4.3 實驗設計
4.3.8 GPPSO 中個機制的收斂速度比較
使用CEC 2005 special session 作為測試函數，比較分別加入三個機制的收斂 速度比較，比較對象分別為 PSO 加上使用直交表選取初始值(PSO + use OA table get initial)，PSO 加上 local search(PSO + local search)，PSO 加上 modified IMM(PSO + modified IMM)以及 GPPSO。圖 4.3 至圖 4.27，為分別對二十五個測試的收斂速 度比較圖。由實驗結果可以得到GPPSO 的收斂速度較佳。 Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
圖 4.4 函數 F 2 收斂比較圖
F 3 Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
s
圖 4.7 函數 F 5 收斂比較圖 Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
圖 4.10 函數 F 8 收斂比較圖 Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
圖 4.13 函數 F 11 收斂比較圖
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
圖 4.16 函數 F 14 收斂比較圖
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
圖 4.19 函數 F 17 收斂比較圖
5.00E+03 1.50E+04 2.50E+04 3.50E+04 4.50E+04 5.50E+04 6.50E+04 7.50E+04 8.50E+04 9.50E+04
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
5.00E+03 1.50E+04 2.50E+04 3.50E+04 4.50E+04 5.50E+04 6.50E+04 7.50E+04 8.50E+04 9.50E+04
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
5.00E+03 1.50E+04 2.50E+04 3.50E+04 4.50E+04 5.50E+04 6.50E+04 7.50E+04 8.50E+04 9.50E+04
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
圖 4.22 函數 F 20 收斂比較圖
F 21
2.00E+02 7.00E+02 1.20E+03
5.00E+03 1.50E+04 2.50E+04 3.50E+04 4.50E+04 5.50E+04 6.50E+04 7.50E+04 8.50E+04 9.50E+04
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
5.00E+03 1.50E+04 2.50E+04 3.50E+04 4.50E+04 5.50E+04 6.50E+04 7.50E+04 8.50E+04 9.50E+04
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
5.00E+03 1.50E+04 2.50E+04 3.50E+04 4.50E+04 5.50E+04 6.50E+04 7.50E+04 8.50E+04 9.50E+04
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
圖 4.25 函數 F 23 收斂比較圖
F 24
2.00E+02 7.00E+02 1.20E+03
5.00E+03 1.50E+04 2.50E+04 3.50E+04 4.50E+04 5.50E+04 6.50E+04 7.50E+04 8.50E+04 9.50E+04
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
圖 4.26 函數 F 24 收斂比較圖
F 25
3.50E+02 4.00E+02 4.50E+02 5.00E+02
5.00E+03 1.50E+04 2.50E+04 3.50E+04 4.50E+04 5.50E+04 6.50E+04 7.50E+04 8.50E+04 9.50E+04
Number of evaluation
Fitness value
PSO+use OA table get initial
PSO + local search PSO + modified IMM GPPSO
圖 4.27 函數 F 25 收斂比較圖
五、結論 5.1 結論
本研究期望提出一種可以通用在一般領域上的粒子群演算法，此通用型粒子 群演算法，包含三個特色，特色一為加入使用多水準直交表求得初始粒子族群，
使用直交實驗均勻的在解空間找出最佳初始點，移除初始解的隨機因素，當問題 有交互作用時，可以更有效率的找到較佳的初始解；特色二為粒子群演算法中加 入local search 機制，local search 針對單獨一個粒子進行，可提供粒子間較高的 變異度，進而可得到較好的搜尋效能；特色三為採用智慧型粒子移動機制，使用 直交實驗系統分析，考慮粒子最佳解跟全域最佳解的直交實驗組合，可以更有效 率的找出潛在最佳解，當面對大參數問題時，此機制可以發揮很大的效能，綜合 三項特性，找出一通用型的粒子群演算法，可以解大參數具交互作用的題目，也 可達到快速收斂的目的。
最後實驗使用CEC 2005 special session 及”OPSO: Orthogonal Particle Swarm Optimization and Its Application to Task Assignment Problems”中的測試函數，實驗 結果顯示提出的演算法有很好的搜尋效率。
5.2 未來展望
未來的研究方向，可找尋更多的測試函數，使用更多不同領域的測試函數幫 助找出更好的演算法預設參數。另一個研究方向可針對直交表初始化進行，目前 初始化時使用多水準直交表，當水準數越高，使用的評估次數越多，可研究使用 兩次的分割解範圍，可以縮小使用直交表的水準數，達到使用更少的評估次數，
也可以有很好的最佳初始解。
參考文獻
[1] S. Forrest and M. Mitchell. “Relative buildingblock fitness and the building-block hypothesis,” FOGA-92. Proc. of Workshop on Foundations of Genetic Algorithms and Classifier Systems, pp.109-126, 1992.
[2] C. Darwin, On the origin of species. London: John Murray, 1859.
[3] J. Kennedy and R. C. Eberhart, “Particle swarm optimization, ” in Proc. IEEE Conf.
Neural Networks IV, Vol. 4, pp. 1942-1948, 1995
[4] J. Kennedy, R. C. Eberhart and Y. Shi, Swarm Intelligence, San Francisco: Morgan Kaufmann, 2001.
[5] H.-M. Chen, B.-F. Liu, H.-L. Huang, S.-F. Hwang, and S.-Y. Ho*, "SODOCK:
Swarm Optimization for Highly Flexible Protein-Ligand Docking," Journal of Computational Chemistry, vol. 28, pp. 612-623, 2007. (SCI)
[6] S.-Y. Ho, H.-S. Lin, W.-H. Liauh and S.-J. Ho, "OPSO: Orthogonal Particle
Swarm Optimization and Its Application to Task Assignment Problems," accepted by IEEE Trans. Systems, Man, and Cybernetics -Part A, Systems and Humans, 2006. (SCI, EI) [7] J. J. Liang and P. N. Suganthan, "Dynamic Multi-Swarm Particle Swarm
Optimizer with Local Search," IEEE Congress on Evolutionary Computation, vol. 1, pp. 522-528, 2005.
[8] M. S. Phadke, Quality Engineering Using Robust Design. Englewood Cliffs. NJ:
Prentice-Hall, 1989.
[9] S. H. Park, Robust Design and Analysis for Quality Engineering. Chapman &
Hall, 1996
[10] Y.-W. Leung and Y. Wang, “An orthogonal genetic algorithm with quantization for global numerical optimization,” IEEE Trans. Evolutionary Computation, vol. 5, no. 1, pp. 41-53, Feb. 2001.
[11] A. Dey, Orthogonal Fractional Factorial Designs. New York:Wiley, 1985.
[12] A. S. Hedayat, N. J. A. Sloane, and J. Stufken, Orthogonal Arrays: Theory and Applications. New York: Springer-Verlag, 1999.
[13] J. J. Liang, and P. N. Suganthan, "Dynamic Multi-Swarm Particle Swarm Optimizer,"
Proc. of IEEE International Swarm Intelligence Symposium, pp. 124-129, 2005.
[14] P. N. Suganthan, N. Hansen, J. J. Liang, K. Deb, Y.-P. Chen, A. Auger, and S. Tiwari,
"Problem Definitions and Evaluation Criteria for the CEC 2005 Special Session on Real-Parameter Optimization," Nanyang Technological University Technical Report,
Singapore May 2005 AND KanGAL Technical Report No. 2005005, IIT Kanpure, India 2005.
Definitions of the 25 CEC’05 Test Functions
Unimodal Functions:
1. F1: Shifted Sphere Function D: dimensions
1 2
[ , ,...,o o oD]
=
o : the shifted global optimum
3. F3: Shifted Rotated High Conditioned Elliptic Function
6 11 2 D: dimensions
1 2
[ , ,...,o o oD]
=
o : the shifted global optimum M: orthogonal matrix
4. F4: Shifted Schwefel’s Problem 1.2 with Noise in Fitness
2 D: dimensions
1 2
[ , ,...,o o oD]
=
o : the shifted global optimum
5. F5: Schwefel’s Problem 2.6 with Global Optimum on Bounds
1 2 1 2
( ) max{ 2 7 , 2 5}, 1,...,
f x = x + x − x +x − i= n, x* =[1,3], f( ) 0x* = Extend to D dimensions:
5( ) max{ i i} _ 5, 1,...,
F x = A x B− + f bias i= D,x=[ , ,...,x x1 2 xD] D: dimensions
A is a D*D matrix, a are integer random numbers in the range [-500, 500], det( )ij A ≠0, Ai is the
Basic Multimodal Functions
6. F6: Shifted Rosenbrock’s Function
1 D: dimensions
1 2
[ , ,...,o o oD]
=
o : the shifted global optimum
7. F7: Shifted Rotated Griewank’s Function without Bounds
2 D: dimensions
1 2
[ , ,...,o o oD]
=
o : the shifted global optimum
M’: linear transformation matrix, condition number=3 M =M’(1+0.3|N(0,1)|)
8. F8: Shifted Rotated Ackley’s Function with Global Optimum on Bounds
2
M: linear transformation matrix, condition number=100 9. F9: Shifted Rastrigin’s Function
2 D: dimensions
1 2
[ , ,...,o o oD]
=
o : the shifted global optimum 10. F10: Shifted Rotated Rastrigin’s Function
2 D: dimensions
1 2
[ , ,...,o o oD]
=
o : the shifted global optimum
M: linear transformation matrix, condition number=2
11. F11: Shifted Rotated Weierstrass Function D: dimensions
1 2
[ , ,...,o o oD]
=
o : the shifted global optimum
M: linear transformation matrix, condition number=5
12. F12: Schwefel’s Problem 2.13
D: dimensions
A, B are two D*D matrix, a ,ij b are integer random numbers in the range [-100,100], ij
1 2
[ ,α α ,...,αD]
=
α ,αj are random numbers in the range [−π π, ].
Expanded Functions
Using a 2-D function ( , )F x y as a starting function, corresponding expanded function is:
1 2 1 2 2 3 1 1
( , ,..., D) ( , ) ( , ) ... ( D , D) ( , )D EF x x x =F x x +F x x + +F x − x +F x x
13. F13: Shifted Expanded Griewank’s plus Rosenbrock’s Function (F8F2) F8: Griewank’s Function:
2
14. F14: Shifted Rotated Expanded Scaffer’s F6 Function
2 2 2
2 2 2
(sin ( ) 0.5) ( , ) 0.5
(1 0.001( )) x y F x y
x y
+ −
= +
+ +
Expanded to
14( ) ( , ,...,1 2 D) ( , )1 2 ( , ) ...2 3 ( D1, D) ( , )D 1 _ 14
F x =EF z z z =F z z +F z z + +F z − z +F z z + f bias , ( )*
= −
z x o M ,x=[ , ,...,x x1 2 xD] D: dimensions
1 2
[ , ,...,o o oD]
=
o : the shifted global optimum
M: linear transformation matrix, condition number=3
Composition functions ( )
F x : new composition function
i( )
f x : ith basic function used to construct the composition function n : number of basic functions
then normalize the weight
1
oi define the global and local optima’s position, bias define which optimum is global optimum. i Using oi, bias , a global optimum can be placed anywhere. i
If ( )fi x are different functions, different functions have different properties and height, in order to get a better mixture, estimate a biggest function value fmax i for 10 functions ( )fi x , then normalize each basic functions to similar heights as below:
'( ) * ( ) / max
i i i
f x =C f x f , C is a predefined constant.
max i
f is estimated using fmax i = (( '/ )*fi x λi M , 'i) x =[5,5…,5].
In the following composition functions, Number of basic functions n=10.
D: dimensions
o: n*D matrix, defines fi( )x ’s global optimal positions
bias=[0, 100, 200, 300, 400, 500, 600, 700, 800, 900]. Hence, the first function f1( )x always the function with the global optimum.
C=2000
15. F15: Hybrid Composition Function
f − x : Weierstrass Function
max max
M are all identity matrices i
16. F16: Rotated Version of Hybrid Composition Function F15
ExceptM are different linear transformation matrixes with condition number of 2, all other i settings are the same as F15.
17. F17: F16 with Noise in Fitness Let (F16 - f_bias16) be ( )G x , then
17( ) ( )*(1+0.2 N(0,1) ) _ 17
F x =G x + f bias
All settings are the same as F16.
18. F18: Rotated Hybrid Composition Function
f − x : Weierstrass Function
max max
19. F19: Rotated Hybrid Composition Function with narrow basin global optimum All settings are the same as F18 except
σ=[0.1, 2, 1.5, 1.5, 1, 1, 1.5, 1.5, 2, 2];,
λ= [0.1*5/32; 5/32; 2*1; 1; 2*5/100; 5/100; 2*10; 10; 2*5/60; 5/60]
20. F20: Rotated Hybrid Composition Function with Global Optimum on the Bounds All settings are the same as F18 except after load the data file, set o1(2 )j = , for 5
1, 2,..., / 2 j= ⎢⎣D ⎥⎦
21. F21: Rotated Hybrid Composition Function
1 2( )
f− x : Rotated Expanded Scaffer’s F6 Function
2 2 2
f − x : Weierstrass Function
max max
22. F22: Rotated Hybrid Composition Function with High Condition Number Matrix
All settings are the same as F21 except M ’s condition numbers are [10 20 50 100 200 1000 i 2000 3000 4000 5000]
23. F23: Non-Continuous Rotated Hybrid Composition Function All settings are the same as F21.
Except 1
1
1/ 2 (2 ) / 2 1/ 2
j j j
j
j j j
x x o
x
round x x o
⎧ − <
= ⎨⎪
− >=
⎪⎩ for 1, 2,..,j= D
1 0 & 0.5
( ) 0.5
1 0 & 0.5
a if x b
round x a if b
a if x b
− <= >=
⎧⎪
=⎨ <
⎪ + > >=
⎩
,
where a is x’s integral part and b is x’s decimal part
All “round” operators in this document use the same schedule.
24. F24: Rotated Hybrid Composition Function
1( )
f x : Weierstrass Function
max max
f x : Rotated Expanded Scaffer’s F6 Function
2 2 2
f x : Rastrigin’s Function
2
f x : Griewank’s Function
2
f x : Non-Continuous Expanded Scaffer’s F6 Function
2 2 2
f x : Non-Continuous Rastrigin’s Function
2
1/ 2
f x : High Conditioned Elliptic Function
1
f x : Sphere Function with Noise in Fitness
2
25. F25: Rotated Hybrid Composition Function without bounds
All settings are the same as F24 except no exact search range set for this test function.