NC. 1. ETSNT 2009 - A Comparison of Multiobjective Evolutionary Algorithms

7/31/2019 NC. 1. ETSNT 2009 - A Comparison of Multiobjective Evolutionary Algorithms

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Manish Khare

School of IT, Centre for Development of Advanced Computing, Noida, India.

[email protected],Tushar Patnaik

School of IT, Centre for Development of Advanced Computing, Noida, India.

[email protected]

Ashish Khare

Department of Electronics and Communication, University of Allahabad, India.

[email protected]

In this paper, a systematic comparison of various

evolutionary approaches to multiobjectiveoptimization using six carefully chosen test functionsis given. Each test function involves a particular

feature that is known to cause difficulty in theevolutionary optimization process, mainly inconverging to the Pareto-optimal front (e.g.,

multimodality and deception). By investigating thesedifferent problem features separately, it is possible topredict the kind of problems to which a certain

technique is or is not well suited. However, incontrast to what was suspected beforehand, theexperimental results indicate a hierarchy of thealgorithms under consideration.

Evolutionary Algorithms,

Multiobjective optimization, Pareto optimality, test

function.

The term evolutionary algorithm (EA) stands for a

class of stochastic optimization methods that

simulate the process of natural evolution. The origin

of EAs can be traced back to the late 1950s, and

since the 1970s several evolutionary methodologies

have been proposed, mainly genetic algorithms,

evolutionary programming, and evolution strategies.

All of these approaches operate on a set of candidate

solutions. Using strong simplifications, this set is

subsequently modified by the two basic principles:

selection and variation. While selection mimics the

competition for reproduction and resources among

living beings, the other principle, variation imitates

the natural capability of creating new living beings

by means of recombination and mutation [1]. The

paper is structured as follows: Section 2 introduces

key concepts of multiobjective optimization and

defines the terminology used in this paper

mathematically. Section 3 gives a brief overview of

the multiobjective EAs. Section 4 describes the testfunctions, their construction, and their choices.

Section 5 gives experimental results and comparison

of different evolutionary algorithm approach. Section

6 gives the result analysis and conclusion.

Optimization problems involve multiple, conflicting

objectives are often approached by aggregating the

objectives into a scalar function and solving the

resulting single-objective optimization problem. In

contrast, in this study, we are concerned with finding

a set of optimal trade-offs, the so-called Pareto-

optimal set. In the following, we formalize this well-known concept and also define the difference

between local and global Pareto-optimal sets.

A multiobjective search space is partially ordered in

the sense that two arbitrary solutions are related to

each other in two possible ways: either one

dominates the other or neither dominates [1].

Let us consider, without loss of

generality, a multiobjective minimization problem

with m decision variables (parameters) and n

objectives:

Minimize y = f(x) = (f1 (x), ----- , fn (x))

Where x = (x1, ----- , xm) X

y = (y1, ---- ,y) Y(1)

and where x is called decision vector, X parameter

space, y objective vector, and Y objective

space. A decision vector a X is said to dominate a

decision vector b X if and only if

I {1, 2, ----, n} : fi (a) fi (b)

j = {1, 2, ----, n) : fi (a) < fi (b) (2)

National Conference on Emerging Trends in Software & Networking Technologies, April 17-18, 2009 (ETSNT'09)

Amity Institute of Information Technology, Amity University, Uttar Pradesh 205


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Let a X be an arbitrary

decision vector.

1. The decision vector a is said to be

nondominated regarding a set X/ X if and only if

there is no vector in X/ which dominates a; formally

(3)If it is clear within the context which set X / is meant,

we simply leave it out.

2. The decision vector a is Pareto-optimal if

and only if a is nondominated regarding X.

Pareto-optimal decision vectors cannot be improved

in any objective without causing a degradation in at

least one other objective; they represent, in our

terminology, globally optimal solutions. However,

analogous to single-objective optimization problems,

there may also be local optima which constitute a

nondominated set within a certain neighborhood.

This corresponds to the concepts of global and local

Pareto-optimal sets introduced in [2].

Consider a set of decision

vectors X/ X.

The set X/ is denoted as a local Pareto-optimal

set if and only if

(4)

Where || . || is a corresponding distance metric and

>0, >0.

The set X/ is called a global Pareto-optimal set

if and only if

(5)

Note that a global Pareto-optimal set does notnecessarily contain all Pareto-optimal solutions. If

we refer to the entirety of the Pareto-optimal

solutions, we simply write Pareto optimal set; the

corresponding set of objective vectors is denoted as

Pareto-optimal front.

Two major problems must be addressed when an

evolutionary algorithm is applied to multiobjective

optimization[4]:

1. How to accomplish fitness assignment and

selection, respectively, in order to guide the searchtowards the Pareto-optimal set.

2. How to maintain a diverse population in order to

prevent premature convergence and achieve a well

distributed trade-off front.

Often, different approaches are classified with regard

to the first issue, where one can distinguish between

criterion selection, aggregation selection, and Pareto

selection. Methods performing criterion selection

switch between the objectives during the selection

phase. Each time an individual is chosen for

reproduction, potentially a different objective will

decide which member of the population will be

copied into the mating pool. Aggregation selection is

based on the traditional approaches to multiobjective

optimization where the multiple objectives are

combined into a parameterized single objectivefunction. The parameters of the resulting function are

systematically varied during the same run in order to

find a set of Pareto-optimal solutions.

Deb [2] has identified several features that may cause

difficulties for multiobjective EAs in

1) converging to the Pareto-optimal front and

2) maintaining diversity within the population.

Concerning the first issue, multimodality, deception,

and isolated optima are well-known problem areas in

single-objective evolutionary optimization. Thesecond issue is important in order to achieve a well

distributed non dominated front. However, certain

characteristics of the Pareto-optimal front may

prevent an EA from finding diverse Pareto optimal

solutions: convexity or non convexity, discreteness,

and non uniformity. For each of the six problem

features mentioned, a corresponding test function is

constructed following the guidelines in Deb [2]. We

thereby restrict ourselves to only two objectives in

order to investigate the simplest case first. In our

opinion, two objectives are sufficient to reflect

essential aspects of multiobjective optimization.

Moreover, we do not consider maximization or

mixed minimization/maximization problems.

Minimize, T (x) = (f1 (x1), f2 (x2))

Subject to, f2 (x) = g ( x2, x3, ..xm)h (f1 (x1),

g ( x2, x3, ..xm))

Where, x = (x1, x2, ..xm ) (6)

The f1 is a function of the first decision variable only,

g is a function of the remaining m-1 variables, h are

the function values of f1 and g. The test functions

differ in these three functions as well as in the

number of variables m and in the values the variables

may take.

DEFINITION 4[2]: Six test functions T1, T2, ----- T6that follow the scheme given in Equation 6:

The test function t1 has a convex Pareto optimal

front

f1(x1) = x1

g (x2,., xm) = 1 + 9

2

/( 1)m

i

i

x m=

h(f1,g)= 1- 1( / )f g (7)




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where m = 30 & x i[0,1]. The Pareto optimal frontis formed with g=1.

The test function t2 has a non convex counter

part of t1f1(x1) = x1

g (x2,.xm) = 1 + 9.

2

/( 1)m

i

i

x m=

h (f1,g) = 1-

2

1( / )g (8)


The test function t3 represents the discreteness

features: its Pareto optimal front consists of

several non- contiguous convex parts.

f1(x1) = x1

g (x2,.xm) = 1 + 9.2

/( 1)m

i

i

x m=

h (f1,g) = 1- 1 /f g - (f1/g) sin (10 xi) (9)


The introduction of the sine function in h causes

discontinuity in the objective space.

The test function t4 represents 219 local Pareto

optimal sets & therefore tests for EAs ability to

deal with multimodality.

f1(x1) = x1

g (x2,.xm) = 1 + 10 (m-1).(2

2

m

i

i=

10 cos (4 xi))

h (f1,g) = 1 - 1( / )f g (10)

where m=10, xi[0,1] & x2, xm[-5,5].The global Pareto optimal front is formed with g=1,

the best local pareto optimal front g=1.25.

The test function t5 describes a deceptive

problem and distinguishes itself from the other

test functions in that x i represents a binary

string:

f1 (x1) = 1+ u (x1)

g (x2,.xm) =2 ( ( ))

m

ii v u x=

h (f1,g) = 1/ f1 (11)

Where u (xi) gives the number of ones in the bit

vector xi (unitation),

v (u (xi)) =

i i

i

2 + u (x ), if u (x ) < 5

1, if u (x ) = 5

and m = 11, x1 {0, 1}30, and x2,.xm {0, 1}5.The true Pareto optimal front is formed with g (x) =

10, while the test deceptive Pareto optimal front is

represented by the solutions for which g (x) =11. The

global Pareto optimal front as well as the local ones

is convex.

The test function t6 includes two difficulties

caused by the non uniformity of the searchspace:

, the Pareto-optimal solutions are non uniformly

distributed along the global Pareto front (the front is

biased for solutions for which f1 (x) is near one);

, the diversity of the solutions is lower near

the Pareto optimal front highest away from the

front:

f1 (x1) = 1 - exp (-4x1) sin6 (6 x1)

g (x2,.xm) = 1 + 9.(( )0.25

2

/( 1)m

i

i

x m=

h (f1,g) = 1-2

1( / )g (12)

where m = 10, xi [0, 1]. The Pareto optimal frontis formed with g (x) = 1 and is non convex.

We compare with six algorithms on the six described

test functions of previous section.

1. Non-dominated Sorting Genetic Algorithm

(NSGA) [3].

2. Elitist Non-dominated Sorting Genetic

Algorithm (NSGA II) [4].

3. Strength Pareto Evolutionary Approach (SPEA)[5].

4. Improved Strength Pareto Evolutionary

Approach (SPEA2) [6].

5. Pareto Archived Evolution Strategy (PAES) [7].

6. Pareto Enveloped based Selection Algorithm

(PESA) [8].

Independent of the algorithm and the test function,

each simulation run was carried out using the

following parameters:

Number of generations: 250,

Population size : 100

Number of objectives: 2,Crossover rate: 0.9

Mutation rate: 0.03,

Niching parameter : 0.48862

Domination pressure tdom: 10

In Fig. 16, the nondominated fronts achieved by the

different algorithms are visualized. Per algorithm and




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test function, the outcomes of the first five runs were

unified, and then the dominated solutions were

removed from the union set; the remaining points are

plotted in the figures. Also shown are the Pareto-

optimal fronts (lower curves), as well as additional

reference curves (upper curves). The latter curves

allow a more precise evaluation of the obtained

trade-off fronts and were calculated by adding0.1.|max{f2(x)} min{f2 (x)}| to the f2 values of the

Pareto-optimal points.

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

F1

F

2

NSGA2

NSGA

SPEA

SPEA2

PAES

PESA

Fig 1. Test function T1 (convex).

0 0.2 0.4 0.6 0.8 1

0

1

2

3

4

F1

F

2

NSGA2

NSGA

SPEA

SPEA2

PAES

PESA

Fig 2. Test function T2 (nonconvex).

0 0.2 0.4 0.6 0.8

0

1

2

3

4

F1

F2

NSGA2

NSGA

SPEA

SPEA2

PAES

PESA

Fig 3. Test function T3 (discrete).

0 0.2 0.4 0.6 0.8 1

0

10

20

30

40

F1

F

2

NSGA

NSGA2

SPEA

SPEA2

PAES

PESA

Fig 4. Test function T4 (multimodal)

0 5 10 15 20 25 30

0

2

4

6

F1

F

2

NSGA

NSGA2

SPEA

SPEA2

PAES

PESA

Fig 5. Test function T5 (multimodal).

0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0

1

2

3

4

5

6

7

8

F1

F

2

NSGA2

NSGA

SPEA

SPEA2

PAES

PESA

Fig 6. Test function T6 (nonuniform).




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We have carried out a systematic comparison of

several multiobjective EAs on six different test

functions. Major results are:

The each test functions require the choice of

three functions, each of which controls a

particular aspect of difficulity for a

multiobjective GA. One function (f1), tests an

algorithms ability to handle difficulties alongthe pareto-optimal region; function (g) tests an

algorithms ability to handle difficulties lateral

to the Pareto-optimal region; and function (h)

tests an algorithms ability to handle difficulties

arising because of different shapes of the Pareto-

optimal region.

It has been seen that for the chosen test problems

and parameter setting NSGA-II & PESA

outperforms other multi objective evolutionary

algorithm. NSGA-II & SPEA2 follows elitism

mechanism to prevent the loss of non-dominated

solution, but then also NSGA-II gives better

result than SPEA, as NSGA-II does not require

specifying external population size that isrequired in SPEA. If we take large external

population size, then there will be a danger of

the external population being overcrowded with

non-dominated solutions. On the other hand, if a

small external population is used, the effect of

elitism is lost. NSGA-II, SPEA, SPEA2 gives

better result than NSGA, as NSGA lacks elitism

property to prevent the loss of non-dominated

solutions. PAES & PESA gives better result than

NSGA, NSGA-II, SPEA, SPEA2 in case of four

test function T1 (Convex), T2 (Non-Convex), T3(Discreteness), & T6 (Non-uniform), and gives

worst result than other method in case of test

function- T4

(Multi model), T5

(Deceptive), but

PAES & PESA have one big disadvantage, these

two algorithms gives good result, when we take

very large number of iteration (20,000 -25,000),

but all other four algorithms (NSGA, NSGA-II,

SPEA, SPEA2) work properly for small iteration

(250). SPEA2 gives better performance than

SPEA, but not give better performance

comparison to all other algorithm (NSGA,

NSGA-II, PAES, and PESA).

Among the considered test function, T4 (Multi

model) & T5 (Deceptive) seem to be hardest

problem, since none of the algorithms was able

to involve a global Pareto optimal set. Finally, it

can be observed that biased search space

together with non-uniform represented Pareto

optimal front (T6) makes it difficult for the

MOEAs to evolve a well distributed non-

dominated set.

I would like to acknowledge Dr. Alok Singh, Reader

University of Hyderabad, Dr. P. R. Gupta, Head

School of IT, CDAC, Noida for the help and

guidance provided by them from time to time. We

also wish to thanks to Amity University Staff for

publishing this paper in conference.

[1]. Eckart Zitzler (1999): EvolutionaryAlgorithm for Multiobjective Optimization

Methods & Application: Ph. D. Thesis, SwissFederal Institute of Technology (ETH), Zurich.

[2]. Deb, K. (1999): Multi-objective geneticalgorithms: Problem difficulties and construction

of test problems. Evolutionary Computation,7(3):205230.

[3]. Kalyanmoy Deb & N. Srinivas (1995):

Multiobjective Optimization Using Non

Dominated Sorting Genetic Algorithm,Evolutionary Computation Journal 2 (3): 221-

248.[4]. Kalyanmoy Deb, Samir Agarwal, Amrit Pratap

& T. Meyarivan (2000): A Fast Elitist NonDominated Sorting Genetic Algorithm for Multi

Objective Optimization: NSGA-II, IndianInstitute of Technology, Kanpur: Kanpur Genetic

Algorithm Laboratory (KanGAL) Report No.-200001.

[5]. Eckart Zitzler & Lothar Thiele (1998): AnEvolutionary Algorithm for Multi Objective

Optimization: The Strength Pareto Approach,Technical Report, 43, Zurich, Switzerland:

Computer Engineering & Networks Laboratory(TIK), Swiss Federal Institute of Technology(ETH).

[6]. Eckart Zitzler, Marco Laumanns & Lothar Thiele

(2001): SPEA2: Improving The StrengthPareto Evolutionary Algorithm, TechnicalReport, 103, Zurich, Switzerland: ComputerEngineering & Networks Laboratory (TIK),

Swiss Federal Institute of Technology (ETH).[7]. J. D. Knowles & D. W. Corne (2000):

Approximating The Non Dominated Front UsingThe Pareto Archived Evolution Strategy,Evolutionary Computation Journal 8 (2): 149-172.

[8]. D. W. Corne, J. D. Knowles & M. J. Ootes(2000): The Pareto Envelope-Based Selection

Algorithm for Multiobjective Optimization. In

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NC. 1. ETSNT 2009 - A Comparison of Multiobjective Evolutionary Algorithms