2009 - Deb - Multi-Objective Evolutionary Algorithms (Slides)

8/6/2019 2009 - Deb - Multi-Objective Evolutionary Algorithms (Slides)

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Multi-Objective Evolutionary Algorithms

Kalyanmoy Deb a

Kanpur Genetic Algorithm Laboratory (KanGAL)

Indian Institute of Technology KanpurKanpur, Pin 208016 INDIA

[email protected]

http://www.iitk.ac.in/kangal/deb.htmlaCurrently visiting TIK, ETH Zurich

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Overview of the Tutorial

Multi-objective optimization Classical methods

History of multi-objective evolutionary algorithms (MOEAs)

Non-elitst MOEAs

Elitist MOEAs

Constrained MOEAs Applications of MOEAs

Salient research issues

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Multi-Objective Optimization

We often face them

B

C

Comfo

rt

Cost10k 100k

90%

1

2

A

40%

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More Examples

A cheaper but inconvenient

flight

A convenient but expensive

flight

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Which Solutions are Optimal?

Domination:

x(1) dominates x(2) if

1. x(1) is no worse than x(2) in all

objectives

2. x(1) is strictly better than x(2)

in at least one objective

f

1

14

3

(maximize)f1

6 102 18

6

3

5

4

1

2

5

2

(minimize)

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Pareto-Optimal Solutions

Non-dominated solutions: Among

a set of solutions P, the non-

dominated set of solutions P

are those that are not dominatedby any member of the set P.

O(M N2) algorithms exist.

Pareto-Optimal solutions: When

P = S, the resulting P is Pareto-

optimal set

(maximize)

14

2

3

f1

6 102 18

1

f

5

(minimize)2

1

4

5

3

Nondominatedfront

A number of solutions are optimal

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Pareto-Optimal Fronts

MinMax

2

f

2

1

1

MinMin

f

MaxMin MaxMax

f2

f

1

f2

f1

f

f

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Preference-Based Approach

optimization problem

Minimize f

Singleobjective


a composite function

or

1 1 2 MM2

Multiobjective

Minimize f

......

Minimize fw

Higherlevel1

2

M (w ..1w2

)

information

M

F = w f + w f +...+ w f

One optimumsolution

optimizer

Singleobjective

Estimate a

relative

importancevector

subject to constraints

Classical approaches follow it

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Classical Approaches

No Preference methods (heuristic-based) Posteriori methods (generating solutions)

A priori methods (one preferred solution)

Interactive methods (involving a decision-maker)

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Weighted Sum Method

Construct a weighted sum of

objectives and optimize

F(x) =M

m=1

wmfm(x).

User supplies weight vector w

Feasible objective space

Paretooptimal front

b

1

a

w

cd

A

w

1

2

f2

f

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Difficulties with Weighted Sum Method

Need to know w Non-uniformity in Pareto-

optimal solutions

Inability to find somePareto-optimal solutions

Paretooptimal front

D

Feasible objective space

B

C

ab

f

2f

1

A

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-Constraint Method

Optimize one objective,constrain all other

Minimize f(x),

subject to fm(x) m, m = ;

User supplies a vector1

B

1

D

1 1

a b c d

C

1

f2

f

Need to know relevant vectors

Non-uniformity in Pareto-optimal solutions

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Difficulties with Most Classical Methods

Need to run a single-

objective optimizer many

times

Expect a lot of problem

knowledge Even then, good distribu-

tion is not guaranteed

Multi-objective optimiza-tion as an application of

single-objective optimiza-

tion

1

f

f

2

frontParetooptimal

D

A

B

C

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0.10 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

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Ideal Multi-Objective Optimization

Minimize f

Step

1

Multiple tradeoffsolutions found

......Minimize fMinimize f

Step 2

subject to constraints

Multiobjective


M

1

2

Choose onesolution

Higherlevel

information

Multiobjectiveoptimizer

IDEAL

Step 1 Find a set of Pareto-optimal solutions

Step 2 Choose one from the set

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Advantages of Ideal Multi-Objective Optimization

Decision-making becomes easier and less subjective

Single-objective optimization is a degen-

erate case of multi-objective optimiza-tion

Step 1 finds a single solution

No need for Step 2 Multi-modal optimization is a special

case of multi-objective optimization

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Two Goals in Ideal Multi-Objective Optimization

1. Converge on the Pareto-

optimal front

2. Maintain as diverse a distri-

bution as possible

f1

2f

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Why Evolutionary?

Population approach suits well to find multiple solutions

Niche-preservation methods can be exploited to find diverse

solutions

2

1

1

0.8

0.6

0.4

0.2

10.80.60.40.200

f

f 1

2f

1

0.8

0.6

0.4

0.2

0

10.80.60.40.20

f

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History of Multi-Objective Evolutionary

Algorithms (MOEAs)

Early penalty-based ap-proaches

VEGA (1984)

Goldbergs suggestion(1989)

MOGA, NSGA, NPGA

(1993-95) Elitist MOEAs (SPEA,

NSGA-II, PAES, MOMGA

etc.) (1998 Present)

Number

of

Studies

Year

0

20

40

60

80

100

120

140

1992 1993 1994 1995 1996 1997 1998 199919911990and before

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What to Change in a Simple GA?

Modify the fitness computation

Initialize Population

Cond?

Begin

Reproduction

Mutation

Crossover

t = t + 1

t = 0

Stop

No

Yes Evaluation

Assign Fitness

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Identifying the Non-dominated Set

Step 1 Set i = 1 and create an empty set P.

Step 2 For a solution j P (but j = i), check if solution j

dominates solution i. If yes, go to Step 4.

Step 3 If more solutions are left in P, increment j by one and go

to Step 2; otherwise, set P = P {i}.

Step 4 Increment i by one. If i N, go to Step 2; otherwise stop

and declare P as the non-dominated set.

O(M N2) computational complexity

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An Efficient Approach

Kung et al.s algorithm (1975)

Step 1 Sort the population in descend-ing order of importance of f1

Step 2, Front(P) If |P| = 1,

return P as the output

of Front(P). Otherwise,

T = Front(P(1)P(|P|/2)) and

B = Front(P(|P|/2+1)P(|P|)).

If the i-th solution ofB is not dom-inated by any solution of T, create

a merged set M = T {i}. Return

M as the output of Front(P).

T1

B1

T2

B2

Merging

O

N(log N)M2

for M 4 and O(Nlog N) for M = 2 and 3

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A Simple Non-dominated Sorting Algorithm

Identify the best non-dominated set

Discard them from population

Identify the next-best non-dominated set

Continue till all solutions are classified

We discuss a O(M N2) algorithm later

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Non-Elitist MOEAs

Vector evaluated GA (VEGA) (Schaffer, 1984)

Vector optimized EA (VOES) (Kursawe, 1990)

Weight based GA (WBGA) (Hajela and Lin, 1993)

Multiple objective GA (MOGA) (Fonseca and Fleming, 1993) Non-dominated sorting GA (NSGA) (Srinivas and Deb, 1994)

Niched Pareto GA (NPGA) (Horn et al., 1994)

Predator-prey ES (Laumanns et al., 1998)

Other methods: Distributed sharing GA, neighborhood

constrained GA, Nash GA etc.

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Non-Dominated Sorting GA (NSGA)

A non-dominated sorting of the population

First front: Fitness F = N to all

Niching among all solutions in first front

Note worst fitness (say F1w)

Second front: Fitness F1w 1 to all

Niching among all solutions in second front

Continue till all fronts are assigned a fitness

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Non-Dominated Sorting GA (NSGA)

f1 f2 Fitness

x Front before after

1.50 2.25 12.25 2 3.00 3.00

0.70 0.49 1.69 1 6.00 6.00

4.20 17.64 4.84 2 3.00 3.00

2.00 4.00 0.00 1 6.00 3.43

1.75 3.06 0.06 1 6.00 3.43

3.00 9.00 25.00 3 2.00 2.00 4

1

52

3

1

6

2 15

20

25

30

0 5

10

25 30

f

5

15100

20

f

Niching in parameter space

Non-dominated solutions are emphasized

Diversity among them is maintained

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Vector-Evaluated GA (VEGA)

Divide population into M equal blocks

Each block is reproduced with one objective function

Complete population participates in crossover and mutation

Bias towards to individual best objective solutions

A non-dominated selection: Non-dominated solutions are

assigned more copies

Mate selection: Two distant (in parameter space) solutions aremated

Both necessary aspects missing in one algorithm

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Multi-Objective GA (MOGA)

Count the number of domi-nated solutions (say n)

Fitness: F = n + 1

A fitness ranking adjust-ment

Niching in fitness space

Rest all are similar toNSGA

F Asgn. Fit.

1 2 3 2.5

2 1 6 5.0

3 2 2 2.5

4 1 5 5.0

5 1 4 5.0

6 3 1 1.0

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Niched Pareto GA (NPGA)

Solutions in a tournament are checked for domination with

respect to a small subpopulation (tdom)

If one dominated and other non-dominated, select second

If both non-dominated or both dominated, choose the one with

smaller niche count in the subpopulation

Algorithm depends on tdom

Nevertheless, it has both necessary components

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NPGA (cont.)

YX

XY

Parameter Space

Check fordomination

t_dom

Population

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Shortcoming of Non-Elitist MOEAs

Elite-preservation is missing

Elite-preservation is important for proper convergence in

SOEAs

Same is true in MOEAs

Three tasks

Elite preservation

Progress towards the Pareto-optimal front

Maintain diversity among solutions

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Elitist MOEAs

Elite-preservation:

Maintain an archive of non-dominated solu-

tions

Progress towards Pareto-optimal front:

Preferring non-dominated solutions

Maintaining spread of solutions:

Clustering, niching, or grid-based competi-

tion for a place in the archive

EliteEA

(maximize)

14

2

3

f1

6 102 18

1

f

5

(minimize)2

1

4

5

3

Nondominatedfront

f2

f1

clusters

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Elitist MOEAs (cont.)

Distance-based Pareto GA (DPGA) (Osyczka and Kundu,

1995) Thermodynamical GA (TDGA) (Kita et al., 1996)

Strength Pareto EA (SPEA) (Zitzler and Thiele, 1998)

Non-dominated sorting GA-II (NSGA-II) (Deb et al., 1999)

Pareto-archived ES (PAES) (Knowles and Corne, 1999)

Multi-objective Messy GA (MOMGA) (Veldhuizen and

Lamont, 1999)

Other methods: Pareto-converging GA, multi-objective

micro-GA, elitist MOGA with coevolutionary sharing

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Elitist Non-dominated Sorting Genetic Algorithm

(NSGA-II)

Non-dominated sorting: O(M N2)

Calculate (ni, Si) for each

solution i

ni: Number of solutions

dominating i Si: Set of solutions domi-

nated by i

f

f

1

2

3

10

11

9

4

8

7

6

5

(5, Nil)

(0, {9,11})

2

1

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NSGA-II (cont.)

Elites are preserved

sortingCrowding

3

t

distancesorting

Nondominated

t+1

F

F1

2

F

Q

R

P

Rejectedt

tP

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NSGA-II (cont.)

Diversity is maintained: O(M Nlog N)

Cuboid

f

f

1

2

ii-1

i+1

0

l

Overall Complexity: O(M N2)

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NSGA-II Simulation Results

1

2

NSGAII (binarycoded)

0.8

0.7

0.4

0.9

0.5

f

1.1

1

0.6

0.3

0.2

0.1

10.90.80.70.60.50.40.30.20.10

f

0

NSGAII

1415161718192012

2

0

2

4

6

8

10

f2

f1

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Strength Pareto EA (SPEA)

Stores non-dominated solutions externally

Pareto-dominance to assign fitness

External members: Assign number of dominated solutions

in population (smaller, better)

Population members: Assign sum of fitness of external

dominating members (smaller, better)

Tournament selection and recombination applied to combined

current and elite populations

A clustering technique to maintain diversity in updated

external population, when size increases a limit

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SPEA (cont.)

Fitness assignment and clustering methods

Function Space

xxxxx

Population Ext_popFitness Assignment

x

x

xx

Function Space

Clustering (d and p_max)

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Pareto Archived ES (PAES)

An (1+1)-ES

Parent pt and child ct are compared with an external archive At

If ct is dominated by At, pt+1 = pt

If ct dominates a member of At, delete it from At and include

ct in At and pt+1 = ct

If |At| < N, include ct and pt+1 = winner(pt, ct)

If |At| = N and c

tdoes not lie in highest count hypercube H,

replace ct with a random solution from H and

pt+1 = winner(pt, ct).

The winner is based on least number of solutions in the hypercube

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Niching in PAES-(1+1)

front

2

Paretooptimal

f

f1

3

1

2

Offspring

Parent

front

Paretooptimal 1

2

f

f

Parent

Offspring

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Constrained Handling

Penalty function approach

Fm = fm + Rm(g).

Explicit procedures to handle infeasible solutions

Jimenezs approach

Ray-Tang-Seows approach

Modified definition of domination

Fonseca and Flemings approach

Deb et al.s approach

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Constrain-Domination Principle

A solution i constrained-

dominates a solution j, if any is

true:

1. Solution i is feasible and so-

lution j is not.

2. Solutions i and j are both in-

feasible, but solution i has a

smaller overall constraint vi-

olation.3. Solutions i and j are feasible

and solution i dominates so-

lution j.

2

f

f

2

1

1

3

5

6

4

3

4

5

0.8 0.9 10.70.60.50.40.30.20.1

10

8

6

4

2

0

Front 1

Front 2

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Constrained NSGA-II Simulation Results

2

1f

10

8

6

4

2

10

8

6

4

2

10.90.80.70.60.50.40.30.20.10

10.90.80.70.60.50.40.30.20.10

f

1

2

1.4

1.2

1

0.8

0.6

0.4

0.2

1.41.210.80.60.40.2

f

0

f

0

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Applications of MOEAs

Space-craft trajectory optimization

Engineering component design

Microwave absorber design

Ground-water monitoring

Extruder screw design

Airline scheduling

VLSI circuit design

Other applications (refer Deb, 2001 and EMO-01 proceedings)

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Spacecraft Trajectory Optimization

Coverstone-Carroll et al. (2000) with JPL Pasadena

Three objectives for inter-planetary trajectory design Minimize time of flight

Maximize payload delivered at destination

Maximize heliocentric revolutions around the Sun

NSGA invoked with SEPTOP software for evaluation

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EarthMars Rendezvous

1 1.5 2 2.5 3 3.50

100

200

300

400

500

600

700

800

900

2236

132

72

Transfer Time (yrs.)

73

44

ass

e

vere

o

ar

ge

g.

Individual 44

Earth

09.01.05Mars

10.16.06

Individual 73

Mars

09.22.07

Earth

09.01.05

Individual 72

Mars

08.25.08

Earth

09.01.05

Individual 36

Mars

02.04.09

Earth

09.01.05

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Salient Research Tasks

Scalability of MOEAs to handle more than two objectives

Mathematically convergent algorithms with guaranteed spread

of solutions

Test problem design Performance metrics and comparative studies

Controlled elitism

Developing practical MOEAs Hybridization, parallelization

Application case studies

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Hybrid MOEAs

Combine EAs with a local search method

Better convergence

Faster approach

Two hybrid approaches

Local search to update each solution in an EA population

(Ishubuchi and Murata, 1998; Jaskiewicz, 1998)

First EA and then apply a local search

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Posteriori Approach in an MOEA

MOEAProblem

local searches

Multiple

NondominationcheckClustering

Which objective to use in local search?

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Proposed Local Search Method

Weighted sum strategy (or a Tchebycheff metric)

F =i

wi fi

fi is scaled

Weight wi chosen based on location of i in the obtained front

wj =(fmaxj fj(x))/(f

maxj f

minj )M

k=1(fmaxk fk(x))/(f

maxk f

mink )

Weights are normalized

i

wi = 1

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Fixed Weight Strategy

Extreme solutions are as-

signed extreme weights

Linear relation betweenweight and fitness

Many solution can converge

to same solution after local

searchlocal search

max

min

set after

MOEA solution set

1f

f

f

maxmin

2

2

f1f1

b

a

A

2f

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Design of a Cantilever Plate

100 mm

60 mm

P

Base plate

2

4

6

8

10

12

14

16

18

20 25 30 35 40 45 50 55 60

Sca

l

ed

de

flect

ion

Weight

Nondominated solutions

2

4

6

8

10

12

14

16

18

20 25 30 35 40 45 50 55 60

Clustered solutions

2

4

6

8

10

12

14

16

18

20 25 30 35 40 45 50 55 60

Sca

le

d

de

fle

ct

ion

Weight

NSGAII

2

4

6

8

10

12

14

16

18

20 25 30 35 40 45 50 55 60

Sca

le

d

de

fle

ct

ion

Weight

Local search

Weight

Sca

le

d

de

flect

ion

Nine trade-off solutions are chosen

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Trade-off Solutions

(1.00, 0.00) (0.60, 0.40) (0.50, 0.50)

(0.43, 0.57) (0.38, 0.62) (0.35, 0.65)

(0.23, 0.77) (0.14, 0.86) (0.00, 1.00)

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Conclusions

Ideal multi-objective optimization is generic and pragmatic

Evolutionary algorithms are ideal candidates

Many efficient algorithms exist, more efficient ones are needed

With some salient research studies, MOEAs will revolutionize

the act of optimization

EAs have a definite edge in multi-objective optimization andshould become more useful in practice in coming years

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Documents

2009 - Deb - Multi-Objective Evolutionary Algorithms (Slides)