1. Comparison of Pulling Back and Penalty Methods for Constraints in DPMBGA
○　Hisashi Shimosaka (Doshisha University)
　　Tomoyuki Hiroyasu (Doshisha University)
　　Mitsunori Miki (Doshisha University)
I am Hisashi Shimosaka.
I am a graduate student of Doshisha University in kyoto, japan.
Now, I will talk about our study, the title is “ Comparison of … “

2. Structural Optimization Problem
is a problem to design the optimum structures.
Objective Function : Volume, Cost, Weight
Constraint : Stress, Displacement, Buckling
Features
The landscape of the objective function has several local optima.
The problems have several types of constraints.
The problems are large-scaled due to many design variables.
The feasible region is very narrow compared to the design field.
Optimization algorithm
should have an efficient searching ability for global search.
should have an efficient mechanism to deal with the constraints.
Our target problem is structural optimization problems.
Structural optimization problem is a problem to design the optimum structures where objective function is often volume, cost, weight and constraints are often stress, displacement, buckling and so on.
This problem has the following features.
Firstly, the landscape of the objective function has several local optimums.
Secondly, the problems have several type of constraints.
Thirdly, the problem are large-scaled due to many design variables.
Finally, the feasible region is very narrow compared to the design field.
Therefore, the optimization algorithm to solve the problems should have an efficient searching ability for global search
and has an efficient mechanism to deal with the constraints.
So, we use Genetic Algorithm(GA) for the optimization algorithm and several mechanisms to deal with the constraints are applied to the GA.
Genetic Algorithm (GA) + Mechanisms to deal with the constraints

3. Target of our research To derive good solutions by GAs
The information of the good parents should be inherited to the children.
The diversity of the population should be maintained during the search.
The children should be generated by consideration of the correlation among the design variables
Mechanism to deal with constraints
Penalty method
Pulling back method
DPMBGA to the structural optimization problems
Penalty method and pulling back method are added to DPMBGA.
The searching abilities of the both methods are compared through the truss structural optimization problems.
Distributed PMBGA (DPMBGA)
To derive good solutions in the GAs, We think these three points are important.
The firstly is that the information of the good parents should be inherited to the children.
The second is that the diversity of the population should be maintained during the GA search.
The third is that the children are generated by consideration of the correlation among the design variables.
Therefore we have already proposed the Distributed PMBGA (DPMBGA)
that is one of the probabilistic model-building GA(PAMBGA).
In this presentation, DPMBGA is applied to solve the structural optimization problems.
However, DPMBGA do not have a mechanism to deal with the constraints,
therefore penalty method and pulling back method are added to DPMBGA, and the searching abilities of the both methods are compared through the truss structural optimization problems.

4. Estimation of the distribution
PMBGA
Probabilistic Model-Building Genetic Algorithm (PMBGA)
Select promising individuals
Estimation of the distribution
Individual
(2) Construct
a probabilistic model
Population
Probabilistic model
(3) Generate new individuals and substitute them for old individuals
Firstly, We talk about the probabilistic model-building GA (PMBGA).
Generally PMBGA repeatedly performs the following three steps.
Firstly, the promising individuals are selected from the population.
Secondly, a distribution of the individuals is estimated and a probabilistic model is constructed by the distribution.
Finally, new individuals are generated by the probabilistic model stochastically,
and some new individuals are substituted for old individuals.
In PMBGA, new individuals are generated from the estimated probabilistic model instead of the crossover and mutation.
Therefore, it is expected that the information of the selected individuals can be inherited to the generated individuals.
New individuals are generated from the estimated probabilistic model instead of the crossover and mutation.
The information of the selected individuals can be inherited to the generated individuals.

5. DPMBGA Distributed PMBGA (DPMBGA) [Hiroyasu,2003]
Real-Coded PMBGA
Island Model (Distributed GA)
Probabilistic model constructed by Principal Component Analysis (PCA) and normal distributions
The diversity of the population can be maintained.
New individuals are generated by consideration of the correlation among the design variables.
Distributed PMBGA is one of the PMBGAs and We have proposed and developed.
DPMBGA is a real-coded PMBGA and uses the island model.
The diversity of the population is maintained by the island model.
In addition, probabilistic model is constructed by principal component analysis (PCA) and normal distributions.
Therefore, new individuals are generated by consideration of the correlation among the design variables.
(The DPMBGA satisfies the three important points for an efficient search by island model and PCA.)
It is already shown that DPMBGA can derive better solutions than UNDX+MGG in some test functions.
DPMBGA can derive better solutions than UNDX+MGG in some test functions.

6. Overview of the DPMBGA operations
(1) Individuals with better fitness values are selected. x2
Island v1 v2 x1
(2) Individuals are transferred into the space where there is no correlation among the design variables using PCA.
(4) New individuals are transferred into the original space. v1 v2
This slide shows the overview of the DPMBGA operations in each island.
In DPMBGA, new individuals are generated by performing these four operations.
Firstly, individuals with better fitness values are selected from the island.
Secondly, Individuals are transferred into the space where there is no correlation among the design variables using PCA.
Thirdly, new individuals are generated from normal distributed model.
Finally, New individuals are transferred into the original space and substituted for the individuals in the island.
Next, we talk about the detail of these operations.
(3) new individuals are generated from normal distributed model.

7. (1) Some individuals are selected
Sampling individuals x2
(1) Some individuals are selected
Sample individuals
Island x1
Sample individuals
Individuals with the best fitness values in the island are chosen as the sample individuals.
The first step of the generation of new individuals is to select some individuals from the island as sample individuals.
In this time, The elite Individuals with the best fitness values in island are chosen as the sample individuals.

8. Individuals are Transferred into the new spacex2
The purpose
New individuals are generated by consideration of the correlation among the design variables.
The flow of the operation
Principal Component Analysis (PCA) is performed and vector V is obtained.
The individuals are rotated into the space where there is no correlation among the design variables using vector V. v1 v2 x1
Second step is to transfer the sample individuals into the new space.
The purpose of this operation is that new individuals are generated by consideration of the correlation among the design variables.
The flow of the operations is the followings.
Firstly, Principal Component Analysis (PCA) is performed and Vector V is obtained.
Secondly, The sample individuals are rotated by vector V into the space where there is no correlation among the design variables.
And next step, new individuals are generated in this new space.

9. New individual generation
Generation of the new individuals
The distribution of the individuals is estimated using normal distributions on each design variable.
The design variables of the new individuals are generated independently.
Substitution for the old individuals
New individuals are moved back to the original space.
They are substituted for all of the old individuals.
Island
(4) New individuals are transferred into the original space.
Third step is to generate new individuals.
The distribution of the sample individuals in the new space is estimated using normal distributions on each design variables.
Using these normal distributions, the design variables of the new individuals are generated independently.
Finally, the new individuals are moved back to the original space by the vector V,
and substituted for all of the old individuals. v1 v2
(3) new individuals are generated from normal distributed model.

10. Features of DPMBGA Features
Real-coded probabilistic model-building GA
The diversity of the population is maintained by island model.
New individuals are generated by consideration of the correlation among the design variables using PCA.
The distribution of the sample individuals is estimated using normal distributions.
DPMBGA is superior to the typical real-coded GA, UNDX+MGG in the several types of test functions.
DPMBGA is applied to structural optimization problems.
DPMBGA does not have a mechanism for dealing with constraints.
These are the features of the DPMBGA.
(DPMBGA is a real-coded probabilistic model-building GA, and the diversity of the population is maintained by Island model.
New individuals are generated by consideration of the correlation among the design variables using PCA.
The distribution of the sample individuals is estimated using normal distributions.
DPMBGA is a real-coded probabilistic model-building GA.
In addtion, DPMBGA uses the island model for the diversity of the population and uses the PCA and the normal distributions for estimating the distribution.)
It is already shown that DPMBGA is superior to the typical real-coded GA, UNDX+MGG in the several types of the test functions.
In this presentation, DPMBGA is applied to structural optimization problems.
However, DPMBGA does not have a mechanism for dealing with constraints.
Therefore, penalty method and pulling back method are added to the DPMBGA.
Penalty method and pulling back method are added to the DPMBGA

11. Penalty method
When an individual violates the constraints, the objective function is added the penalty.
The method is easy to implement.
It is difficult to obtain the appropriate penalty parameter.
When the feasible region is very narrow, it is difficult to search effectively.
Minimize
Such that
The objective function is modified.
Minimize
In the penalty method, when an individual violates the constraints, the objective function is added the penalty.
Suppose there is an optimization problem like this, in this time the penalty method modifies the objective function like this.
P is more than 0 and called penalty parameter.
The penalty method is easy to implement, but it is difficult to obtain the appropriate penalty parameter.
In addition, when the feasible region is very narrow, it is difficult to search effectively.

12. Problem of the penalty method
Penalty method is the most popular method to deal with constraints.
Estimation of the distribution
Generation of
new individuals
The penalty method is the most popular method to deal with constraints.
However, when the penalty method is applied to the DPMBGA, there is one problem.
Suppose some promising individuals are selected like this.
In the DPMBGA, the distribution of the promising individuals are estimated by normal distributions and new individuals are generated like this.
In this time, some new individuals violates the constraints.
This means that an effective probabilistic model can not be constructed because the feasible region is very narrow.
Therefore, the penalty method may not be performed an efficient search in the DPMBGA.
Some new individuals violates the constraints.
An effective probabilistic model can not be constructed because the feasible region is very narrow near the boundary.
The penalty method may not be performed an efficient search in the DPMBGA.

13. Pulling back method
An individual that violates the constraints is moved to the nearest point in the feasible region. [Mimura,2002]
The violated constraints are linearized.
The distance to the feasible region is minimized.
Terminal criteria of the pulling back
All the constraints are satisfied.
The number of the operation exceeds a certain number.
The distance after the operation is smaller than the preset distance.
minimize
Such that
On the other hand, in the pulling back method,
an individual that violates the constraints is moved to the nearest point in the feasible region.
(Suppose there is an optimization problem like the previous one,)
When an individual (X out) violates one or more constraints,this quadratic programming problem should be solved in the pulling back method.
(This problem can be solved by several optimization techniques.)
In this problem, the violated constraints are linearized and the distance to the feasible region is minimized.
(However, because the pulling back method linearizes the constraints, the individual may not be satisfy all the constraints
after just one pulling back operation. Therefore, )
This pulling back operation is repeatedly performed until one of the following terminal criteria is met.
(The terminal criteria of the pulling back are the followings.)
Firstly, all the constraints are satisfied.
Secondly, the number of the pulling back operation exceeds a certain number.
Finally, the (moving) distance after the operation is smaller than the preset distance.
However, when one of the terminal criteria is met, if the individual violates the constraints, the penalty method is applied.
If the individual still violates the constraints, the penalty method is applied.

14. Numerical example
The penalty method and the pulling back method are added to the DPMBGA
Comparison of the searching ability of both methods
Discussion on the comparison of the results
Truss structural optimization problem
Design variables : Areas of each member
Objective function : Minimization of the volume
Constraints : Stress is less than 4e+7 [Pa] Buckling should not occur.
I would like to talk about the numerical example.
In the numerical example, the penalty method and the pulling back method are added to the DPMBGA and DPMBGA is applied to structural optimization problems.
The purposes of the numerical example are to compare of the searching ability of both methods and to discuss on the comparison of the results.
Target problems are the 2-stages and the 3-stages truss structural optimization problem.
Truss structure consists of nodes and members.
This 2-stages truss structure consists of 6-nodes and 10-members and this 3-stages truss structure consists of 8-nodes and 15-members.
In the both structures, two nodes at the ground are fixed and several nodes are loaded as 5000N.
Design variables of the problem are areas of each member.
Objective function is to minimize the volume of each structure under the following conditions on each member.

15. Comparison of the penalty and pulling back method
Discussion on the affect of the number of islands
Number of individuals per island is 16
Number of islands is 1,2,4,8,16 and 32.
The searching ability of the both method is determined by
the number of the optimum found
the number of the evaluations when the optimum is found.
Population size is changed from 16 to 512.
Parameters of the DPMBGA
Number of elites 1
Sampling Rate
0.25
Amp. of variance
2.0
Mutation rate
0.1/ (Dim. of function)
Migration interval 5
Migration Rate
0.0625
Terminal criterion (Number of evaluations)
1.0e+6 2.0e+6
Number of trials 25
Parameters of the both method
Maximum time of pulling back 20
Minimum distance of one pulling back
1e-8 (m)
Penalty parameter（ρ）
1e+6
This example discusses on the affect of the number of the islands when the number of the individuals is fixed as 16 and
number of islands is changed from 1 to 32. This means that population size is changed from 16 to 512.
The searching abilities of the both methods are determined by the number of the optimum found and by the number of the evaluations when the optimum is found.
This table shows the parameters of the DPMBGA.
In the 2-stages truss, terminal criterion is 1 million evaluations and in 3-Stages truss, terminal criterion is 2 million evaluations.
Number of trials is 25.
This table shows the parameters of the pulling back method and the penalty method.

16. Number of the optimum found
2-Stages Truss (10 design variables)
25 trials
The penalty method can derive the optimum with the large number of islands.
The pulling back method can derive the optimum with not only the large number but also the small number of islands.
Islands (Population size)
Penalty method
Pulling back method
　　　　1 (16) 25
　　　　2 (32) 6
　　　　4 (64) 14
　　　　8 (128) 18
16 (256) 22
　　　 32 (512)
This slide shows the number the optimum found in the 2-stages truss.
The 2-stages truss is 10 design variables. Number of trials is 25.
This left row shows the number of the islands and the population size.
This row shows the result of the penalty method and this row shows the result of the pulling back method.
from this result, the penalty method can derive the optimum with the large number of the islands.
On the other hand, the pulling back method can derive the optimum with not only the large number but also small number of the islands.

17. Number of the optimum found
3-Stages Truss (15 design variables)
25 trials
The pulling back method can derive the optimum with only the small number of island.
In the difficult problem, the pulling back method with the small number of islands can derive the better solution than the penalty method.
Islands (Population size)
Penalty method
Pulling back method
　　　　1 (16) 24
　　　　2 (32) 1 19
　　　　4 (64) 4
　　　　8 (128)
16 (256) 5
　　　 32 (512) 12
And this is the result of the 3-stages truss.
The 3-stages truss is 15 design variables.
from this result, the penalty method can not perform an efficient search and
the pulling back method can derive the optimum with only the small number of the islands.
However, as the number of the islands becomes larger, the searching ability of the pulling back method is reduced.
Therefore, in the difficult problem, it is said that the pulling back method with the small number of the islands can derive the better solution than the penalty method.

18. Average number of evaluations
Average number of evaluations required to find the optimum.
2-Stages Truss (10 design variables)
The pulling back method can derive the optimum with larger number of evaluations as the number of islands becomes larger.
Pulling back operation
In order to explain that the searching ability of the pulling back method is reduced as the number of the islands becomes larger,We show the result of the average number of evaluations.
This figure shows the average number of evaluations required to find the optimum in 2-stages truss.
This axis shows the number of evaluations and this axis shows the number of islands.
The blur bar is the result of the penalty method and the red bar is the result of the pulling back method.
The smaller number of evaluations is better.
From this result,
//it is found that the penalty method can derive the optimum with smaller evaluation times.
//On the other hand,
The pulling back method can derive the optimum with bigger evaluation times as the number of islands becomes bigger.

19. Average number of evaluations
Average number of evaluations required to find the optimum.
3-Stages Truss (15 design variables)
The pulling back method with the small number of islands is effective because the pulling back operation requires many evaluations.
Pulling back operation
This is the result of the 3-stages truss.
From this result, the pulling back method requires many more evaluations as the number of islands becomes larger.
In the pulling back method, the violated constraints are linearized and the pulling back operation is repeatedly performed.
In this time, to obtain the differential value of the violated constraints, many evaluations are required.
Therefore,
// it is said that for simple problems, the penalty method with the large number of the islands is effective.
//On the other hand, for difficult problems,
It is said that the pulling back method with the small number of the island is effective because the pulling back operation required many evaluations.
many evaluations are required.

20. Discussion on the comparison of the results
From the numerical examples,
The pulling back method with the small number of islands can derive the better solution than the penalty method.
In the pulling back method, the small number of island is effective.
The large number of island is not effective because the pulling back operation requires many evaluations.
Why is the small number of islands effective?
Target of the comparison of the result for the discussion
2-Stages Truss (10 design variables)
32 islands and 512 individuals
Median value of 25 trials
- Rate of the individuals that violates the constraints
- Search mechanism of the both methods for constraints
Next, we discuss on the comparison of the results of the both methods.
From the numerical example, it is already found that the…
//This means that the diversity of the population is maintained by the island model in the DPMBGA.
In addition, in the pulling back method, The large number of the island is not effective because the pulling back operation requires many number of evaluations.
However, we have one question. Why is …
To solve this problem, we focus on the rate of the … and the search mechanism …
(In this discussion, the rate of the individuals that violated the constraints in each generation and the average evaluation value of the individuals that violate the constraints in each generation are discussed.)
The target of the comparison result for the discussion is the result of the 2-stages truss and 32 islands and the median value of 25 trials.

21. Rate of the individuals that violates the constraints
Pulling back operation
The violated constraints
are linearized.
Terminal Criteria of the pulling back operation
- No violated constraint
- Maximum times:20
- Minimum distance:1e-8
This figure shows the rate of the individuals that violate the constraints in each generation.
This axis shows the rate of the invalid individuals and this axis shows the number of evaluations.
The blue line shows the result of the penalty method and the red line shows the result of the pulling back method.
In the penalty method, 60 % of the population violates the constraints.
Therefore the penalty method requires many individuals for an efficient search.
On the other hand, in the pulling back method, the invalid individuals are only 30%.
Therefore the pulling back method can use more individuals for …
In the penalty method, 60% of the population violates the constraints.
In the pulling back method, the invalid individuals are only 30%.
More individuals can be used for an efficient search than the penalty method.

22. Search mechanism for constraints
In the optimization problem with constraints, the optimum often exists on the boundary of the feasible region.
Next, we focus on the search mechanism of the both methods for constraints.
In the optimization problem with constraints, the optimum often exists on the boundary of the feasible region.
Therefore, the search of this region is very important.
Suppose new individuals are generated like this.
In the penalty method, the most individuals that violates the constraints are dead and the population is early converged when the population size is small.
On the other hand, in the pulling back method, the pulling back operation pulls back these individuals near the boundary of the feasible region and creates new search points that are difference from the parent points.
Therefore, this operation can keep the diversity of the population and an effective search is expected even when the population size is small.
In the penalty method, the most invalid individuals are dead and the population is early converged when the population size is small.
The pulling back operation pulls back these individuals near the boundary of the feasible region and creates good new search points that are different from the parent points.

23. Conclusion Structural optimization problem
Distributed PMBGA（DPMBGA）
Penalty method and pulling back method
Comparison of the searching ability
The pulling back method with the small number of island is effective than the penalty method in the difficult problem.
In the pulling back method, The large number of island is not effective because the pulling back operation requires many evaluations.
Discussion on the comparison of the result
The pulling back method can use more individuals for an efficient search than the penalty method.
The pulling back operation can keep the diversity of the population even when the population size is small.
I would like to conclude.
In this study, DPMBGA is applied to structural optimization problems.
In order to deal with the constraints, the penalty method and the pulling back method are added to DPMBGA.
Through the truss structural optimization problems, it is shown that the penalty method with the large number of the islands is effective in the simple problem and the pulling back method with the small number of the island is effective in the difficult problem.
In addition, we discusses on the comparison of the result and it is found that the pulling back method can use more individuals for an efficient search than the penalty method and the pulling back operation can keep the diversity of the population even with the small number of the islands.