1. Parallel Simulated Annealing with Adaptive Temperature Determined by Genetic Algorithm
Thank you chairperson for introducing me.
I am Takeshi Yoshida from Doshisha University.
Now I will talk about our study ”Parallel Simulated Annealing with adaptive temperature determined by Genetic algorithm".
Doshisha University, JAPAN
Takeshi Yoshida, Mitsunori Miki,
Tomoyuki Hiroyasu, Toshihiko Fushimi

2. Background Application of parallel processing
Many studies on parallelization of numerical analysis.
Few studies on parallel optimization.
Parallel optimization is Promising.
Two types of studies on parallel optimization
Parallelization of mathematical programming
Parallelization of analysis is very popular.
New approach : parallelization of sensitivity analysis.
In the area of applications of parallel processing, the parallelization of numerical analysis has been well investigated. However, the studies on parallel optimization have not been well performed. We think that studies on parallel optimization can be considered to be promising.
There are two types of studies on parallel optimization. One is the parallelization of mathematical programming. In this area, the parallelization of analysis is very popular, but this is not a parallel optimization.
Another direction of parallel optimization is the parallelization of heuristic search methods. Heuristic search methods include Simulated Annealing, Genetic Algorithm, and so on. These methods need high computational power, so the parallelization of these methods is very important. And fortunately, these methods have natural parallelism in their approach.
Parallelization of heuristic search methods
Simulated Annealing(SA), Genetic Algorithms(GA), Neural Network(NN)
Parallelization is needed to reduce high computational costs.
Heuristic methods have natural parallelism.

3. Simulated Annealing (SA)
An algorithm to solve combinational optimization problems.
Based on the simulation of the physical process “annealing”.
Algorithm
Cooling Schedule : The procedures for updating temperature.
Disadvantages :
Move of a solution
Change of the “energy”, E.
“Temperature”, T, is decreased.
High temperature
Local minimum
Global minimum
Low temperature
Decreased
Always accept.
Increased
Accepted in a
certain probability, “P”.
Simulated Annealing is an algorithm to solve combinational optimization problems. It is based on the simulation of physical process “annealing” of metal. After this time, I’ll call simulated annealing SA.
This algorithm has three important procedures.At first, a current solution moves to another state.If the value of the objective function called “energy” is decreased by this move, the move will be always accepted since we want to minimize it.Otherwise, if the energy is increased, the move will be accepted in a certain probability that is derived from this equation.After these procedures are repeated several times, a parameter called “temperature” is decreased.
The procedures for updating T is called the cooling schedule.This temperature is the parameter,which controls this probability.So,the cooling schedule is very important for searching solutions.
SA has good results for many combinational optimization problems.But it requires high computational costs and it is extremely difficult to determine a proper cooling schedule.
This temperature is the parameter, which controls this probability, and the procedure for updating T is called the cooling schedule.SA searches the solution by repeating these procedures 1,2 and 3. The feature of SA is that SA probably allows escaping local minima with uphill move.
At the beginning, higher temperature is set, so it is easy to escape from local minima.As the temperature is gradually decreased, SA can search global minima.The advantage of SA is that the algorithm is useful for solving complex multimodal problems.But it requires high computational power and it is extremely difficult to determine a proper cooling schedule for the global minima.
High computational costs. Difficult determination of a proper cooling schedule.

4. Parallelization of SA Classification of PSA [D.R.Greening,1990]
Temperature Parallel SA : TPSA
Serial-Like : maintains the convergence properties of SA.
Altered generation : several SA generate their own state.
Asynchronous : searching with asynchronous communication.
Altered generation
Asynchronous
Serial-Like
For these disadvantages, a lot of research about Parallel SA has been achieved.In this presentation, I will call parallel SA PSA.Here, these figures show three types of PSA.
The first one is serial-Like type.This type only parallels some procedures of SA such as evaluating cost function. It maintains the convergence properties of SA.The second one is altered generation. In this type, several SA generate their own state,and communicate with each other synchronously while searching.And the third one is a PSA with asynchronous communication.
Some PSA methods have been proposed for each type respectively until now.And these results show reduction of the computational time. But most methods have the same problem that it is difficult to determine a proper cooling schedule.By the way, a new method called temperature parallel SA was proposed. And this algorithm has been applied to standard combinatorial optimization problems.
Reduction of the computational time.
Difficult to determine a proper cooling schedule.
Application : Traveling Salesman Problems, LSI allocation problems
Graph partition problems

5. Temperature Parallel SA(TPSA)
Algorithm
Features
The different temperatures are assigned to different processors.
Each SA searches solution with a constant temperature.
Exchanging solution.
Better solutions move to SA at lower temperature with a certain probability.
High T
Low T SA
This figure shows the TPSA algorithm.At first, the different temperatures are assigned to different processors, like this, the highest temperature , higher,middle,lower,lowest.And then each processor performs a sequential SA with a constant temperature during the entire annealing time.After a certain interval of annealing time, two solutions in two processors with adjacent temperatures are exchanged in a certain probability that is derived from the differences in temperature and energy.After repeating this cycle, then finally this method can get the optimum solution at the lowest temperature.
I’d like to show the features of TPSA.The first feature is automatic determination of a cooling schedule, while, in the case of sequential SA you have to determine a proper cooling schedule.Another feature is that the algorithm is easy to be implemented to a parallel computer.But assigning the initial temperature on each processor is difficult as well as other PSA.
TPSA
Automatic determination of a cooling schedules.
Facility of implementation to a parallel computer.
Assigning the initial temperature on each processor is difficult.

6. Purpose of this study
Parallel SA is effective in reduction of computational time.
It is difficult to determine a proper cooling schedule of PSA.
A new parallel SA
Performance of PSA/AT is evaluated with Traveling Salesman Problems.
The searching ability of TPSA depends on assigning the initial temperatures.
Parallel SA
Parallel SA with Adaptive Temperature
Determined by Genetic Algorithm (PSA/AT(GA)) +
GA to Temperature
So far the features of PSA have been shown. Parallel SA is effective in reduction of computational time. But the determination of a proper cooling schedule in PSA is difficult. That goes for TPSA, the searching ability of TPSA depends on the assigning the initial temperatures.
So, the purpose of this study is the proposal of a new PSA. The method is called Parallel SA with adaptive temperature determined by Genetic algorithm. We will call this method PSA/AT-GA.PSA/AT is a hybrid method of SA with GA operation.The cooling schedule of PSA/AT is automatically determined by GA．In this study, We evaluate the performance of PSA/AT with Traveling Salesman Problems.
PSA/AT is a hybrid method of SA with GA operation.
The cooling schedule of PSA/AT is automatically determined by GA.

7. n : synchronous interval
PSA / AT
SA + evaluate Fitness 1
GA to temperature 2
Set new temperature 3
GA operation
Solution
GA operation
Temperature
(individual)
E N D
GA operation
Now, I’d like to talk about PSA/AT.This figure shows PSA/AT algorithm. PSA/AT is based on Parallel SA. The different solutions and temperatures are assigned to different processors.
And , the each processor performs a sequential SA with a constant temperature and evaluating fitness for the synchronous interval.This fitness value is not the energy of solution.This value is used to evaluate the move of solution in PSA/AT. n n
PSA/AT is based on Parallel SA.
The different solutions and temperatures are assigned to different processors.
Sequential SA and evaluating fitness are performed on each processor.
The fitness value is used to evaluate the move of solution in PSA/AT.
n : synchronous interval

8. Evaluate Fitness If the fitness is a high value…
Fitness value is the summation of the differential energy value lowered below a baseline.
Baseline is an average of the energies of all SA processes performed in parallel.
Solution
Baseline
Energy
Steps
If the fitness is a high value…
In this slide,I show evaluating fitness value.This value is used to evaluate the move of solution.This figure shows the move of solution in sequential SA.This axis(X) shows the number of searching steps, and this axis(Y) shows the value of solution energy. In SA, the solution moves up and down like this. Then, the fitness value is the summation of the differential energy value lowered below a baseline.This function shows how to evaluate fitness.
This Baseline is an average of the energies of all SA processes performed in parallel.If the fitness is high value, the number of solutions lowered below a baseline must be frequent or the differential energy value is high. This means the solution moves within the limits of good energy and the temperature is suitable for searching solution.
The solutions moves below a baseline frequently.
The differential energy value is large.
The solutions moves within the limits of good energy.
The temperature is suitable for searching solution.

9. n : synchronous interval
PSA / AT
SA + evaluate Fitness 1
GA to temperature 2
Set new temperature 3
GA operation
Solution
GA operation
Temperature
(individual)
E N D
GA operation
In PSA/AT, sequential SA and evaluating fitness are performed on each processor for a synchronous interval.after these operations all temperatures are adjusted by GA synchronously. n n
Sequential SA and evaluating fitness are performed on each processor for synchronous interval .
All temperature are adjusted by GA synchronously.
n : synchronous interval

10. Genetic Algorithm (GA)
Optimization method based on the mechanism of natural selection and natural genetics.
Searching points : Individuals
The new searching points are generated by GA operator.
Useful for various problems.
GA operator
Individual
Selection
Individuals with
high fitness survive.
Crossover
Perform direct
information exchange
between Individuals.
Mutation
Change information
of individuals.
Here I’ll talk about Genetic Algorithm,GA.Genetic algorithms is a optimization method based on the mechanism of natural selection and natural genetics.In GA, there are many searching points, called individuals. The new searching points are generated by genetic operators such as selection, crossover, mutation, for each generation. After some generations, we may get a good solution as a result. GA has many searching points and,GA is useful for various problems.
Population

11. Coding of temperature PSA/AT use GA to optimize a cooling schedule.
Individual : Temperature on each processor
Design variable : The exponent of temperature function.
The expression of temperature in PSA/AT is suitable for the exponential cooling schedule in SA.
Real Value
Bit Array
Encoding
Temperature
= exp(X)
PSA/AT use GA to optimize a cooling schedule.A temperature on each processor operates as an individual.In GA, each individual has design variables，and these variables are encoded from real value to binary.In PSA/AT , a temperature is expressed by this function, this X is treated as a design variable of an individual.
For example, these are several individuals of bit array ,which are decoded between minimum temperature “-2” and max temperature “6” .Then this bit array “ ” is decoded to “2” ,that is,this X value is 2 and the temperature is
Generally, SA has a exponential cooling schedule by using this function. The expression of temperature in PSA/AT is suitable for exponential cooling schedule.
Decoding
SA cooling method

12. PSA / AT SA + evaluate Fitness 1 GA to temperature 2
Set new temperature 3
GA operation
GA operation
E N D
GA operation
Well , in the GA operation these individuals of temperature are regenerated by GA operator such as selection,crossover,mutation.In selection individuals with a high fitness value probably are selected. And in crossover and mutation, various individuals of temperature are generated.
Selection
In selection, individuals with a high fitness value are probably selected.
In crossover & mutation, various individuals of temperature are generated
Crossover
Mutation

13. Effective of a selection
Selection of individuals are performed with the fitness values.
The fitness value is used to evaluate the moves of solution.
The temperature is concerned with the moves of solution.
Searching at a proper temperature, the fitness value is high.
Individuals of proper temperatures are probably selected in PSA/AT.
High temperature
Proper temperature
Low temperature
Energy
Steps
Baseline
Energy
Steps
Baseline
Energy
Steps
Baseline
In PSA/AT, the selection is very important GA operator to generate individuals of a proper temperature.this selection of individuals are performed with the fitness values.As you know, fitness value is used to evaluate the moves of solution.and the temperature is concerned with the moves of solution.So,here I’d like to talk about the relation between the fitness and the temperature.
Here,these three figure shows the moves of solution at a high temperature , a proper temperature and a low temperature.
At high temperature, a solution moves up and down hard like this. But the energies of solution are mostly above the baseline.
So , in this case,the fitness value is very low.And at low temperature, a solution can’t escape from local minima like this.so the fitness value is very low too.On the other hand, at proper temperature, a solution is moderate in moving up and down around the baseline.so, the fitness value is high.
In selection of PSA/AT,the individuals with higher fitness are probability selected.As a result, individuals of proper temperatures are probably selected in PSA/AT.
Fitness : low
Fitness : high
Fitness : low

14. n : synchronous interval
PSA / AT
SA + evaluate Fitness 1
GA to temperature 2
Set new temperature 3
GA operation
Solution
GA operation
Temperature
(individual)
E N D
GA operation
PSA/AT repeats this cycle , 1 ,2 and 3,to the end.Finally all temperatures of each processor are automatically adjusted to a proper temperature for searching solution. n n
n : synchronous interval
PSA/AT repeats this cycle to end.
All temperatures of processors are automatically adjusted to a proper temperature for searching solution.

15. Traveling Salesman Problems (TSP)
Typical combinatorial problem.
Finding a minimum tour to round all cities.
Problem
City size
Opt
a280
280
2579
d198
198
7542
fl417
417
11861
gil262
262
2378
kroA200
200
29368
lin318
318
42029
pr264
264
49135
rat195
195
2323
Now, I will go on to numerical examples to evaluate the proposed algorithm.We use Traveling Salesman Problem as the objective problem.I will call traveling salesman problem TSP.
TSP is a typical combinational problem. The objective of this problem is finding a minimum tour to cover all cities.and this is cost function. We try eight TSP problems in this table. These are selected from TSPLIB .
Σd(vπ(i),vπ(i+1)+d(vπ(N), vπ(1))
v(i) : i-th city
π : tour
N : city size
d(v(i),v(j)) : distance between two point

16. Parameters used in the experiments
Algorithm
PSA/AT
TPSA
Single SA
Number of Processes 32 1
Annealing Step
Cooling Cycle × 160
Cooling Cycle
N(city size) × 20
N × 640
Number of trials 20
Max Temperature
The maximum energy of uphill move is accepted
at a rate of 50%.
Min Temperature
One uphill move of solution is accepted for the cooling cycle.
This slide shows the parameters we use in experiments. We tried three methods PSA/AT,TPSA,and Single SA to evaluate the searching ability.To perform the same annealing steps, we adjusted the cooling cycle on each method.
The maximum temperature and minimum temperature are closely related with the cooling schedule, so we adjusted these values for each problem.These parameters in red cell, bit Length selection Method,Crossover Rate, and Mutation Rate are only used in PSA/AT.
Bit Length 10 -
Selection Method
Roulette -
Crossover Rate
0.1 -
Mutation Rate
0.01 -

17. Experimental results (Energy)
- kroA200 -
- rat195 -
TSP Distance
TSP Distance
I’d like to talk about the experiments results. These figures show the history of best energy values for the TSP,kroA200 and rat195.These results are averages of 20 trials.This(X) axis shows the number of cooling cycle, and this(Y) axis shows TSP Distance, The solutions ,which have lower energy, show better solutions. The red line in figure shows the result of PSA/AT, The green line shows the results of TPSA, the blue line shows the results of Single SA.
From these result,PSA/AT has a good searching ability for both problems. For the speed of convergence of solution,PSA/AT and TPSA are fast , single SA is slow. Because PSA/AT and TPSA search solutions with a proper cooling schedule.
Cooling Cycle
Cooling Cycle
PSA/AT has a good searching ability.
Convergence of solution
Fast : PSA/AT and TPSA
( with a proper cooling schedule )
Slow : Single SA

18. Experimental results (Error Rate)
Next figure shows the error rate from optimum for each problem.This(X) axis shows TSP problem,and this(Y) axis shows Error rate which means the difference of energy between the solution derived by three methods and optimum.The value which is lower shows better result. From this result, PSA/AT provides the best result in the whole problem.
TSP Problem
PSA/AT provides the best result in the whole problems.

19. History of temperature
- PSA/AT -
- TPSA -
Temperature
Temperature
And next,these figures show the history of temperatures of all solutions in the case where PSA/AT and TPSA are performed for the TSP called kroA200. This(X) axis shows the number of annealing steps,this(Y) axis shows temperature,and one line in this figure shows the history of temperature of one solution.In this experiment,the number of processor which has one solution is 32,so these figures show 32 lines.
From this result, In PSA/AT all temperatures of solution converge on the specific temperature region.Histories like this result for other TSP problems.On the other hand, In TPSA, exchanging solution are performed within parts of temperatures, that is, all solutions are not effective in searching solution.Considering from the point of view of searching solution, the cooling schedule which PSA/AT automatically determines is more proper than TPSA’s.
Annealing Step
Annealing Step
A line : History of temperature of one solution.
PSA/AT : Convergence on the specific temperature region.
TPSA : Exchanging solutions are only performed within parts of temperatures.
The cooling schedule of PSA/AT is proper for searching solution.

20. Conclusions This study show a new hybrid method,
Parallel Simulated Annealing with Adaptive Temperature
Determined by Genetic Algorithm(PSA/AT).
PSA/AT has the following features
Based on Parallel SA, so the calculation time is less than sequential SA.
Use GA to determine temperature on each SA.
PSA/AT automatically determine a proper cooling schedule for searching solution.
We apply PSA/AT,TPSA,Single SA to TSP.
The result shows that PSA/AT has better searching ability
than the other methods.

21. Fin

22. Cooling Schedule The move of solution depends on the temperature.
The cooling schedule
The proper cooling schedule is required for each objective problem.
Max Temperature
Parameter
Max temperature : High Value
Min temperature : Low Value
Cooling rate
Cooling cycle
Temperature
Min Temperature
PSA/ATについて説明する前に，SAにおける温度スケジュールについて説明します．
先程も述べましたようにSAにおける解の動きは温度と密接に関係しており，良好な解を得るためには最適な温度スケジュールがひつようになります．
しかし，この図に示すような一般的な温度スケジュールを決定するには，最高温度，最低温度，冷却率，冷却サイクルなど複数のパラメータを設定する必要があります．
また，一般的にゆっくりと冷却する事が必要とされ，温度スケジュールの決定は非常に困難とされています．
Time
Cooling schedule is determined by these parameters.
To get a good solution, program has to reduce temperature slowly form max T to min T.
It is difficult to determine the proper cooling schedule.

23. Important Temperature Region
Max Temperature
Sequential SA with a constant temperature of important temperature region has a good searching ability.[Mark00]
It is difficult to determine important temperature region, because it depends on each objective problem.
Temperature
Min Temperature
Time
Experiments to determine Region
Max Temperature
Different sequential SAs with the different temperatures are performed.
Comparing the solutions, the temperature with which a SA is performed is the important temperature region.
一方，近年の研究において，一定温度の解探索で良好な解探査能力を示す事が報告されています．
ここでは，この温度のことを重要温度領域と呼びます．
この温度がわかれば，先程述べた温度スケジュールを設定する必要がないのですが，この温度の値は対象問題に依存しており，一意の決定方法を見つけるためには多くの予備実験が必要となります．
そこでPSA/ATではGAによって重要温度領域を探索し，PSA/ATの温度スケジュールを最適化する
重要温度領域は対象問題に依存しており，決定することが難しい．
It’s difficult to find important temperature region because those values are dependent on problem
Temperature
Min Temperature
Time

24. Important Temperature Region
- kroA200 -
Problem
Size
Opt
Imp Temp
A280
280
2579
2-3
d198
198
15780
4-13
fl417
417
11861
1-3
gil262
262
2378
kroA200
200
29368
30-50
lin318
318
42029
20-30
pr264
264
49135
40-60
rat195
195
2323
2-5
There is the important temperature region for each problem.
The values of important temperature region are different.

25. Parameters used in the experiments 2
Algorithm
PSA/AT
TPSA
Single SA
Number of Processes 32 1
Annealing Step
Cooling Cycle × 160
Cooling Cycle
N(city size) × 20
N × 640
Number of trials 20
Max Temperature
Min Temperature
0.01
This slide shows the parameters we use in experiments. We tried three methods PSA/AT,TPSA,and Single SA to evaluate the searching ability.To perform the same annealing steps, we adjusted the cooling cycle on each method.
The maximum temperature and minimum temperature are closely related with the cooling schedule, so we adjusted these values for each problem.These parameters in red cell, bit Length selection Method,Crossover Rate, and Mutation Rate are only used in PSA/AT.
Bit Length 10 -
Selection Method
Roulette -
Crossover Rate
0.1 -
Mutation Rate
0.01 -

26. Experimental results2 (Error Rate)
Next figure shows the error rate from optimum for each problem.This(X) axis shows TSP problem,and this(Y) axis shows Error rate which means the difference of energy between the solution derived by three methods and optimum.The value which is lower shows better result. From this result, PSA/AT provides the best result in the whole problem.
TSP Problem
PSA/AT provides the best result in the whole problems.

27. GA operator Selection of individuals Send Information Crossover &2
Selection of individuals
Individuals with higher fitness survive. 1
Send Information
Individuals of a proper temperature for searching probably survive.
Individuals of temperature
and fitness are gathered to
one processor.
At synchronized time, a temperature and a fitness on each processor are gathered to one processor.I will call the processor master node, the other processors slave nodes.Next step, the individuals of temperature are selected.In selection, the individuals with higher fitness value are probably selected ,as the result, individuals of a proper temperature for searching probably survive .
Next Crossover and Mutation are performed, these operations generate various individuals of temperature.At Last,Master Node sends individuals to slave nodes for updating temperature on each processor. 3
Crossover &
Mutation
Various individuals of
temperature are generated. 4
Update temperature