1. Temperature Parallel Simulated Annealing with Adaptive Neighborhood for Continuous Optimization problem
Mitsunori MIKI
Tomoyuki HIROYASU
Masayuki KASAI
Keiko ONO
Takeshi Jitta
Thank you, Chairperson.
Now I will talk about our study “Temperature Parallel Simulated Annealing with Adaptive Neighborhood for Continuous Optimization problem”.
I am Takeshi JITTA, Graduate student of Doshisha University, Japan.
Doshisha University,
Kyoto, Japan

2. Background Applications 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
Parallelization of heuristic search methods
Simulated Annealing(SA)，Genetic Algorithms(GA)， Cellular Automata(CA)，Neural Network(NN)
Parallelization is needed to reduce high computational costs.
Natural parallelization can be realized.
Studies on such parallelization are not yet enough.
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.
Fortunately, these methods have natural parallelism in their approach.
But, the studies on such parallelization are not yet enough.

3. Simulated Annealing (SA)
An algorithm to solve combinatorial optimization problems.
Based on the simulation of the physical process “annealing”.
Algorithm
Advantages
Useful for solving complex multimodal problems.
Disadvantages
High computational costs
Difficult determination of the proper cooling schedule.
1. Move of a solution.
2. Change of the “energy”，E.
Decreased The move is always accepted.
Increased The move is accepted in a certain
probability, P =
3. “Temperature”, T, is decreased.
－(Enext－Ecurrent) T
exp
local
minimum
Global minimum
Simulated Annealing is an algorithm to solve combinatorial optimization problems.
It is based on the simulation of physical process “annealing” of metal.
This algorithm has three important procedures.
At first，a current solution moves to another state.
If the value of 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 not always be rejected, that is, 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.
This “temperature” is the parameter, which controls this probability.
Simulated Annealing searches the solution by repeating these procedures 1,2 and 3.
At the beginning, higher temperature is set. And then, temperature is gradually decreased.
Simulated Annealing allows escaping form local optima and then finds the global optimum by this cooling schedule.
The higher the temperature is, the higher this probability in which the solution escape from local optima becomes.
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 the proper cooling schedule for finding the global optimum.

4. Parallelization of SA Reduction of the computational time
Conventional methods
Simple parallel SA (Individual trial)
Parallel SA with periodical synchronization
Parallel SA with synchronization on new state
New parallelization method
Temperature Parallel Simulated Annealing (TPSA)
Applications
Travelling salesman problems
LSI allocation problems
Graph partition problems
For high computational cost, several parallel algorithms of SA have been proposed.
The first one is simple parallel SA.
This algorithm assigns tasks of sequential SA to each processor of parallel computer, independently.
And it is executed without communications between processors.
The second one is a parallel SA, which synchronizes periodically, and the information about the solutions is exchanged.
And the third one is a parallel SA, which synchronizes when a new state is accepted.
By the way, a new parallelization method of SA was proposed a few years ago.
It is called the “Temperature Parallel Simulated Annealing” “TPSA”.
And this algorithm has been applied to standard combinatorial optimization problems.

5. Temperature Parallel SA
Algorithm
Sequential SA
With constant Temp.
High
Temp
Exchanging
solution
Better solutions move to
processors which work at
lower temperature with a
certain probability
Low
Temp
Automatic determination of cooling schedule
Temperature
Temperature SA
TPSA
Now I will explain the algorithm of TPSA in the case of using 5 processors.
At first, the different temperatures are assigned to different processors, like this, the highest temperature, higher temperature, middle, lower and the lowest one.
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 the sequential SA at constant temperature and exchanging the solutions, then finally this method can get the optimum solution at the lowest temperature.
TPSA has two attractive features.
The first feature is automatic determination of the 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.
Initial
Solution T1 T1
T on Proc1 T2 T2
T on Proc2 T3 T3
Initial
T on Proc3 T4
Solution T4
T on Proc4 T5 T5
T on Proc5 T6
T on Proc6
Time
Time

6. Purpose of this study
TPSA has been applied to only combinatorial optimization problems.
The purpose of this study is a proposal of a new approach of TPSA for continuous optimization problems.
Temperature Parallel Simulated Annealing with Adaptive Neighborhood (TPSA/AN).
Performance of TPSA/AN is evaluated with the standard test functions.
TPSA has been applied to combinational optimization problems until now.
However, this sophisticated algorithm, TPSA, has not been applied to continuous optimization problems.
So, the purpose of this study is proposal of a new approach of TPSA for continuous optimization problems.
We propose a method called “ Temperature Parallel Simulated Annealing with Adaptive Neighborhood (TPSA/AN), which is suitable for continuous optimization problems.
In this study, we evaluate the performance of TPSA/AN with the standard test function.

7. (adjusting neighborhood range)
Approach
Parallel algorithm
Sequential algorithm
TPSA
Corana’s method
(adjusting neighborhood range) ＋
Extended algorithm for solving continuous optimization problems
Uses the Corana’s method for the moves of solutions in continuous problem spaces.
Uses the information of landscape of objective functions.
Adjusts the neighborhood range by keeping the rate of acceptance constant.
Our approach is the combination of TPSA and Corana’s method for adaptive neighborhood.
The features of our proposed algorithm are these points.
This algorithm is TPSA with an extended algorithm for solving continuous optimization problems.
This algorithm uses Corana’s method for the moves of a solution in continuous problem space.
Therefore, the algorithm uses the information of the landscape of an objective function and adjusts the neighborhood range adaptively by keeping the rate of acceptance constant.

8. Adjusting neighborhood range for temperature －Corana’s method－
Neighborhood range(higher temperature) x2
Neighborhood range affects the accuracy of solutions.
When the temperature is high, the neighborhood range is adjusted to be large.
When the temperature is low, the neighborhood range is adjusted to be small.
Here I will talk about the method combined in TPSA for adjusting the neighborhood range.
It is easy to understand that the neighborhood range affects the accuracy of solutions.
If the algorithm uses a large neighborhood, it can search an optimum in the global area efficiently.
Otherwise, if the algorithm uses a small neighborhood, it can search an optimum in the local area efficiently.
Therefore, the algorithm uses a larger neighborhood range at higher temperature, and the smaller neighborhood at lower temperature.
Neighborhood range
(lower temperature) x1

9. Adjusting neighborhood range for the rate of acceptance －Corana‘s method－x2 x1
The range have to
be decreased
be increased
The neighborhood range is adjusted by using the rate of acceptance at each temperature.
When the rate of acceptance is small
the range have to
be decreased
On the other hand, with a large neighborhood range in a steep area, the moves are apt to be rejected frequently.
It is not efficient for the search.
So, the neighborhood range has to be decreased in this case.
Otherwise, with a small neighborhood in a gentle area, the neighborhood range have to be large.
For realizing this mechanism, the algorithm keeps the rate of acceptance constant, which means the rate of acceptance is maintained to be 50%.
When the rate of acceptance is large
the range have to
be increased
Keep the rate of acceptance 50%

10. Formulation of adjusting neighborhood range
0.4
0.6
1/(c+1)
c+1 1
m, m’: Neighborhood range.
g(p) : Coefficient for adjusting
neighborhood range.
p : Rate of acceptance.
p = n / N
N : Interval for adjusting
neighborhood range
n : Number of acceptance
in interval N
The adaptive adjustment of neighborhood range mentioned here can be expressed as this formulation.
The rate of acceptance “p” is calculated from the interval for adjusting the neighborhood range and the number of acceptance in interval N.
If this value “p” is more than 60%, the neighborhood range will be increased.
And if it is less than 40%, the neighborhood range will be decreased.
In this study, empirically
c=2，N = 8.

11. Determination of control parameter －Maximum temperatureー
Determine the maximum temperature for considering the trade-off between calculation time and coverage of searches.
Preliminary experiments.
1. Temporary maximum temperature is set.
2. Perform each sequential SA at a constant temperature.
3. Evaluate the calculation time and the coverage of search. .
1.00E-01
1.00E+00
1.00E+01
1.00E+02
1.00E+03 5 10 15 16 17 18
Max temperature
Calculation time(sec)
1.00E-04
1.00E-03
1.00E-02
1.00E-01
1.00E+00
1.00E+01 5 10 15 16 17 18
Max Temperature
Unsearchd area
In TPSA, you do not have to determine the cooling schedule, but maximum and minimum temperature should be determined.
The proposed algorithm uses a new method to determine them.
For the maximum temperature, both the coverage of searches and the calculation time are considered.
We carrid out the preliminary experiments for finding the proper maximum temperature.
This figure (left one) shows the relation between the size of unsearchd area and the maximum temperature.
If you have high maximum temperature, you can get the small unsearchd area.
However, you can see that it takes a long computational time from this figure (right one).
This figure shows the relation between the calculation time and maximum temperature.
From these two figures, we can determine the proper maximum temperature based on its trade-off.
In this figure, we used this temperature, 10, as the maximum temperature. .

12. Determination of control parameter －Minimum temperatureー
Determine the minimum temperature based on the solution accuracy.
Preliminary experiments.
1. Temporary maximum temperature is set.
2. Perform each sequential SA at a constant temperature.
3. Average of neighborhood ranges are derived.
4. Choose the best temperature base on the required accuracy.
Required accuracy
The minimum temperature is determined by considering the accuracy of solutions.
We carrid out the preliminary experiments for finding the proper minimum temperature.
This figure shows the relation between the minimum temperature and the neighborhood range.
You can see that the decreasing in the minimum temperature provides small neighborhood range, which relates the accuracy of solutions.
When the required accuracy is here, we can set the minimum temperature to be 0.01. .

13. Standard test function
Rastrigin Function
Griewank Function
Now, I will go on to the evaluation of the proposed algorithms.
We use the Rastrigin function and Greiwank function as the test function for evaluating the proposed method.
Rastrigin function has a lot of large local minima and Griewank function has a lot of small local minima.
Global minimum point is here, and here, the origin of this coordinate.

14. Parameters used in the experiments
Functions
Rastrigin
Griewank
Algorithms SA
TPSA SA
TPSA
Number of Processes 1 32 1 32
Max. temperature 10 20
Min. temperature
0.01
0.01
Number of iterations
10240×32
10240
30720×32
30720
Markov length
10240 -
30720 -
Cooling rate
0.8 -
0.73 -
In this experiment, we use these parameters.
The number of processes for TPSA is 32 processes.
Maximum temperatures and Minimum temperature for both SA and TPSA are same.
The number of iterations of SA is 32 times as many as TPSA.
Therefore total computational resources for both SA and TPSA are same.
The Markov Length for SA is steps for Rastrigin function and steps for Griewank function.
The Cooling rate for SA is 0.8 and 0.73.
The exchange interval for TPSA is 32 steps.
The neighborhood adjust interval is 8.
Exchange interval - 32 - 32
Neighborhood adjust interval 8 8

15. Experimental results Rastrigin Function
This is the result of Rastrigin function.
This shows the experimental results over 10 trials.
This light red line shows the results by the sequential SA, and this blue one shows those by TPSA.
This axis (the axis of abscissa) shows the neighborhood range configurations, these are constant range configurations, that is, it means the neighborhood range is fixed during entire annealing time.
And this axis (the axis of ordinate) shows the energy of the function.
The solutions, which have lower energy, show better solutions.
For constant neighborhood ranges, you can see the appropriate neighborhood ranges for providing better solutions in SA and TPSA.
For sequential SA, the range of 1 is proper.
For TPSA, the range of 0.5 is proper.
And the most important thing is that the results by TPSA with fixed ranges are worse than those by sequential SA in total.
However, this red line is the result by TPSA/AN, so, TPSA/AN provides the best result in the whole configuration.
TPSA/AN provides the best result in the whole configuration

16. Experimental results Griewank Function
This is the result of Griewank function.
You can see the almost same result as Rastrigin function.
This result shows that the appropriate neighborhood range is 5 both SA and TPSA
And TPSA/AN provides the best result in the whole configuration.
TPSA/AN provides the best result in the whole configuration

17. Histories of temperature and neighborhood range
(Max. T)
Here you can see the history of temperature, with the best solution experienced on TPSA/AN.
And you can also see the history of the neighborhood range of the same solution.
The red line shows the temperature and green line shows the neighborhood range.
From this figure, you can see the automatic determination of cooling schedule and the proper neighborhood range.
(Min. T)
Automatic determination of cooling schedule
Proper neighborhood range

18. Histories of Energy of the best solutions
0.1
0.5 1 5
This figure shows the histories of energy of the best solution for various neighborhood ranges.
This axis shows the annealing steps and this shows the energy.
The red line is TPSA/AN, and other ones are TPSA with fixed neighborhood range.
The solution obtained with the large fixed neighborhood range “5” can reach a better solution quickly.
But it cannot become an accurate solution in the latter step.
The solution obtained with the small fixed range “0.1” converged to a worse solution quickly, because such a small neighborhood range cannot search solutions globally.
However, the adaptive neighborhood can search solutions both globally and locally. AN

19. History of moves of a solution
Global
optimum
This is the locus of the move of the best solution on 2-dimensional problem space.
In these areas, the solution is moving locally.
Here you can also see the escaping form local minima of the solution
Initial
solution

20. Benefits of TPSA/AN Annealing steps of sequential SA ・・・
Parallelization in temperature
A solution of sequential SA moves L times.
Each solution of TPSA moves L/n times．
Appropriate moves at each temperature are needed in TPSA for such short annealing.
Good result by TPSA/AN shows such an appropriate move.
You can see the benefits of TPSA/AN here.
Assume that a solution of sequential SA can move L times.
When the algorithm is parallelized in n processors, each solution of TPSA can move L/n times.
The number of moves is very few in this case.
So, appropriate moves at each temperature are needed in TPSA and the good result by TPSA/AN shows such an appropriate move.

21. Conclusions
TPSA/AN which adjusts the neighborhood range at each temperature is proposed.
The optimum solution obtained by TPSA/AN is better than those obtained by SA with fixed range, SA/AN and TPSA with fixed range.
TPSA/AN has both the high ability of global searching
These are our conclusions.
Thank you for your kind attention.

22. appendix

23. Accept ratio
・Changed every 8 steps.
・Accept ratio is 0 to 1.

24. Result of SA/AN and TPSA/AN
100000
200000
300000
Annealing steps
Energy
search solutions
both globally and locally
converged to a worse solution

25. Parallel implementation
PC-cluster
CPU： Pentium II 233MHz× 8
Memory： 64MB×8
OS： Linux
Parallel library： PVM
Network： 100Base-T (Fast Ethernet)
Implementation
One temperature stage / One process
We used a PC-cluster, which include 8 Pentium II 233MHz processors and 128MB memories.
The operating system of each PC is Linux.
We used PVM for communication library.
The interconnection network is 100 Base-T, “Fast Ethernet”.
We assigned one procedure corresponding to a certain temperature to one process.

26. Computation time
Speed-up by parallelization is not large due to the small size problem.
The total experiment time can be reduced remarkably with TPSA/AN, because of its few preliminary experiments for determination of cooling schedule.
今回の例では，2次元のRastrigin関数といった問題サイズの小さい問題を解いているため，並列処理による大幅な高速化は得られていません．
しかしながら，今後より大きな問題を解くことで容易に並列処理効率が得られることが予想できます．
そして，今回何よりも注目すべきことは，TPSA/ANは計算時間の短縮が得られるアルゴリズムであるにもかかわらず，解の精度も向上させることのできるアルゴリズムであるということです．

27. Background Heuristics search methods Advantages Disadvantages
Simulated Annealing (SA), Genetic Algorithms (GA), Cellar Automata (CA), Neural Network (NN)
Advantages
These methods can solve complex problems.
Disadvantages
High computational cost
Difficult determination of several parameter (e.g. cooling schedule for SA)
Several heuristics search methods have been proposed for optimization problems which contain a lot of local optima.
For example, simulated annealing, genetic algorithms, cellar automata, neural network is favored by many researchers.
One of the most attractive advantages of these methods is that it can process cost functions possessing quite arbitrary degrees of nonlinearities.
But these methods also have disadvantages.
They need high computational costs.
And it is difficult to determine parameters peculiar to each methods.
(In the case of SA, you have to determine the parameters called “cooling schedule”, but it is difficult.)

28. Simulated Annealing (SA)
An algorithm to solve combinatorial optimization problems using simulation of annealing metal.
The algorithm has control parameter called “temperature”.
The algorithm allows to move to uphill direction of cost functions with a certain probability, P = exp(－ΔE／T), controlled by “temperature”.
Simulated Annealing is an algorithm to solve combinatorial optimization problems using simulation of annealing metal.
This algorithm has control parameter called “temperature”.
And it allows to move to uphill direction of cost functions with a certain probability controlled by “temperature”
For example, if the temperature is high, it will be easy to escape from local minima.
And if the temperature is low, the algorithm search the solution locally.
At the first of the algorithm, higher temperature is set. And then, the temperature is decreased.
local minimum
Global minimum

29. Feature of SA Advantages Disadvantages It can solve complex problems.
It can be implemented quite easily.
It can statistically guarantee finding an optimal solution.
Disadvantages
A lot of computation time to find the optimum solution.
Difficult determination of the proper cooling schedule.
The advantages of SA are these ones.
SA can process cost functions possessing quite arbitrary degrees of nonlinearities.
It can be implemented quite easily.
It can statistically guarantee finding an optimal solution.
The disadvantages are these ones.
SA needs a lot of computation time to find the optimum solution.
And it is difficult to determine the proper cooling schedule which defines the behavior of the algorithm.

30. Temperature Parallel SA (TPSA)
Better solutions migrate to processors
of lower temperature with a certain probability.
Exchanging
solution
Exchanging
solution
Exchanging
solution
The highest
temperature
Higher
temperature
Lower
temperature
The lowest
temperature
温度並列SAのアルゴリズムの概念図は，この図のようになります．
この図は，4つの温度用に4つのプロセッサを用いる例です．
まず，4つの初期解を生成しそれぞれのプロセッサに1つずつ与えます．
そして，それぞれのプロセッサに異なる温度を担当させます．
それぞれのプロセッサは，それらの温度を一定に保ち，解摂動と受理判定による通常の逐次SAを行っていきます．
そして，ある回数の反復計算を行うと，それぞれがお互いの解を交換します．解を交換するルールは，良い解が高温プロセッサに存在した場合には必ず交換し，そうでない場合でもある確率にしたがって交換することにします．
これらの動作により，結果として良質な解が低温プロセッサの方に集まり，
最終的に最低温度プロセッサから解を得ます．
Move &
Metropolis criterion
Move &
Metropolis criterion
Move &
Metropolis criterion
Move &
Metropolis criterion

31. Feature of TPSA Automatically determined cooling schedule
Easy implementation to parallel computer
Temperature
Time
Parallelize in Temperature
TPSA has two attractive features.
The first feature is automatically determined cooling schedule.
In the case of sequential SA, users have to determine a proper cooling schedule. But for TPSA, user do not have to do it, and each solution on each processor find a cooling schedule which is appropriate to each solution.
Another one is that it is easy to implement the algorithm to parallel computer.
Annealing on T1
Annealing on T2
Annealing on T3
Annealing on T4
Networked
processors

32. Sequential SA for continuous optimization problems
Corana‘s method : adaptive neighborhood
Dekkers & Aarts‘s method : quasi-Newton
Very Fast Simulated Re-annealing : sensitivity
(Adaptive Simulated Annealing)
Fast Annealing : Cauchy distribution
Boltzmann Annealing : Boltzmann distribution
miscellaneous
feature of these methods
using structural information of objective function
adjusting neighborhood range
in the phase of generating a next state.
There are several sequential SA algorithms for continuous optimization problems.
The features of these methods are using structural information of objective functions and adjusting neighborhood range in the phase of generating a next state.
For example, Corana’s method uses adaptive neighborhood system. Dekkers & Aarts’s method uses quasi-Newton.

33. Temperature Parallel Simulated Annealing with Adaptive Neighborhood (TPSA/AN)
Features
With an extended algorithm for solving continuous optimization problems.
Uses the Corana’s method for the moves of solutions in continuous problem spaces.
Uses the information of landscape of objective functions.
Adjusts the neighborhood range by keeping the rate of acceptance constant.
The features of our proposed algorithm are these. (ここでスライドを指す)
This algorithm uses corana’s method for local search in continuous problem spaces.
Therefor the algorithm uses the information of landscape of objective functions and adjusts the neighborhood range for maintaining a rate of accepted and rejected moves.

34. Determination of control parameter －Maximum temperatureーB x2
Determine the max. temperature for considering the trade-off between calculation time and coverage of searches.
Preliminary experiments
1. Temporary maximum temperature
is set.
2. Perform each sequential SA at a constant
temperature.
3. Evaluate the calculation time and the
coverage of search. C A x1
Problem
space D
シミュレーテッドアニーリングにおいては，“温度”と呼ばれる重要なパラメータを決定しなければなりませんが，本研究のアルゴリズムでは独自の方法でそれらを決定することができます．
最高温度は，計算時間と探索能力のトレードオフから決定します．
具体的には，候補となるいくつかの温度を用意し，それらの一定温度のもとにSAを実行させ，その時の計算時間と，その時に問題空間の境界線に近づいた解，例えば2次元では右上の図のように4つの境界に最も近づいた4つの解のことであり，それらの解と境界線の距離の平均，つまりA,B,C,Dの距離の平均を記録するという予備実験を行います．
計算時間は右の図のように温度が上がれば増加していきます．
またA,B,C,Dの平均は

35. Experimental result

36. History of energy

37. History of Energy on T0

38. procedure SIMULATED ANNEALING:
begin
Initialize(i(0), T(0));
k := 0;
i := i(0);
repeat
for l := 1 to L(k) do
Generate(j);
if f(j) ≦ f(i) then i := j;
else
if exp(-(f(i)-f(j))/T(k)) > random[0,1)
then i := j;
end;
k := k + 1;
UpdateTemp(T(k));
until stopcriterion

39. Conclusion
TPSA/AN which adjusts the neighborhood range at each temperature is proposed.
The neighborhood range affects the search abilities of SA and TPSA in continuous optimization problems.
The determination of the best neighborhood range is very important.
The optimum obtained by TPSA with the best constant neighborhood range is worse than the one obtained by SA with the best constant neighborhood range. This is due to the short annealing period.
The optimum solution obtained by TPSA/AN is better than those obtained by SA with fixed range, SA/AN and TPSA with fixed range.
TPSA/AN has both the high ability of global searching and quick convergence to the global optimum.
まとめは，このようになっています．

40. Experimental results on Griewangk