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Distributed Probabilistic Model-Building Genetic Algorithm

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## Distributed Probabilistic Model-Building Genetic Algorithm

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### Distributed Probabilistic Model-BuildingGenetic Algorithm

Tomoyuki Hiroyasu

Mitsunori Miki

Masaki Sano

Hisashi Shimosaka

Shigeyoshi Tsutsui

Jack Dongarra

(Doshisha University)

(Doshisha University)

(Doshisha University)

(Doshisha University)

(Hannan University)

(University of Tennessee)

DPMBGA

- Probabilistic Model Building GA
- Principle Component Analysis (PCA)
- Distributed Population Scheme
- Distributed Environment Scheme

Genetic Algorithm

New search points are generated by Crossover and Mutation.

Good characteristics of parents should be inherited to children.

Evaluation

Selection

Crossover

Mutation

Probabilistic Model-Building GA

(1) Select better individuals

Estimation of the Distribution

Individual

(2) Construct a probabilistic model

Population

Probabilistic Model

(3) Generate new individualsand substitute them for old individuals

New individuals are generated from estimated probabilistic model instead of crossover and mutation.

Classification of PMBGAs (Pelikan, 1999)

Type of design variables

f (x1, x2)

0

1

0

1

0

1

Bit Strings

Real Vectors

x1

x2

Correlation among the design variables

- The method does not care for the correlation.
- The method cares the correlation between the two design variables.
- The method cares the correlation among more than three design variables.

F(x1, x2, …, xn)=F(x1)+F(x2)+…+F(xn)

F(x1, x2, x3)=F(x1,x2)+F(x3)

Classification of PMBGA

DPMBGA

For the effective search

- Maintaining the diversity of the solutions
- Consideration of the correlation among the design variables

Distributed Population Scheme

x2

The selected individuals are transferred into another space by PCA.

x1

Probabilistic model is constructed.

New points are generated. These points are transferred back to the original space.

v2

The overflow of the operationsx2

(1) Individuals who have better fitness values are selected.

Population

v1

v2

x1

(4) New individuals are transferred into the original space.

(2) Individuals are transferred into the space where there is no correlation among the design variables.

(3) new individuals are generated from normal distributed model.

Selected Population

x2

Some individuals are selected.

Population

Sample population

x1

Best m individuals are chosen from the population.

v2

x1

Individuals are transferred into the new spacex2

- The Goal
- New individuals are generated by consideration of the correlation among the design variables.

- The flow of the operations

1. The archive for the PCA is renewed.

2. The individuals in the archive are analyzed by PCA.

3. The Individuals are transferred into the space where there is no correlation among the design variables.

Generation two

Generation three

Archive for CPAPopulation

Archive

- The best individuals in each generation are restored in the archive.
- Archive has the size

v2

x1

Individuals are transferred into the new spacex2

- The Goal
- New individuals are generated by consideration of the correlation among the design variables.

- The flow of the operations

1. The archive for the CPA is renewed.

2. The individuals in the archive are analyzed by CPA.

3. The Individuals are transferred into the space where there is no correlation among the design variables.

v1

v2

Distribution of individuals

x1

Principle Component Analysis- PCA analysis for individuals in the archive
- Define the Covariance Matrix S in the design field.
- Derive the eigen vectors V = (V1, V2, …, VD, ) of S
- When one eigen value is bigger than others, the distribution is biased to the direction that is corresponding to the eigen value.
- This means that there is strong correlation is existed.

v2

New individual generationPopulation

New individuals are substituted for some old individuals

Moved back to the original space

generate new individuals

Distribution in the new space

Normal distribution

Probabilistic Model-Building GA

(1) Select better individuals

Estimation of the Distribution

Individual

(2) Construct a probabilistic model

Population

Probabilistic Model

(3) Generate new individualsand substitute them for old individuals

Because the model is constructed with the elite individuals, early convergence sometimes happens.

The mechanism that keeps the diversity of the solution is needed.

Distributed Population Scheme

- Distributed GA（DGA）island model
- Total population is divided into sub populations.
- GA operations are performed in each sub population.
- Migration
- Parallel Efficiency
- Ability to keep the diversity of the solutions.
- High searching capability.

Results

Optimum value： 1.0E-10 ，Terminal condition ： number of evaluations 3.0E+06

Rastrigin, Rosenbrock

- History of the number of renewed individuals in the archive

Archive is eliminated every 10 generation

- Rastrigin ： erase/10 is better
- Rosenbrock ： normal is better

When the number of renewed individuals becomes small, PCA does not work well.

without PCA

Distributed Environment Scheme (DES)- In some sub populations, PCA is performed
- In some sub populations, PCA is not performed

Results

Optimum value： 1.0E-10 ，Terminal condition ： # of evaluations 3.0E+06

DPMBGA

- Probabilistic Model Building GA
- Principle Component Analysis (PCA)
- Distributed Population Scheme
- Distributed Environment Scheme

P2

C2

P1

Parents

Children

Comparison with UNDX+MGG- Unimodal Normal Distribution Crossover

（UNDX）（ Ono et al., 1999）

- Typical Real-Coded GA
- It has a strong search capability.

- Minimal Generation Gap （MGG）（Sato et al., 1997）
- Generation Alternate Model
- MGG can maintain the diversity of the solutions

Extended Region

Feasible region

x

Functions whose optimums locate near the boundary- Problems in Real-Coded GAs
- The searching capability may decrease for the problems whose optimum locates near the boundary of the feasible region.

- Boundary Extension by Mirroring (BEM) （Tsutsui,1998）

- Semi-feasible region is prepared
- It is reported that BEM is useful for the problems whose optimum locates near the boundary.

Handling the constraints

- The operation for the individuals that violate the constraints in DPMBGA
- The corresponding individual is pulled back to the edge of the feasible field.
- When the optimum point locates the near the boundary, there is a possibility that the probabilistic model cannot be constructed correctly.

x2

Out of constraints

Feasible field

x1

x2

Optimum

x1

x1

x2

Test Problems- Test problems
- The range of design variable is modified.
- The optimum locates on the boundary of the feasible region.

Example）Rastrigin, Ridge

x1

History of the search （Rastrigin, Schwefel）

- Problems that have not the correlation

- The model without BEM derived the better results.

History of the search（Rosenbrock, Ridge）

- Problems that have the correlation

- The model without BEM derived the better results.
- The proposed model works for these problems

Conclusions

- DPMBGA
- The diversity of the population is maintained by the distribute population scheme.
- The correlation among the design variables are considered by using PCA.
- Effectiveness of PCA
- Because of the individuals in the archives, sometimes PCA does not work well.
- Distributed Environment Scheme is useful.
- Comparison with UNDX+MGG
- DPMBGA derived the better solutions.
- Problems where the solution locates the edge of the design field
- BEM or other special mechanism is not necessary.s

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