유전자알고리즘(Genetic Algorithm)1853623F… · · 2015-04-29Center for Advanced e-System Integration Technology 유전자알고리즘(Genetic Algorithm) [목차] Concept

Center for Advanced eCenter for Advanced e--System Integration TechnologySystem Integration Technology

유전자 알고리즘 (Genetic Algorithm)유전자 알고리즘 (Genetic Algorithm)

[목 차]

Concept of Genetic AlgorithmTerminologyGenetic Operator Examples of Simple Genetic AlgorithmsApplication for GA

Center for Advanced eCenter for Advanced e--System Integration Technology, System Integration Technology, KonkukKonkuk Univ.Univ.

Principal Heuristic AlgorithmsPrincipal Heuristic Algorithms

Genetic Algorithms (Holland – 1975) : Today’s issue√ Inspired by genetics and natural selection

Simulated Annealing (Kirkpatrick – 1983)√ Inspired by molecular dynamics – energy minimization

Particle Swarm Optimization (Eberhart and Kennedy -1995)√ Inspired by the social behavior of swarms of insects or flocks of

birds

These techniques all use a combination of randomness and heuristic “rules” to guide the search for global maxima or minima


What is a Heuristic?What is a Heuristic?

A Heuristic is simply a rule of thumb that hopefully will find a good answer. Why use a Heuristic? √ Heuristics are typically used to solve complex (large, nonlinear, nonconvex

(ie. contain many local minima)) multivariate combinatorial optimization problems that are difficult to solve to optimality.

Unlike gradient-based methods in a convex design space, heuristics are NOT guaranteed to find the true global optimal solution in a single objective problem, but should find many good solutions (the mathematician's answer vs. the engineer’s answer) Heuristics are good at dealing with local optima without getting stuck in them while searching for the global optimum.


1) Concept of Genetic Algorithm1) Concept of Genetic Algorithm


Concept of Genetic Algorithm (1)Concept of Genetic Algorithm (1)

Genetic Algorithm √ 생물학적진화이론과유전학에기반

√ 우수한형질의개체가자연계에잘적응하여우수한후손을생성한다는원리

이용

√ 전통적인최적화알고리즘과는달리도함수(Gradient)를이용하지않음√ 이진수의조합으로구성된개체(individual 혹은염색체)들의집단(population)을가지고선택(selection)

Genetic Algorithm의 도입대규모 조합의 최적화 문제제한조건이 많은 이산화문제의 최적화

Objective

Goldberg & Santani (1989): GA의 구조물설계적용

도입

적용Holland (1975)“Adaptation in Naturaland Artificial Systems”

http://cs.felk.cvut.cz/~xobitko/ga/dnapic.html



Genetic Algorithm의 특징 및 장점√ 일반적인공학적인최적화문제의경우그현상이비선형적인거동을보이는경우가많기때문에

전통적인함수의구배를이용하는탐색기법에서는 Local Optimum에도달하는경우가많이존재함

√ Genetic Algorithm의경우, 구배정보를이용하지않으며주어진설계공간전역에대한탐색을수행하기때문에 Global Optimum을도출할수있는확률이구배법에비해많으며초기치에의존하는경우가없음

√ 특히최적해는알지못하지만평가는할수있는 Black Box형태의공학적문제에서는대단히유용하게적용될수있다.

√ Ex. Maximize F(x)=-cos(x)*cos(x/20)

* GA Results

Evaluation Count: 560

;Best Ref. Val = 0.987719237804

X=-3.133696794510

* GBM Results

Initial Point(12.)

Function Call 22

;F(x) = 0. 891295

X= 9.39938

* GBM Results

Initial Point(-4.)

Function Call 17

;F(x) =0. 987719

X= -3.13369

Global Optimum

Local Optimum



분류

Evolutionary Algorithm(진화 알고리즘)

Genetic Algorithm (GA)- 가장 대중적으로 알려짐

Evolutionary Programming (EP)(GA+Data Structure)

Evolution Strategies



TerminologyTerminology

1 1 0 0 0 0 1 1

Solutions

유전자형(Genotype)

표현형(Phenotype)

Decoding Encoding

유전자좌(Locus): 염색체 상에서 유전자의 위치

염색체(Chromosome):2진수로 표현된 String

유전자(Genes, Bit) : 2진수로 표현된 값


Genetic Optimizer Terminology Genetic Optimizer Terminology

0101:1101:1101:0001

1001:1001:1101:0001

0101:1101:1111:1011

1001:1001:0001:0001

0101:1101:1100:0000

1000:0001

3.교 차(Crossover)

3.변 이(Mutation)

2.재 생 산(Reproduction)

1.개 체 군(Population)

4.적합도 판단(Fitness Value)

5.세 대(Generation)


Structure of Genetic AlgorithmStructure of Genetic Algorithm

Procedure GABegin

t→0initialize P(t)evaluate P(t)while (not termination condition) do

recombine P(t) to yield C(t) // selection, crossover, mutation

evaluate C(t)select P(t+1) from P(t) and C(t)t←t+1

endEnd


Encoding - DecodingEncoding - Decoding

Biology

Design

Genotype(유전자형) Phenotype(표현형)

Coded Domain Decision Domain

“Blue Eyes”

Radius R= 2.57[m]

Expression

Sequencing

Decoding

Encoding


DecodingDecoding

E.g. binary encoding of integers:

Genotype(유전자형)

Phenotype(표현형)


Binary Encoding IssuesBinary Encoding Issues

Number of bits dedicated to a particular design variable is veryimportant.

Resolution depends on:√ upper and lower bounds xLB, xUB

√ number of bits

Example

Loss in precision !!!


Genetic OperatorGenetic Operator

GA에 적용되는 중요한 연산자

연산자 선 택(Selection) 교 차(Crossover) 변 이(Mutation)

-다음세대로 전달하기위한 유전자의 선택

-선택된 염색체들을조합,다음세대의염색체를 생성

-유전자를 일정한 확률로 변화시키는 조작

기 능

-다음세대로 높은 적합도를 가지는 유전자의특징을 전달

-수렴속도 가속화-높은 최적치를 남길가능성 부여

-전역적 탐색효과의극대화

효 과

-적합도 비례전략-순위전략-토너먼트 선택전략-엘리트 보존 전략

-단순 교차-복수점 교차-일점 교차

-정적변이-동적변이

전 략



Genetic Operator : Selection (1)Genetic Operator : Selection (1)

선택 (Selection)√ Goal is to select parents for crossover

√ Should create a bias towards more fitness

√ Must preserve diversity in the population

This scheme tends to favor the fittest individuals in

a population more than the ranking-scheme, faster

convergence, but can also be a disadvantage.

1. 순위전략(Selection according to RANKING)

•적합도(fitness)의해 각 개체에 순위를

부여하여 그 순위에 의해 사전에 결정된

확률로 자손을 남김

(Better ranking has a higher probability of being

chosen)

•적합도 와 순위에 의해 부여되는 확률이 차이

15 %

10 %

32 %

8 %

25 %10 %

10 %

1st: 40%

2nd : 30%3rd : 15%

4th : 10%5th : 5%



2. 적합도비례전략(Proportional to FITNESS Value Scheme)

•기본 모델(Classical Selection Model)

•적합도에 비례 하여 자손을 남김

(Better fitness value has a higher probability of being

Chosen)

•룰렛(Roulette Model) , 몬테카를로 모델(Monte Carlo Model)

15 %

10 %

32 %

8 %

25 %10 %

10 %

3. 엘리트 보존 전략

-각 집단 중에서 가장 적합도가 높은 개체

를 다음세대로 그대로 넘김 N-th Generation

(N+1)-th Generation

Best

Best

(N-1)-th Generation



4. 토너먼트 선택전략(Tournament Selection)

-임의의 수의 개체를 무작위 선택

(2 members of current population

Chosen randomly)

-그 가운데 적합도가 높은 개체를

다음 세대로 넘김

(Dominant performer placed in

intermediate population of

survivors)

N-th Generation

(N+1)-th Generation

(N-1)-th Generation

superior

superior

superior

superior


Genetic Operator : CrossoverGenetic Operator : Crossover

교차(Crossover)

1. 단순교차(Simple Crossover)

-하나의 교차위치 설정,

그 전후로 부모의 유전자형 교환

2. 복수점교차(Multi-point Crossover)

-교차위치가 복수인 경우

3. 일정교차(Uniform Crossover)

-마스크를 사용하여 어느쪽의 유전자를

받아 들일지 결정

가령 마스크의 비트가 ‘0’일 경우에는

그대로, ‘1’일 경우에는 두 부모의

유전자를 교환

개체 A

개체 B

1 0 0 1 1 1 1

0 0 1 1 0 0 0

1 0 0 1

1 1 10 0 1 1

0 0 0

개체 A

개체 B

1 0 0 1 1 1 1

0 0 1 1 0 0 0 0 1 1 0 00 0

1 11 1 01 0

개체 A

개체 B

0 0 1 1 1 1

1 1 1 1 0 0

개체 A

개체 B

0 1 1 1 1 0

1 0 1 1 0 1

mask 0 1 0 1 0 1

-일반적인 임의탐색 기법과 구별되는 큰 특징

exchange1 point

exchange

filtering


Genetic Operator : Mutation (1)Genetic Operator : Mutation (1)

변이 (혹은 돌연변이, Mutation)√ 유전자를일정한확률로변화시키는조작

√ 전역적탐색효과의극대화(Maximize Global Search), 수렴속도가속(Increase Convergence Rate)

√ 집단의다양성증대 ( Increase Diversification of Population )

Strategy 1. 정적변이

2. 동적변이

-돌연변이의 확률을 일정하게 고정

-적응변이(Adaptive Mutation)

-돌연변이의 확률이 경우에 따라 변화


GAs versus Traditional MethodsGAs versus Traditional Methods

Differ from traditional search/optimization methods:

GAs search a population of points in parallel, not only a single pointGAs use probabilistic transition rules, not deterministic onesGAs work on an encoding of the parameter set rather than the parameter set itselfGAs do not require derivative information or other auxiliary knowledge - only the objective function and corresponding fitness levels influence search


ReferencesReferences

A. Zalzala, P.J. Fleming, “Genetic Algorithms in Engineering Systems”Control Engineering Series 55, The Institution of Electrical Engineers (IEE), 1997Gen, Mitsuo, Cheng, Runwei., “Genetic Algorithms and Engineering Optimization” , Wiley, New York , 2000.Back, Thomas, “Evolutionary Algorithms in Theory and Practice :Evolution Strategies, Evolutionary Programming, Genetic Algorithms”, Oxford University Press,Oxford, 1996.Michalewicz, Z., “Genetic Algorithm + Data Structures = Evolution Programs”, Spinger-Verlag, 1996.

* For Korean Students :조성배 역, “GA의 기초이론, 공학응용 및 인공생명 – 유전자알고리즘”, 대청 컴퓨터월드, 1996년


2) Examples withSimple Genetic Algorithms

2) Examples withSimple Genetic Algorithms


Examples with Simple Genetic AlgorithmsExamples with Simple Genetic Algorithms

Optimization Problem

8.51.41.120.3

)20sin()4sin(5.21),(

2

1

221121

≤≤≤≤−++=

xx

xxxxxxf ππ

Object Function

Maximize

f(x1 , x2)

x1

x2



Representations√ Encode decision variables into binary strings.√ The precision requirement implies that the range of domain of each

variable should be divided into at least (domain of is and the required precision is 4 after the decimal point.) size ranges.

√ Required bits( mj )

√ The mapping from a binary string to a real number

4ij 10)ab( ×−jx]b,a[ jj

1210)ab(2 jj m4jj

1m −≤×−<−

12

ab)substring(decimalax

jmjj

jjj−

−×+=

331518mmm15m2170002

1700010)1.48.5(x

18m21510002

15100010))0.3(1.12(x

21

21514

42

11817

41

=+=+==≤<

=×−

=≤<

=×−−



√ The total length of a chromosome is 33 bits

Binary Number Decimal Number----------------------------------------------------------------

111101011110111001010010000010101v j

361653.5121.48.5243181.4x

687969.212

)0.3(1.1254170.3x

24318111101011110111x5417001010010000010101x

152

181

2

1

=−−

×+=

−=−−−

×+−=



Initial Population√ Randomly generated as follows :

]000010011010110010010010111011001[v]001010100010110011010001011001000[v]000001010100110011010000011101001[v]011111011011101010011010000101010[v]101010100010110011011001101001000[v]000001010100110011001100011101011[v]100111011100001010011010000101010[v]101010100010110011011001101001000[v]000001010100110011000000011101011[v]100111011101101010011010000101010[v

10

9

8

7

6

5

4

3

2

1

=

=

==

=

=

=

==

=

]361653.5,687969.2[]x,x[v]873501.4,671267.11[]x,x[v]694977.4,01330256[]x,x[v

]121702.5,342067.9[]x,x[v]174346.4,788084.11[]x,x[v]477282.4,301286.2[]x,x[v

]109598.4,159951.6[]x,x[v]661461.4,419457.10[]x,x[v

]170144.4,474101.0[]x,x[v]361653.5,687969.2[]x,x[v

2110

219

218

217

216

215

214

213

212

211

−==

==

−==

==

==

−==

==

==

==

−==



Evaluation √ Step 1. Convert the chromosome’s genetype to its phenotype.

√ Step 2. Evaluate the objective function .√ Step 3. Convert the value of objective function into fitness. For the maximization

problem, the fitness is simply equal to the value of objective function

.size_pop,,,2,1k),x,x(x k2

k1

k ==)x(f k

.size_pop,...,2,1k),x(f)v(eval kk ==

264971.20)361653.5,687969.2(f)v(eval159724.15)873501.4,671267.11(f)v(eval763190.19)694977.4,01330256(f)v(eval

958717.17)121702.5,342067.9(f)v(eval900541.11)174346.4,788084.11(f)v(eval686091.15)477282.4,301286.2(f)v(eval

406122.29)109598.4,159951.6(f)v(eval590546.9)661461.4,419457.10(f)v(eval370896.17)170144.4,474101.0(f)v(eval

805119.19)361653.5,687969.2(f)v(eval

10

9

8

7

6

5

4

3

2

1

=−=

==

=−===

==

=−=

==

==

===−=



Selection√ In most practices, a roulette wheel approach is adopted as the

selection procedure.1. Calculate the fitness value eval(vk) for each chromosome vk :

eval(vk)=f(x), k=1,2,…,pop_size2. Calculate the total fitness for the population:

3. Calculate selection probability pk for each chromosome vk :

4. Calculate cumulative probability qk for each chromosome vk :

∑=

=size_pop

1kk )v(evalF

size_pop,...,2,1k,F

)v(evalp kk ==

kv

size_pop,...,2,1k,pqk

1jjk ==∑

=



√ Each time, a single chromosome is selected for a new population in the following way :

− Step 1. Generate a random number r from the range[0,1].− Step 2. If r≤q1 , then select the first chromosome v1 ; otherwise, select the kth

chromosome vk(2≤k≤pop_size) such that qk-1<r≤qk

√ The probability of a selection pk for each chromosome vk (k=1,…,10) is as follows :

p1=0.111180, p2=0.097515, p3=0.053839…

√ The cumulative probabilities qk for each chromosome vk is as follows :q1=0.111180, q2=0.208695, q3=0.262534…

√ Now we are ready to spin the roulette wheel 10 times. Let us assume that a random sequence of 10 numbers from the range [0,1] is as follows:

0.301431 0.322062 0.766503…



√ The first number r1=0.301431 is greater than q3 and smaller than q4, meaning that the chromosome v4 is selected for the new population; and so on. The new population consists of the following chromosomes :

.

.

.)v(]100011001100100001100010010110101[v)v(]001000001011100101101001001101101[v)v(]001000001011100101101001001101101[v

83

42

41

=′=′=′



Crossover√ One-cut-point method : randomly selects one cut-point and

exchanges the right parts of two parents to generate offspring.√ Consider two chromosomes as follows, and the cut-point is

randomly selected after the 17th gene:

√ The resulting offspring by exchanging the right parts of their parents would be as follows:

√ The probability of crossover is set as pc=0.25 (25% of chromosomes undergo crossover.).

]100011001100100001100010010110101[v]001000001011100101101001001101101[v

2

1

=

=

]001000001011100001101000010110101[v]100011001100100101100011001101101[v

2

1

=′=′



Procedure : Crossover

Generate a random integer number pos from the range [1, 32]end

end;1kk

end;crossoverforparentoneasvselect

then)25.0r(if];1,0[fromnumberrandomr

do)10k(while;0k

begin

k

k

k

+←

<←

≤←

Assume that the sequence of random number is0.625721 0.266823 0.288644 0.295114 0.163274 0.567461 0.085940 0.392865 0.770714 0.548656

This means that the chromosomes and were selected for

crossover. 5v′ 7v ′

]001000001011100101101000001101101[v]001000001011100101101001001101101[v

]001000001011100101101000001101101[v]001000001011100101101001001101101[v

7

5

7

5

=′=′

=′=′



Mutation√ Alters one or more genes with a probability equal to the mutation rate.

√ Assume that the 18th gene of the chromosome v1 is selected for a mutation. Since the gene is 1, it would be flipped into 0.

The probability of mutation is set as pm =0.01, so 1% of total bit of population would undergo mutation.

√ Need to generate a sequence of random numbers rk (k=1,…,330) from the range[0,1].

Bit_pos chrom_num bit_no random_num105 4 6 0.009857164 5 32 0.003113199 7 1 0.000946329 10 32 0.001282

111001]10000000101001011000[100110110v

111001]10000000101001011010[100110110v

1

1

=′

=



Final population

.

.Corresponding decimal values of variables [x1,x2]

f (6.159951,4.109598)=29.406122f (6.159951,4.109598)=29.406122f (-0.330256,4.694977)=19.763190

.

.√ Just completed one iteration of genetic algorithm.The best chromosome in the 419th generation as follows:

]100011001100100001100010010110101[v]001000001011100101101001001101101[v]001000001011100101101001001101101[v

3

2

1

=′=′=′

818208.38)x,x(f

724824.5x

631407.11x818208.38)724824.5,631407.11(f)v(eval

)110110100101000111000111111100000(v

*2

*1

*2

*1

*

*

=

=

=

==

=


3) Application of GA3) Application of GA

[목 차]

High Bypass Ratio Jet Engine Design


Application of GA (1)Application of GA (1)

√ Powell 등, “ EnGENEous : Domain independent, machine learning for design optimization,” Proc. Of ICGA-89,1989

√ 이문제의경우, 가능한해집합의수는 10의 387승개에이름√ 통상의방법으로는두명의설계자가한달이상의작업을요하는문제

High Bypass Ratio Jet Engine Design

Genetic Algorithm

+ Expert System EnGENEous




ExpertSystem

ExpertSystem

ExpertSystem

ExpertSystem

ExpertSystem

ExpertSystem

EnGENEous System

Genetic Algorithm

GBM

: 초기해집합에

대한 국소탐색

초기 해집합

Next Generation




Results적응도 향상 정도 Man-Month RatioMethod

1 2개월인 간

2 8일Expert System Only

3 17일GA(임의의 초기치)

3 9일GA+Expert System

Initial

Manual Design

Expert System Only

수계산 최적값

GA+Expert System

93% 94% 95% 96% (엔진효율)



플라이휠 설계√ 형상의용이한설계를위해대개플라이휠은금속재료로제작된다. 플라이휠은스스로의관성을이용하여속도의변동을조절할뿐아니라, 에너지를저장하는역할을한다. 원판형의플라이휠이회전하게되면마찰손실이존재하더라도어느수준정도의

운동에너지를생성하게되고모멘텀을발생시킨다. 또한, 저장된에너지는기계적, 전기적작용에의해방출된다. 플라이휠형배터리는화학적배터리보다에너지저장능력이뛰어나다. 특히, 크기가작아질수록이러한플라이휠의요구가더욱절실해지는데전기자동차(self-contained electric powered vehicle)인경우에서그우수성을발견할수있다.

Hybrid 전기자동차

GA에 의해 최적화된 복합재료 플라이휠의 3차원 형상,



병렬 유전자알고리즘을 이용한 헬리콥터 블레이드의 공력/공력소음 최적화√ 헬리콥터와같이회전날개항공기에서는로터시스템과와류의충돌에의해비행성능에지장을

주게되고또한회전유동에의한소음이발생한다. 저진동, 저소음(BVI :blade-vortex interactions)의로터및블레이드를설계하기위해유전알고리즘이적용되고있다. 전산유체역학적해석방법에의해유전자알고리즘설계해를탐색하는경우에는계산시간및

비용이엄청나게증가하므로이러한단점을최적화기법에서해결하기위해병렬형

유전알고리즘을개발하여전역설계의생성뿐아니라병렬연산기술을활용하여설계기간을

단축시킬수있다.

병렬유전자알고리즘을 이용한 각 Processor간의 Data 교환


Simulated AnnealingSimulated Annealing

2006. 4. 1


Principal Heuristic AlgorithmsPrincipal Heuristic Algorithms

Genetic Algorithms (Holland – 1975) √ Inspired by genetics and natural selection

Simulated Annealing (Kirkpatrick – 1983) : Today’s issue√ Inspired by molecular dynamics – energy minimization

Particle Swarm Optimization (Eberhart and Kennedy - 1995)√ Inspired by the social behavior of swarms of insects or flocks of birds

These techniques all use a combination of randomness and heuristic “rules” to guide the search for global maxima or minima


Origin of Simulated Annealing (SA)Origin of Simulated Annealing (SA)

Definition: A heuristic technique that mathematically mirrors the cooling of a set of atoms to a state of minimum energy. Origin: Applying the field of Statistical Mechanics to the field of Combinatorial Optimization (1983) Draws an analogy between the cooling of a material(search for minimum energy state) and the solving of an optimization problem. Original Paper Introducing the Concept √ Kirkpatrick, S., Gelatt, C.D., and Vecchi, M.P., “Optimization by

Simulated Annealing,” Science, Volume 220, Number 4598, 13 May 1983, pp. 671680.


Simulated Annealing(SA)Simulated Annealing(SA)

Like GA, SA is a stochastic process.SA mimics the physical process of heating then slowly cooling a metal during processing to receive thermal/structure stresses.

√ Randomly generates design points√ Downhill point movement always o.k√ Uphill movement depends on the

statistics (90% No, 10%Yes)

√ More limit changes ?√ High/low temperature criterion &

steps ?x*low temperature

F

x

F*

high temperature

F

x

More limit

2nd iteration


Simulated Annealing(SA)Simulated Annealing(SA)

1) Start with initial selection of point, 2) Pick an initial high temperature T3) For each design

4) If is accepted, next Else, accept an”uphill move”with the following probability

5) Tnew = 80% of Told

Repeat the process

x2

pxx oldnewρρρ

±= Move limit

)()( oldnew xFxF ρπ

ρ

xρ

newxρ xρ

⎥⎦⎤

⎢⎣⎡ −−

=TK

xFxFP oldnewaccept '

|)()(|expρρ

Constant used for scaling “Big deal”simulated annealing temp

Get next xρ

x1

downhill movement

Uphill movementAccept ?

**


Accepting a New Current Solution Accepting a New Current Solution

Why move to a worse current solution?√ To avoid getting trapped in a local optimum.

Local Optimum√ A solution is locally optimal if there is no neighbor who has a better

objective function value.

Global Optimum√ A solution is globally optimal if there is no other solution in the

entire feasible trade space that has a better objective functionvalue.

√ Note: We are only talking about single objective problems.


Accepting a New Current Solution (Continue) Accepting a New Current Solution (Continue)

Local Optima vs. the Global OptimumExample

f(x) = cos(14.5x – 0.3) + (x + 0.2)x

Minimize f(x)


Accepting a New Current Solution (Continue) Accepting a New Current Solution (Continue)

Local Optima vs. the Global OptimumExample

Minimize f(x) = cos(14.5x – 0.3) + (x + 0.2)x

Subject to -2.2 ≥ x ≥ -3


Summary: Steps of SASummary: Steps of SA

The Simulated Annealing Algorithm1) Choose a random X, select the initial system temperature, and outline the

cooling (ie. annealing) schedule

2) Evaluate E(Xi) using a simulation model

3) Perturb Xi to obtain a neighboring Design Vector (Xi+1)

4) Evaluate E(Xi+1) using a simulation model

5) If E(Xi+1)< E(Xi), Xi+1 is the new current solution

6) If E(Xi+1)> E(Xi), then accept Xi+1 as the new current solution with a probability e(-Δ/T) where Δ = E(Xi+1) -E(Xi).

7) Reduce the system temperature according to the cooling schedule.

8) Terminate the algorithm.


Example : Traveling Salesman ProblemExample : Traveling Salesman Problem

Initial (Random) RouteLength: 17.43

Final (Optimized) RouteLength: 8.24


Research in SAResearch in SA

Alternative Cooling Schedules and Termination criteria Adaptive Simulated Annealing (ASA) √ determines its own cooling schedule

Hybridization with other Heuristic Search Methods √ GA, Tabu Search …

Multiobjective Optimization with SA


Summary of stochastic Methods (GA,SA)Summary of stochastic Methods (GA,SA)

suitable for problems with many local minimadesign space can be non-smooth and discontinuouscan handle discrete variables (x’s)Function calls can be large compared topath –building methods(be careful not to exceed grid search number of functions)


SA code : simman.fSA code : simman.f

simman.f√ Simple SA code √ Format : FORTRAN 77 √ Optimization problem type : Unconstrained

Optimization Problem Only √단, penalty function을이용, pseudo-objective형태로전환하여 Constrained Problem을해결하는방식이가능

√ Reference : Goffe, Ferrier and Rogers, "Global Optimization of Statistical Functions with Simulated Annealing," Journal of Econometrics, vol. 60, no. 1/2, Jan./Feb. 1994, pp. 65-100.



Input Parameter

Internal Parameter for SA

√ parameter RT

− The temperature reduction factor. The value suggested by Corana et al. is .85

parameter type definitionN Integer Number of Design Variables

X(I) real Initial Value Vector

LB(I) real Lower Boundary Vector

UB(I) real Upper Boundary Vector

parameter type definition(default)RT real Temperature Reduction Factor (0.1~0.95)

MAX logic True = max. / False = min.

EPS real Error tolerance for termination(1.0E-06)

NT integer Number of iterations before temperature reduction(20)

NEPS integer Number of final function values used to decide upon termination(4)

MAXEVL real The maximum number of function evaluations(100000)



Program Layout



procedure File : simman.f

Step 1 : parameter definition

No. of Design Variable

Problem Type , tolerance, RT

NT, NS,

Upper & Lower Bound

Initial Value



Step 2 : objective definition

File : simman.f

Object Function



Step 3

√Compiling and running the system:

Data from Optimization Processing

Optimal Solution

: Optimum X vectors

Optimal Function Value,

Optimum Results


2-Bar Truss Design : Structural Optimization2-Bar Truss Design : Structural Optimization

Description √ The symmetric 2-bar truss design shown in below has been studied by several

researchers√ Balling and Clark(1992), Schmit, (1981), Sobieszczanski-Sobieski et al(1982)

√ The objective of this optimum problem is to minimize the weight of truss systemsubject to behavioral constraints

√ Related parameters - B = 30 in - t = 0.1 in - ρ = 0.3 lbs/in3 - σy = 60,000 psi- E = 30E6 psi


2-Bar Truss Design : Structural Optimization2-Bar Truss Design : Structural Optimization

Formulation for Optimization√ Minimize W(X)=2ρπDt(B2+H2)1/2

− Minimize the weight of truss system √ Subject to

g1(X) = σe - σ ≥0− the first constraint prevents failure due to Euler buckling

g2(X) = σy - σ ≥0− the second constraint prevents failure due to yield stress

Where,0.5≤D≤5.0 (in) ⇒ X(1) : mean tube diameter5.0≤H≤50.0 (in) ⇒ X(2) : height of the truss

√ The resulting optimum value from (Schmit, 1981) for W(x) is 19.8 lbs.

- W* = 19.8 lbs (at D* = 2.47 in , H* = 30.15 in )


SA Application : 2-Bar Truss DesignSA Application : 2-Bar Truss Design

Algorithm과 initial Value에 따른 결과 비교rp = 1.0E2 , initial values X(1) =0. , X(2)= 0.

DOTProgram

Constrained Unconstrained

Algorithm MMFD SLP SQP BFGS F-R SA -

X1 (D) 2.481 2.476 2.476 4.558 4.558 - 2.47

X2 (H) 29.870 29.992 30.00 15.031 15.031 - 30.15

Opt. val. (W) 19.800 19.800 19.800 28.828 28.828 - 19.8

Num. of Iter. 9 20 11 3 3 - -

Num. of fun. eval. 77 69 45 24 24 - -

simman true

rp = 1.0E2 , initial values X(1) =1. , X(2)= 5.

-22012626406626Num. of fun. eval.

--3310205Num. of Iter.

19.819.80423.60123.60119.80019.80022.690Opt. val. (W)

30.1530.58217.82017.82030.04129.99017.588X2 (H)

2.472.4523.5883.5882.4742.4803.462X1 (D)

-SAF-R BFGSSQPSLPMMFDAlgorithm

UnconstrainedConstrainedtruesimman

DOTProgram

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