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
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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.