresponding to the solutions in SP , i.e.,

1 ( ) ( ( ), , ( )) | F k SP F x f x f x x P= = ∈ .

3. Elitist multiobjective evolutionary algorithm As opposed to single-objective optimization, where the best solution is always copied

into the next population, the incorporation of elitism in MOEAs is substantially more

complex. Often used is the concept of maintaining an external archive of solutions that

are nondominated among all individuals generated so far. Another promising elitism approach provides the so-called ( )µ λ+ selection, where parents and offspring compete

against each other. In the study of Zitzler et al. [29], it was clearly shown that elitism

helps in achieving better convergence in MOEAs. Among the existing elitist MOEAs,

Deb et al.’s NSGA-II [25], Zitzler et al.’s SPEA [26] and Zitzler et al.’s SPEA2 [27], Knowles

and Corne’s Pareto-archived PAES [24] enjoyed more attention.

1) Nondominated Sorting Genetic Algorithm II (NSGA-II): This algorithm is proposed

by Deb et al. [25]. NSGA-II is a revised version of the nondominated sorting genetic

algorithm (NSGA) proposed by Srinivas and Deb [30]. The original NSGA is based on

several layers of classifications of the individuals as suggested by Goldberg [31]. Before

selection, all nondominated individuals are classified into one category. This group of

classified individuals is then ignored and another layer of nondominated individuals is

considered. The process continues until all individuals in the population are classified.

Since individuals in the first front have the maximum fitness value, they always get more

chance of surviving than the remainder of the population. The NSGA-II is more efficient

in terms of computational complexity than the original NSGA. To keep diversity, NSGA-

II uses a crowded comparison operator without specifying any additional parameters, while the original NSGA used fitness sharing. Besides, NSGA-II adopts ( )µ λ+ -selection

as its elitist mechanism [32].

2) Strength Pareto Evolutionary Algorithm (SPEA) and Strength Pareto

Evolutionary Algorithm 2 (SPEA2): Zitzler et al. [26] suggested the SPEA. They

suggested maintaining an external population at every generation storing all

nondominated solutions discovered so far. This external population participates in all

genetic operations. Presented by Zitzler et al. [27], SPEA2 is an improvement of the

SPEA. In contrast to SPEA, SPEA2 uses a fine-grained fitness assignment strategy which

incorporates density information. Furthermore, the archive size in SPEA2 is fixed, i.e.,

whenever the number of nondominated individuals is less than the predefined archive

size, the archive is filled up by dominated individuals. In addition, the clustering

technique, which is used in SPEA as the archive truncation method, has been replaced by

an alternative truncation method which has similar features but does not lose boundary

points. Finally, another difference to SPEA is that only members of the archive

participate in the mating selection process.

3) Pareto Archived Evolution Strategy (PAES): Knowles and Corne [24] suggested a

simple MOEA using an (1+1) evolution strategy (a single parent generates a single

offspring). PAES is purely based on the mutation operator to search for new individuals.

An archive is used to store the nondominated solutions found in the evolutionary process.

Such a historical archive is the elitist mechanism adopted in PAES. An interesting aspect

of this algorithm is the adaptive grid method used as the archive truncation method. They also presented (1+ λ ) and ( µ λ+ ) variations of the basic approach. The former is

identical to PAES (1+1) except that λ offspring are generated from the current solution. The ( µ λ+ ) version maintains a population of size µ from which λ copies are made.

The fittest µ from the µ λ+ solutions then replace the current population.

4. Extremal optimization 4.1. Bak-Sneppen model EO is based on Bak-Sneppen (BS) model of biological evolution, which simulates far-from

equilibrium dynamics in statistical physics. BS model is one of the models that show the nature of

SOC. The SOC means that regardless of the initial state, the system always tunes itself to a critical

point having a power-law behavior without any tuning control parameter. In the BS model,

species are located on the sites of a lattice. Each species is randomly assigned a fitness value with

a uniform distribution. At each update step, the worst adapted species is always forced to mutate.

The change in the fitness of the worst adapted species will cause the alteration of the fitness

landscape around it. Thus the fitness value of the neighbors of the weakest one will also be

changed together, even if they are well adapted. After a number of iterations, the system evolves

to a highly correlated state known as SOC. In that state, almost all species have fitness values

above a certain threshold, and a little change of one species will result in co-evolutionary chain

reactions called “avalanches”. The probability distribution of the sizes “ K ” of these avalanches is

depicted by a power law ( )P K K τ−∼ , where τ is a positive parameter. That is, the smaller

avalanches are more likely to occur than those bigger ones, but even the avalanches as big as the

whole system may occur with a small but non-negligible probability. Therefore, the large

fluctuation makes any possible configuration accessible.

4.2. Extremal optimization Inspired by the BS model, Boettcher and Percus proposed the EO algorithm for a minimization

optimization problem as follows [8]: 1) Generate a solution S randomly. Set the optimal solution bestS S= .

2) For the current solution S , a) evaluate the fitness iλ for each component (i.e., decision variable) ix , b) rank all the components by their fitness values and find the component jx with the

“worst fitness”, i.e., j iλ λ≤ for all i,

c) choose one solution 'S in the neighborhood of S , i.e., ' ( )S N S∈ , such that the

worst component jx must change its state,

d) accept 'S S= unconditionally, e) if the current cost function value is less than the so-far minimum cost function value,

i.e. ( ) ( )bestC S C S< , then set bestS S= ,

3) Repeat at Step 2) as long as desired. 4) Return bestS and ( )bestC S .

Note that in the EO algorithm, each decision variable in the current solution S is

considered “species”. In this study, we adopt the term “component” to represent

“species” which is usually used in biology. From the above algorithm, it can be seen that

unlike genetic algorithms which work with a population of candidate solutions, EO

evolves a single solution S and makes local modification to the worst components. This

requires that a suitable representation be selected which permits the solution components

to be assigned a quality measure (i.e. fitness). This differs from holistic approaches such

as evolutionary algorithms that assign equal-fitness to all components of a solution based

on their collective evaluation against an objective function.

It is important to note that the governing principle behind the EO algorithm is the improvement

through successively removing low-quality species and changing them randomly. This is

obviously at odds with genetic algorithms, which select good solutions in an attempt to make

better solutions. By always mutating the worst adapted species, EO could evolve solutions quickly

and systematically, and at the same time preserve the possibility of probing different regions of

the design space via avalanches.

5. Multiobjective extremal optimization In order to further extend EO to solve the multiobjective problems, in this work, we propose a

novel elitist multiobjective optimization algorithm, so-called Multiobjective Extremal

Optimization (MOEO), through introducing Pareto dominance strategy to EO. Similar to EO, the

MOEO performs only one operation, i.e. mutation, on each decision variable of the current

solution.

It is well known that there exist two fundamental goals in multiobjective optimization

design: one is to minimize the distance of the generated solutions to the Pareto-optimal

set, the other is to maximize the diversity of the achieved Pareto set approximation. In

this study, MOEO mainly consists of three components: fitness assignment, diversity

preservation and external archive. A good fitness assignment is beneficial for guiding

the search towards the Pareto-optimal set. In order to increase the diversity of the

nondominated solutions, we introduce the diversity-preserving mechanism to our

approach. For the sake of preventing nondominated solutions from being lost, we also

adopt an external archive to store the nondominated solutions found in the evolutionary

process. In the following sections, main algorithm, fitness assignment, diversity

preservation, external archive and mutation operator will be addressed in some details.

5.1. Main algorithm

For a multiobjective optimization problem, the proposed MOEO algorithm works as Figure 1. At

1) Randomly generate an initial solution 1 2( , , , )nS x x x= . Set the external archive empty. Set 0iteration = .

2) Generate n offspring of the current solution S by performing mutation on each decision variable one by

one. 3) Perform dominance ranking on the n offspring and then obtain their rank numbers, i.e.,

[0, 1]jr n∈ − , 1, , j n∈ .

4) Assign the fitness j jrλ = for each variable jx , 1, , j n∈ .

5) If there is only one variable with fitness value of zero, the variable will be considered the worst component; otherwise, the diversity preservation mechanism is invoked. Assuming that the worst component is wx with

fitness 0wλ = , 1, , w n∈ .

6) Perform mutation only on wx while keeping other variables unchanged, then get a new solution wS .

7) Accept wS S= unconditionally.

8) Apply UpdateArchive(S, archive) to update the external archive (see Figure 4).

9) If the iterations reach the predefined maximum number of the generations, go to Step 10); otherwise, set 1iteration iteration= + , and go to Step 2).

10) Output the external archive as the Pareto-optimal set.

Figure 1. The pseudo-code of MOEO algorithm

the same time, we give the flowchart of MOEO in Figure 2.

No

gen=gen+1

No

gen<maxgen?

output external archive as Pareto optimal set

STOP

Yes

Randomly generate a solution, set archive empty, gen=0

START

Evaluate fitness value for each component in the current solution

number of worst components>1?

Perform mutation on the worst component

Update the external archive using Archiving Logic

invoke diversity preservation

Yes

5.2. Fitness assignment In this study, to extend EO to solve multiobjective optimization problems efficiently, we

introduce the Pareto-based fitness assignment strategy to EO. We use the dominance

ranking method proposed by Fonseca and Fleming [33], i.e., the fitness value of one solution

equals to the number of other solutions by which it is dominated, to determine the fitness value for

each solution. Therefore, the nondominated solutions are ranked as zero, whilst the worst possible

ranking is the number of all the solutions minus one. It is important to note that there exists only

one solution in the search process of MOEO. In order to find out the worst component via fitness

assignment, we generate a population of new offspring by performing mutation on the components

of the current solution one by one. Then the dominance ranking is carried out in the newly

generated offspring. In this work, the component corresponding to the nondominated offspring (i.e.,

its fitness value equals to zero) is considered the weakest one. Then the nondominated offspring

will replace the current solution in the next generation. From the aforementioned, we can see that,

Figure 2. The flowchart of MOEO algorithm

similar to (1+λ ) PAES, MOEO is also a (1+λ ) multiobjective optimization approach. That is, λ

offspring can be generated from the current solution via performing mutation on all the components

in turn, and then the offspring with the lowest fitness will replace the current solution. Here, λ

equals to the number of the decision variables.

To be clearer, we illustrate the process of dominance ranking in MOEO using Figure 3 (a).

Assume that this is a two-objective optimization problem and both objectives are to be minimized.

The locations of the current solution in the objective space (marked with black solid circle) will

change to new one in the next generation via mutating its weakest component. For example, given the current solution 1 2 3 4( , , , )iS x x x x= , of which the location in the objective space is denoted by

the circle i , we can identify the weakest component by mutating the four variables one by one

and then performing dominance ranking on the newly generated offspring. First, a new offspring '1 2 3 4( , , , )iAS x x x x= can be obtained by mutating 1x to '

1x , keeping other variables unchanged. Similarly, the other three offspring, i.e.,

2

'1 3 4( , , , )iBS x x x x= ,

3

'1 2 4( , , , )iCS x x x x= and

4

'1 2 3( , , , )iDS x x x x= , are generated. The four white dashed circles (i.e., A, B, C, D) in Figure 3(a)

stand for the locations of four offspring (i.e., iAS , iBS , iCS , iDS ) in the objective space,

respectively. The next location of the current solution iS in the objective space depends on the

dominance ranking number of the four newly generated offspring. It can be seen from Figure 3 (a)

that for the circle i , circle A is nondominated by any of the other three dashed circles (i.e., B, C, D). The rank numbers of A, B, C and D are 0, 1, 1, 3, respectively. Hence, the component 1x

corresponding to A is considered the weakest one and the current solution iS will change to iAS

in the next generation.

If there exists more than one worst component, i.e., at least two components with the same

fitness value of zero, then the following diversity-preserving mechanism will be invoked.

5.3. Diversity preservation The goal of introducing the diversity preservation mechanism is to maintain a good spread of

solutions in the obtained set of solutions. In this study, we propose a new approach to keep good

diversity of nondominated solutions in the search process. It is worth pointing out that our

i

Figure 3. (a) Dominance ranking in MOEO; (b) diversity preservation in MOEO

F1

F2

C

A

B D

(a) (b)F1

F2 I

II III

IV

i

B

D

C

A

approach does not require any user-defined parameter for maintaining diversity among population

members. Note that this diversity preservation mechanism is only invoked when more than one

nondominated offspring has been generated. Another diversity preservation strategy will be used

when archiving the Pareto set in Section 5.4.

As can be seen from Figure 3 (b), the locations of the nondominated offspring (marked with

white dash circle) relative to that of the parent i (indicated by black circle) can be categorized

into four cases, i.e. regions I, II, III and IV. The offspring (circle B) residing in the region II

dominates its parent, those (circles A and C) in the regions I and III do not dominate the parent

and vice versa, and the one (circle D) in the region IV is dominated by the parent. It should be

pointed out that those offspring lying in region IV will increase the searching time to approach the

Pareto-optimal set. Thus we will not choose the nondominated offspring residing in region

IV as the new solution in the next generation, unless there exists no nondominated

offspring lying in other three regions. In addition, for the purpose of keeping a good spread

of nondominated solutions, we not only regard the nondominated offspring lying in region I but

also those in regions II and III as the candidates of the new solution in the next generation. If there

are more than one nondominated offspring lying in the above three regions, we can randomly

choose one as the new solution in the next generation.

In a word, if there exists more than one nondominated offspring, MOEO will first pick one

randomly from those nondominated offspring which do not reside in region IV. The worst case is

that all the nondominated offspring reside in region IV, and then MOEO will choose one from

them randomly.

5.4. External archive The main objective of the external archive is to keep a historical record of the nondominated

solutions found along the search process. Note that the archive size is fixed in MOEO. This external

archive provides the elitist mechanism for MOEO. The archive in our approach consists of two

main components as follows.

1) Archiving logic: The function of the archiving logic is to decide whether the current solution

should be added to the archive or not. The archiving logic works as follows. The newly

generated solution S is compared with the archive to check if it dominates any member

of the archive. If some members of the archive are dominated by S, all the dominated

solutions are eliminated from the archive and S is added to the archive. If at least one

member of the archive dominates S, the archive does not need to be updated and the

iteration continues. However, if S and any member of the archive do not dominate each

other, there exist three cases:

a) If the archive is not full, S is added to the archive.

b) If the archive is full and S resides in the most crowded region in the objective space

among the members of the archive, the archive does not need to be updated.

c) If the archive is full and S does not reside in the most crowded region of the

archive, the member in the most crowded region of the archive is replaced by S.

Based on the above archiving logic, the pseudo-code of function UpdateArchive(S,archive) is

shown in Figure 4.

2) Crowding-distance metric: In our approach, we adopt the crowding-distance metric proposed

by Deb et al. [25] to judge whether the current solution resides in the most crowded region of the

archive. It is interesting to note that this metric is capable of preserving the boundary solutions in

the archive. Via sorting the current solution S and all members of the archive together according

to their crowding distances in an ascending order, the one which resides in the most crowded

region can be found out. If S is the one with the lowest rank, S will be considered lying in the

most crowded region. Otherwise, the solution with the lowest rank will be removed from the

archive and S will be added to the archive. As can be seen from the literature [25], the complexity of the crowding-distance computation is ( (2 ) (2 ))O k A log A , where A is the archive size and k is the

number of objectives. Thus the complexity of crowding-distance computation is not large. For

more details about crowding-distance metric, the interested readers are referred to [25].

5.5. Mutation operator Since there is merely mutation operator in MOEO, the mutation plays a key role in MOEO search

that generates new solutions through adding or removing genes at the current solutions, i.e.,

chromosomes. In this work, we present a new mutation method based on mixing Gaussian

mutation and Cauchy mutation and choose it as the mutation operator in MOEO.

The mechanisms of Gaussian and Cauchy mutation operations have been studied by Yao et al.

function UpdateArchive(S, archive)

begin function if the current solution S is dominated by at least one member of the archive, then

the archive does not need to be updated; else if some members of the archive are dominated by S , then

remove all the dominated members from the archive and add S to the archive; end if else

if the archive is not full, then add S to the archive; else

if S resides in the most crowded region of the archive, then the archive does not need to be updated; else replace the member in the most crowded region of the archive by S; end if end if end if end function

Figure 4. Pseudo-code of function UpdateArchive(S, archive)

[35]. They pointed out that Cauchy mutation can get to the global optimum faster than Gaussian

mutation due to its higher probability of making longer jumps. However, they also indicated that

Cauchy mutation spends less time in exploiting the local neighbourhood and thus has a weaker

fine-tuning ability than Gaussian mutation from small to mid-range regions. Therefore, Cauchy

mutation is better at coarse-grained search while Gaussian mutation is better at fine-grained

search. At the same time, they pointed out that Cauchy mutation performs better when the current

search point is far away from the global optimum, while Gaussian mutation is better at finding a

local optimum in a good region. Thus, it would be ideal if Cauchy mutation is used when search

points are far away from the global optimum and Gaussian mutation is adopted when search

points are in the neighbourhood of the global optimum. Unfortunately, the global optimum is

usually unknown in practice, making the ideal switch from Cauchy to Gaussian mutation very

difficult. A new method based on mixing (rather than switching) different mutation operators has

been proposed by Yao et al. [35]. The idea is to mix different search biases of Cauchy and

Gaussian mutations. The method generates two offspring from the parents, one by Cauchy

mutation and the other by Gaussian mutation. The better one is then chosen as the offspring.

Inspired by the above idea, we present a new mutation method based on mixing Gaussian

mutation and Cauchy mutation. For the sake of convenience, we call it “hybrid GC mutation”. It

must be indicated that, unlike the method in [35], this mutation doesn’t compare the anticipated

outcomes between Gaussian mutation and Cauchy mutation due to the characteristics of EO. In

the hybrid GC mutation, the Cauchy mutation is first adopted. It means that the large step size will

be taken first at each mutation. If the new generated variable after mutation goes beyond the

interval of the corresponding decision variable, the Cauchy mutation will be used repeatedly for

some times (TC), until the new generated offspring falls into the interval. Otherwise, Gaussian

mutation will be carried out repeatedly for another some times (TG), until the offspring satisfies

the requirement. That is, the step size will become smaller than before. If the new generated

variable after mutation still goes beyond the interval of the corresponding decision variable, then

we choose the upper or lower bound of the decision variable as the new generated variable. Thus,

our approach combines the advantages of coarse-grained search and fine-grained search. The

above analyses show that the hybrid GC mutation is very simple yet effective. Unlike some

switching algorithms which have to decide when to switch between different mutations during

search, the hybrid GC mutation does not need to make such decisions.

In this paper, the Gaussian mutation performs with the following representation: ' (0,1)k k kx x N= + (2)

where kx and 'kx denote the k-th decision variables before mutation and after mutation,

respectively, (0,1)kN denotes the Gaussian random number with mean zero and standard

deviation one and is generated anew for k-th decision variable. The Cauchy mutation

performs as follows: 'k k kx x δ= + (3)

where kδ denotes the Cauchy random variable with the scale parameter equal to one and is

generated anew for the k-th decision variable.

In the hybrid GC mutation, the values of parameters TC and TG are set by the user beforehand.

The value of TC decides the coarse-grained searching time, while the value of TG has an effect on

the fine-grained searching time. Therefore, both values of the two parameters can not be large

because it will prolong the search process and hence increase the computational overhead.

According to our experimental experience, the moderate values of TC and TG can be set to 2~4.

6. Experiments and test results 6.1. Test problems We choose five problems out of six benchmark problems proposed by Zitzler et al. [29] and call

them ZDT1, ZDT2, ZDT3, ZDT4, and ZDT6. All problems have two objective functions. None of

these problems have any inequality or equality constraints. We describe these problems in Table 1.

The table also shows the number of variables, their bounds, the Pareto-optimal solutions, and the

nature of Pareto-optimal front for each problem. All objective functions in Table 1 are to be

minimized.

Deb et al. have compared the performance of NSGA-II with PAES and SPEA and obtained

their experimental results in [25]. In this paper, we apply the MOEO to solve five difficult test

problems in Table 1 and compare our simulation results with those obtained in [25] under the

same conditions. Furthermore, our approach is also compared with SPEA2 proposed by Zitzler et

al. [27], which is the improvement of SPEA. To be fair, we adopt the identical parameter settings

as suggested in [25]. All approaches are run for a maximum of 25 000 fitness function evaluations

(FFE). In addition, 10 independent runs are carried out. The source codes of all experiments are

coded in JAVA.

For MOEO, the algorithm is encoded in the floating point representation and uses an archive of

size 100. The maximum number of generations for ZDT1, ZDT2 and ZDT3 is 830

(FFE=maximum generation×number of decision variables=830×30=24 900 ≈ 25 000) and for

ZDT4 and ZDT6 is 2500 (FFE=2500×10=25 000). The parameters in the hybrid GC mutation,

i.e., TC and TG, are both set to 3. For SPEA2, the algorithm is also encoded in the floating point

representation. We use a population of size 80 and an external population of size 20 (this 4:1 ratio

is suggested by the developers of SPEA2 to maintain an adequate selection pressure for the elite

solutions), so that the overall population size becomes 100. We use the simulated binary crossover (SBX) operator and polynomial mutation [34]. The crossover probability of 0.9cp = and a

mutation probability of 1/mp n= (where n is the number of variables) are used. The distribution

indexes [34] for crossover and mutation operators are 20cη = and 20mη = are adopted,

respectively. The maximum number of generations for all the test problems is 250

(FFE=maximum generation×population size=250×100=25 000).

Table 1. Test problems used in this study, n is the number of decision variables

Problem n Variable bounds Objective functions Optimal

solutions Pareto front

ZDT1 30 [0,1] ( )1 1( ) ( )[1 / ( )]2 1

( ) 1 9( ) /( 1)2

f X xf X g X x g X

ng X x ni i

== −

= + −∑ =

1 [0,1]

0,2, ,

i

xxi n

∈

=

=

convex

ZDT2 30 [0,1] ( )1 1

2( ) ( )[1 ( / ( )) ]2 1( ) 1 9( ) /( 1)2

f X x

f X g X x g Xng X x ni i

=

= −

= + −∑ =

1 [0,1]

0,2, ,

i

xxi n

∈

=

=

nonconvex

ZDT3 30 [0,1]

( )1 11( ) ( )[1 / ( ) sin(10 )]2 1 1( )

( ) 1 9( ) /( 1)2

f X xx

f X g X x g X xg X

ng X x ni i

π

=

= − −

= + −∑ =

1 [0,1]0,

2, ,i

xxi n

∈

=

=

convex, disconnected

ZDT4 10 1 [0,1][ 5,5],2, ,

i

xxi n

∈∈ −=

( )1 1( ) ( )[1 / ( )]2 1

2( ) 1 10( 1) [ 10cos(4 )]2

f X xf X g X x g X

ng X n x xi i iπ

== −

= + − + −∑ =

1 [0,1]

0,2, ,

i

xxi n

∈

=

=

nonconvex

ZDT6 10 [0,1]

6( ) 1 exp( 4 )sin (6 )1 1 12( ) ( )[1 ( ( ) / ( )) ]2 1

0.25( ) 1 9[( ) /( 1)]2

f X x x

f X g X f X g Xng X x ni i

π= − −

= −

= + −∑ =

1 [0,1]

0,2, ,

i

xxi n

∈

=

=

nonconvex,

nonuniformlyspaced

6.2. Performance measures In this article, we use two performance metrics proposed by Deb et al. [25] to assess the

performance of our approach. For more details, the readers can refer to [25]. The first metric ϒ

measures the extent of convergence to a known set of Pareto-optimal solutions. In all simulations, we address the average ϒ and variance σϒ of this metric obtained in ten independent runs.

Deb et al. [25] has pointed out that even when all solutions converge to the Pareto-optimal front,

the convergence metric does not have a value of zero. The metric will yield zero only when each

obtained solution lies exactly on each of the chosen solutions.

The second metric ∆ measures the extent of spread of the obtained nondominated solutions. It

is desirable to get a set of solutions that spans the entire Pareto-optimal region. The second metric

∆ can be calculated as follows:

1

1

( 1)

Nf l ii

f l

d d d d

d d N d

−

=+ + −

∆ =+ + −∑

(4)

where, fd and ld are the Euclidean distances between the extreme solutions and the boundary

solutions of the obtained nondominated set, id is the Euclidean distance between consecutive

solutions in the obtained nondominated set of solutions, and d is the average of all distances

id ( 1,2, , 1)i N= − , assuming that there are N solutions on the best nondominated front. Note

that a good distribution would make all distances id equal to d and would make 0f ld d= =

(with existence of extreme solutions in the nondominated set). Consequently, for the most widely

and uniformly spreadout set of nondominated solutions, ∆ would be zero.

6.3. Discussion of the results Table 2 shows the mean and variance of the convergence metric ϒ obtained using five

algorithms, i.e., MOEO, NSGA-II (real-coded), PAES, SPEA and SPEA2. In this study, all the

experimental results of NSGA-II (real-coded), SPEA and PAES shown in Table 2 and Table 3

come from [25].

Table 2. Mean (first rows) and variance (second rows) of the convergence metric ϒ

Algorithm ZDT1 ZDT2 ZDT3 ZDT4 ZDT6

0.001277 0.001355 0.004385 0.008145 0.000630 MOEO

0.000697 0.000897 0.00191 0.004011 3.26E-05

0.033482 0.072391 0.114500 0.513053 0.296564 NSGA-II (real-coded) 0.004750 0.031689 0.007940 0.118460 0.013135

0.082085 0.126276 0.023872 0.854816 0.085469 PAES

0.008679 0.036877 0.00001 0.527238 0.006664

0.001799 0.001339 0.047517 7.340299 0.221138 SPEA

0.000001 0 0.000047 6.572516 0.000449

0.001448 0.000743 0.003716 0.028492 0.011643 SPEA2

0.000317 8.33E-05 0.000586 0.047482 0.002397

It can be observed from Table 2 that the MOEO is able to converge better than any other

algorithm on three problems (ZDT1, ZDT4 and ZDT6). For problem ZDT2, the MOEO performs

better than NSGA-II and PAES, but a little worse than SPEA and SPEA2 in terms of convergence.

For problem ZDT3, the MOEO outperforms NSGA-II, PAES and SPEA with respect to the

convergence to the Pareto-optimal front, where SPEA2 performs the best. In all cases with MOEO,

the variance of the convergence metric in ten runs is very small.

Table 3 shows the mean and variance of the diversity metric ∆ obtained using all the

algorithms. As can be seen from Table 3, MOEO is capable of finding a better spread of solutions

than any other algorithm on all the problems except ZDT3. This indicates that our approach has

ability to find a well-distributed set of nondominated solutions than many other state-of-the-art

MOEAs. In all cases with MOEO, the variance of the diversity metric in ten runs is also small. It

is worth noting that for those problems with discontinuous front, the distribution of the

nondominated solutions found by our approach is not very good. We will improve our approach to

handle this problem in the future studies.

Table 3. Mean (first rows) and variance (second rows) of the diversity metric ∆


0.32714 0.285062 0.965236 0.275664 0.225468 MOEO

0.065343 0.056978 0.046958 0.183704 0.033884


1.229794 1.165942 0.789920 0.870458 1.153052 PAES

0.004839 0.007682 0.001653 0.101399 0.003916

0.784525 0.755148 0.672938 0.798463 0.849389 SPEA

0.004440 0.004521 0.003587 0.014616 0.002713

0.472254 0.473808 0.606826 0.705629 0.670549 SPEA2

0.097072 0.0939 0.191406 0.266162 0.077009

It is interesting to note that MOEO can perform well with respect to convergence and diversity

of solutions in problem ZDT4, where there exist 921 different local Pareto-optimal fronts in the search space, of which only one corresponds to the global Pareto-optimal front. This indicates that

MOEO is capable of escaping from the local Pareto-optimal fronts and approaching the global

nondominated front. Furthermore, MOEO is suitable to deal with those problems with

nonuniformly-spaced Pareto front, e.g., problem ZDT6.

For illustration, we also show one of ten runs of MOEO on three test problems (ZDT1, ZDT2

and ZDT3) in Figure 5~ Figure 7, respectively. The figures show all nondominated solutions

obtained after 25 000 fitness function evaluations with MOEO. From Figure 5~Figure 7, we can

see that MOEO is able to converge to the true Pareto-optimal front on all the three problems.

Moreover, MOEO can find a well-distributed set of nondominated solutions on ZD1 and ZDT2.

But the diversity of the nondominated solutions found by MOEO on ZDT3 is not very

good.

Figure 8, which is taken from the literature [25], shows one of ten runs with NSGA-II

and PAES on ZDT4. From Figure 8, we can see that NSGA-II finds better convergence

and spread of solutions than PAES on ZDT4, but both of them can not converge to the

true Pareto-optimal front. Figure 9 shows one of ten runs with MOEO and SPEA2 on

ZDT4. As can be observed from Figure 9, MOEO is able to find a better convergence

and spread of solutions than SPEA2 on ZDT4. It is interesting to notice that MOEO can

converge to the true Pareto-optimal front and find a good spread of solutions that cover

the whole front. From Figure 8 and Figure 9, we can see that MOEO outperforms

NSGA-II, PAES and SPEA2 in terms of convergence and diversity of solutions on

ZDT4.

Figure 5. Nondominated solutions with MOEO on ZDT1



Figure 8.[9] NSGA-II finds better convergence and spread of solutions than PAES on ZDT4, but they can not

converge to the true Pareto-optimal front.

Figure 9. MOEO converges to the true Pareto-optimal front and finds a better spread of solutions than SPEA2

on ZDT4.

Taken from the literature [25], Figure 10 shows one of ten runs with NSGA-II and

SPEA on ZDT6. As can be observed from Figure 10, NSGA-II finds better spread of

solutions than SPEA on ZDT6, but SPEA has a better convergence. Figure 11 shows one

of ten runs with MOEO and SPEA2 on ZDT6. From Figure 11, we can see that MOEO is

able to find a better convergence and spread of solutions than SPEA2 on ZDT6. It is

worth noting that our approach is capable of converging to the true Pareto-optimal front

and finding a well-distributed set of nondominated solutions that cover the whole front

on ZDT6. It can be seen from Figure 10 and Figure 11 that MOEO performs better than

NSGA-II, SPEA and SPEA2 with respect to convergence and diversity of solutions on

ZDT6.

Figure 10.[9] NSGA-II finds better spread of solutions than SPEA on ZDT6, but SPEA has a better convergence.

Figure 11. MOEO can converge to the true Pareto-optimal front and finds a better spread of solutions than

SPEA2 on ZDT6.

In order to compare the running times of MOEO with those of other three algorithms,

i.e., NSGA-II, SPEA2 and PAES, we also show the mean and variance of running times

of each algorithm in ten runs in Table 4. To avoid any bias or misinterpretation when

implementing each of the three other approaches, we adopted the public-domain versions

of NSGA-II, PAES and SPEA2 (the former two can be available at http:// www.lania.mx/

~ccoello/EMOO/EMOOsoftware.html and the last one can be available at http://

www.tik.ee.ethz.ch/pisa/selectors/spea2/spea2_c_source.html). Note that all the

algorithms were run on the same hardware (i.e., Intel Pentium M with 900 MHz CPU and

256M memory) and software (i.e., JAVA) platform. The fitness function evaluations for

all the algorithms is 25 000. It is known that SPEA2 and PAES adopt the clustering

method [27] and the adaptive grid method [24] as the archive truncation method,

respectively. Although there is no archive used in NSGA-II, NSGA-II adopts the

crowding-distance metric [25] to keep diversity of solutions. In this paper, MOEO adopts

the crowding-distance metric to truncate the archive. Thus, to compare the running times

of each algorithm fairly, we use the crowding-distance metric as the archive truncation

method for MOEO, SPEA2 and PAES.

Table 4. Mean (first rows) and variance (second rows) of the running times (in milliseconds)


427.93 436.20 330.81 1971.27 2741.30 MOEO

35.56 61.17 39.66 466.79 100.26


2782.37 2802.36 2884.46 2303.01 2687.53 SPEA2

303.95 167.87 474.73 267.51 302.44

12051.63 12088.77 8890.13 2661.80 25716.33 PAES

1044.91 240.52 357.17 139.36 1685.56

From Table 4, we can see that MOEO is the fastest algorithm in comparison with

other three algorithms on problems ZDT1, ZDT2, ZDT3 and ZDT4. It is important to

notice that MOEO is about 6 times faster than NSGA-II and SPEA2, even

approximately 30 times as fast as PAES on problems ZDT1, ZDT2 and ZDT3. For

problem ZDT6, MOEO runs almost as fast as NSGA-II and SPEA2, nearly 10 times as

fast as PAES. MOEO runs much faster than PAES when the same archive truncation

method is adopted. The results in Table 4 are remarkable if the NSGA-II and the

SPEA2 are normally considered “very fast” algorithms. Therefore, MOEA may be

considered a very fast approach. 6.4. Advanced features of the proposed approach From the above analysis, it can be seen that our approach has the following advanced

features:

Similar to multiobjective evolutionary algorithms, our approach is not susceptible to

the shape of Pareto front.

Only one operator, i.e., mutation operator, exists in our approach, which makes our

approach simple and convenient to be implemented.

The hybrid GC mutation operator suggested in our approach combines the advantages of

coarse-grained search and fine-grained search.

The historical external archive provides the elitist mechanism for our approach.

Our approach provides good performance in both aspects of convergence and

distribution of solutions.

Our approach is capable of handling those problems with multiple local Pareto-

optimal fronts or nonuniformly-spaced Pareto-optimal front.

Compared with three competitive MOEAs in terms of running times on five test

problems, our approach is testified very fast.

7. Conclusions and future work In this paper, we present a novel elitist Pareto-based multiobjective algorithm, called

Multiobjective Extremal Optimization (MOEO). The proposed algorithm extends EO to

handle multiobjective optimization problems with satisfactory results. The fitness

assignment of our approach is based on the Pareto dominance strategy. We also present a

new hybrid mutation operator that enhances the exploratory capabilities of our algorithm.

The experimental simulation studies with five benchmark problems show that MOEO is

highly competitive to the state-of-the-art MOEAs. Moreover, compared with three

competitive MOEAs in terms of running times on five test problems, our approach is

validated very fast. Thus, MOEO may be a good alternative to deal with multiobjective

optimization problems. It is worth pointing out that the problem dimension of the five

test problems studied in this work is not very large. We will explore the efficiency of our

approach on those problems with large number of decision variables in the future. The

future work also includes the studies on how to extend MOEO to solve constrained or

discrete multiobjective optimization problems. It is desirable to further apply MOEO to

solving those complex engineering optimization problems in the real world.

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a novel elitist multiobjective optimization algorithm

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