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MGA.rar （18个子文件）

MGA

license.txt 1KB

MGA

plot_objective.m 476B

non_domination_sort_mod.m 5KB

evaluate_objective.m 969B

initialize_variables.m 2KB

NSGA_2.pdf 365KB

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tournament_selection.html 6KB

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genetic_operator.html 10KB

initialize_variables.html 4KB

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crowding_distance.m 3KB

MULTI-OBJECTIVE OPTIMIZATION USING EVOLUTIONARY

ALGORITHMS (MOEA)

ARAVIND SESHADRI

1. Multi-Objective Optimization Using NSGA-II

NSGA (Non-Dominated Sorting in Genetic Algorithms [5]) is a popular non-

domination based genetic algorithm for multi-objective optimization. It is a very

eﬀective algorithm but has been generally criticized for its computational com-

plexity, lack of elitism and for choosing the optimal parameter value for sharing

parameter σ

. A modiﬁed version, NSGA-II ( [3]) was developed, which has a

better sorting algorithm , incorporates elitism and no sharing para meter needs to

be chosen a priori. NSGA-II is discussed in detail in this report and two sample

test functions are optimized using it.

2. General Description of NSGA-II

The population is initialized as usual. Once the population in initialized the

population is sorted based on non-domina tion into each front. The ﬁrst front being

completely non-dominant set in the current population and the second front being

dominated by the individuals in the ﬁrst front only and the front goes so on. Each

individual in the each front are assigned rank (ﬁtness) values or based on front in

which they belong to. Individuals in ﬁrst front are given a ﬁtness value of 1 and

individuals in second are ass igned ﬁtness value as 2 and so on.

In addition to ﬁtness value a new parameter called crowding distance is cal-

culated for each individual. The crowding distance is a measure of how close an

individual is to its neighbors. Larg e average crowding distance will result in better

diversity in the population.

Parents are selected from the population by using binary to urnament selection

based on the rank and crowding distance. An individual is selected in the rank is

lesser than the other or if crowding distance is gr e ater than the other

. The selected

population generates oﬀsprings from crossover and mutation operators, which will

be discussed in deta il in a later section.

The population with the current population a nd current oﬀsprings is sorted again

based on non-domination and only the best N individuals are selected, where N is

the population size. The selection is based on rank and the o n crowding distance

on the last front.

3. Detailed Description of NSGA-II

3.1. Population Initialization. The population is initialized based on the prob-

lem range and cons traints if any.

Crowding distance is compared only if the rank for b oth individuals are same

2 ARAVIND SESHADRI

3.2. Non-Dominated sort. The initialized population is so rted based on non-

domination

. The fast sort a lgorithm [3] is described as below for each

• for each individual p in main population P do the following

– Initialize S

= ∅. This set would contain all the individuals that is

being dominated by p.

– Initialize n

= 0. This would be the number of individuals that domi-

nate p.

– for each individual q in P

∗ if p dominated q then

· add q to the set S

i.e. S

= S

∪ {q}

∗ else if q dominates p then

· increment the domination c ounter for p i.e. n

= n

+ 1

– if n

= 0 i.e. no individuals dominate p then p belong s to the ﬁrst

front; Set rank of individual p to one i.e p

rank

= 1. Update the ﬁrst

front set by adding p to fro nt one i.e F

= F

∪ {p}

• This is carr ie d out for all the individuals in main population P .

• Initialize the front counter to one. i = 1

• following is carr ied out while the i

front is nonempty i.e. F

6= ∅

– Q = ∅. The set for storing the individuals for (i + 1)

front.

– for each individual p in front F

∗ for e ach individual q in S

is the set of individuals do mina ted

by p)

· n

= n

−1, decrement the domination count for individual

· if n

= 0 then none of the individuals in the subsequent

fronts would dominate q. Hence set q

rank

= i + 1. Update

the set Q with individual q i.e. Q = Q ∪ q.

– Increment the front counter by one.

– Now the set Q is the next front and hence F

= Q.

This algorithm is better than the original NSGA ( [5]) since it utilize the infor-

mation about the set that an individual domina te (S

) and number of individuals

that dominate the individual (n

3.3. Crowding Distance. Once the non-dominated sort is complete the crowding

distance is assigned. Since the individuals are selected based on rank and crowding

distance all the individuals in the population are assigned a c rowding distance value.

Crowding distance is assigned front wise and comparing the crowding dis tance

between two individuals in diﬀerent front is meaning le ss. The cr owing distance is

calculated as below

• For each front F

, n is the number of individuals.

– initialize the distance to be zero for all the individuals i.e. F

) = 0,

where j corresponds to the j

individual in front F

– for each objective function m

∗ Sort the individuals in front F

based on objective m i.e. I =

sort(F

, m).

An individual is said to dominate another if the objective fu nctions of it is no worse than the

other and at least in one of its objective functions it is better than the other

NSGA-II 3

∗ Assign inﬁnite distance to boundary values for each individual

in F

i.e. I(d

) = ∞ and I(d

) = ∞

∗ for k = 2 to (n − 1)

· I(d

) = I(d

) +

I(k + 1).m − I(k − 1).m

max

− f

min

· I(k).m is the value of the m

objective function of the k

individual in I

The basic idea behind the crowing distance is ﬁnding the euclidian distance

between each individual in a front based on their m objectives in the m dimensional

hyper spac e . The individuals in the boundary are always selected since they have

inﬁnite distance assignment.

3.4. Selection. Once the individuals are sorted based on non-do mina tion and

with crowding distance assigned, the selection is carried out using a crowded-

comparison-operator (≺

). The comparison is ca rried out as below based on

(1) non-domination rank p

rank

i.e. individuals in front F

will have their rank

as p

rank

= i.

(2) crowding distance F

)

• p ≺

q if

– p

rank

< q

rank

– or if p and q belong to the same front F

then F

) > F

) i.e. the

crowing distance should be more.

The individuals ar e selected by using a binary tournament selection with crowed-

comparison- operator.

3.5. Genetic Operators. Real-coded GA’s use Simulated Binary Crossover

(SBX) [2], [1] operator for cross over and polynomial mutation [2], [4].

3.5.1. Simulated Binary Crossover. Simulated binar y cro ssover simulates the bi-

nary crossover observed in nature and is give as below.

1,k

[(1 − β

1,k

+ (1 + β

2,k

]

2,k

[(1 + β

1,k

+ (1 − β

2,k

]

where c

i,k

is the i

child with k

component, p

i,k

is the s e le c ted parent and β

(≥ 0) is a sample from a random number generated having the density

p(β) =

(η

+ 1)β

, if 0 ≤ β ≤ 1

p(β) =

(η

+ 1)

, if β > 1.

This distribution can be obtained from a uniformly sampled random number u

between (0, 1). η

is the distribution index for crossover

. That is

β(u) =(2u)

(η+1)

β(u) =

[2(1 − u)]

(η+1)

This determine how well spread the children will be from their parents.

4 ARAVIND SESHADRI

Population Generations Po ol Size Tour Size η

200 500 50 2 20 20

Table 1. MOP1- Parameters for NSGA-II

3.5.2. Polynomial Mutation.

= p

+ (p

− p

)δ

where c

is the child and p

is the parent with p

being the upper bound

the parent compo nent, p

is the lower bound and δ

is small variation which is

calculated from a polynomial distribution by using

=(2r

)

+ 1

− 1, if r

< 0.5

=1 − [2(1 − r

)]

+ 1

if r

≥ 0.5

is an uniformly sampled random number between (0, 1) a nd η

is mutation

distribution index.

3.6. Recombination and Selection. The oﬀspring population is combined with

the current generatio n population and selection is performed to set the individuals

of the next generation. Since all the previous and current best individuals are

added in the population, elitism is ensured. Population is now sor ted based on

non-domination. The new generation is ﬁlled by each front subsequently until the

population size exceeds the current population size. If by adding all the individuals

in front F

the population exceeds N then individuals in front F

are selected based

on their crowding distance in the descending order until the population size is N.

And hence the process repeats to generate the subsequent generations.

4. MOP1 (Example Problem - 1)

This problem is to ﬁnd the global pareto front for a discontinuous function given

by objective function as in [3]

(x) =1 − e

−4x

sin

(6πx

)

(x) =g(x)(1 − (

(x)

g(x)

)

where,

g(x) =1 + 9(

i=1

)

0.25

subject to 0 ≤ x

≤ 1, i = 1, . . . , 6. This set of objective function resulted in the

following solution as shown in Figure 1.

The decision space upper bound and lower bound for that particular component.

NSGA-II 5

0.2 0.4 0.6 0.8 1

0.2

0.4

0.6

0.8

1.2

1.4

f(x

)

f(x

)

Pareto Optima for MOP1 (Objective Space Solution)

Figure 1. MOP1

Population Generations Po ol Size Tour Size η

200 1000 100 2 20 20

Table 2. MOP2- Parameters for NSGA-II (Figures 2, 3,4)

5. MOP2 (Example Problem - 2)

This problem is to ﬁnd the global pareto front for the 3 dimensional objective

space for the following function.

(x) =(1 + g(x)) cos(0.5πx

) cos(0.5πx

)

(x) =(1 + g(x)) cos(0.5πx

) sin(0.5πx

)

(x) =(1 + g(x)) sin(0.5πx

)

(x) =

i=1

− 0.5)

subject to 0 ≤ x

≤ 1, i = 1, . . . , 12. The simulation results are shown in Figures

2 to 9.

A modiﬁcation in the objective function resulted in similar result as shown in [6].

g(x) =

i=3

− 0.5)

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