第五章 總結與未來工作
5.3 主要總結
本文主要的目的為針對傳統 ES 的突變機制上做改良，由於傳統 ES 的 突變機制上，主要是以隨機亂數來決定其突變的行為，self-adaptation 雖然 能夠在演化過程中適度地調整策略參數以降低其亂數擾動的性質，然而對 關聯式突變中，由於 rotation angle 此部分的策略參數與問題維度呈現二次 平方項次成長，造成self-adaptation 對此部分的調整能力上有所限制，rotation angles 決定個體所能夠突變的方向，因而我們可以說傳統的突變機制，對於 突變方向的決定上仍存在大量的隨機因子所造成的不穩定性。在本文中提 出了一個引導式突變的機制，其主要精神為擷取PSO 中群體間相互合作的 群體智能，以此為「概念」來對於偏向隨機搜尋的ES 進行方向上的引導，
進而降低突變方向的決定上造成的不穩定性，由視覺化的實驗觀察，此引 導的概念將有效地使個體在突變的過程中，達到更有具系統化的突變能 力，因此能有效率地縮短實驗時間及成本，進而達到改善並穩定其搜尋能 力。經由實驗結果的分析與探討，結合了引導式突變的PSGES，於 2005 年 CEC Special Session 所提供的 benchmark 下所表現出來的結果，確實能大幅 度增進傳統 ES 的搜尋效能，且與目前一些進階演算法在求解問題的能力 上，在尚未對參數做特別調整上，已具有可競爭性 (comparable)的表現。
本研究對於演化計算領域所做的主要貢獻，為將PSO 的群體智能(swarm
intelligence)概念與傳統 ES 中的演化機制做一個本質及概念層次上的結 合，此種結合的方式保留了 ES 以 self-adaptation 來對突變行為進行調整的 優勢，更可充分地發揮群體智能對於搜尋行為上所增進的效益，相較於過 去僅僅在計算層次上的結合，本研究提供了另一個可加以深入研究的思維 方向。
附錄一
3. F3: Shifted Rotated High Conditioned Elliptic Function
( )
i-14. F4: Shifted Schwefel’s Problem 1.2 with Noise in Fitness
5. F5: Schwefel’s Problem 2.6 with Global Optimum on Bounds
{
1 2 1 2}
* *f(x) = max x + 2x - 7 , 2x + x - 5 , i = 1, ,n, x = [1,3], f(x ) = 0… Extend to D dimensions:
A is a D D matrix, a are integer random number in the range [-500, 500], det(A) 0, A is the i row of A
Basic Multimodal Function 6. F6: Shifted Rosenbrock’s Function
6 D-1
( (
2i i+1)
2(
i)
2)
67. F7: Shifted Rotated Griewank’s Function without Bounds
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
( )
After load the data file, set o = -32o are randomly distributed in the search range, for j = 1,2, , D/2
inear transformation matrix, condition number = 100
9. F9: Shifted Rastrigin’s Function
D 2
10. F10: Shifted Rotated Rastrigin’s Function
M: linear transformation matrix, condition number = 2
∗ …
…
11. F11: Shifted Rotated Weierstrass Function
( )
M: linear transformation matrix, condition number = 5
…
…
12. F12: Schwefel’s Problem 2.13
( ( ) )
A, B are two D D matrix, a ,b are integer random number in the range [-100,100], = [ , ,
Expanded Functions
1 2 D 1 2 2 3 D-1 D D 1
Using a 2-D function F(x, y) as a starting function, corresponding expanded function is:
EF(x ,x , ,x )=F(x ,x )+F(x ,x )+ +F(x ,x )F(x ,x )… …
13. F13: Expanded Extended Griewank’s plus Rosenbrock’s Function (F8F2)
2 D
14. F14: Shifted Rotated Expanded Scaffer’s F6
2 2 2
M: linear transformation matrix, condition number = 3
∗ …
…
Composition functions
th i
i i
i
F(x): new comosition function
f (x): i basic function used to construct the composition function n: number of basic functions
D: dimensions
M : linear transformation matrix for each f (x) o : new shifted optimum position for each f (x)i
n
i i i i i
i=1
F(x) =
∑
{w [f '((x - o)/∗ λ ∗M ) + bias ]} + f_biasi i
w : weight value for each f (x), calculated as below:
( )
: used to control each f (x)'s coverage range, a small give a narrow range for that f (x) : used to stretch compress the function, >1 means stretch, <1 means compress
o define the global and
σ σ
λ λ λ
i
i i
local optima's position, bias define which optimum is global optimum.
Using o , bias , a global optimum can be placed anywhere.
i
i
max i
If f (x) are different functions, fifferent functions have different properties and height, in order to get a better mixture, estimate a biggest function value f for 10 function f (x), then normali
ze each basic functions to similar heights as below:
f (x) = C f (x)/ f , C is a predefined constant.
f is estimated using f = f ((x'/ ) M ), x' = [5,5, ,5].λ
∗
∗ …
i
In the following composition functions, Number of basic functions n=10.
D: dimensions
o: n D matrix, defines f (x)'s global optimal positions bias = [0, 100, 200, 300, 400, 500, 600, 700, 800, 900].
Hence,
∗
the first function f (x) always the function with the global optimum1
C=2000
15. F15: Hybrid Composition Function
max max
f (x): Rastrigin's Function
f (x) = (x - 10cos(2 x ) + 10) f (x): Weierstrass Function
f (x) = a cos(2 b (x + 0.5)) - D a cos(2 b 0.5) , f (x): Griewank's Function
x x
f (x) = - cos( ) + 1
4000 i
f (x): Ackley's function
1 1 f (x): Sphere Function
f (x) = x M are all identity matrices
σ λ
…
16. F16: Rotated Hybrid Composition Function
i
15
Except M are different linear transformation matrixes with condition number of 2, all other setting are the same as F .
17. F17: Rotated Hybrid Composition Function with Noise in Fitness
18. F18: Rotated Hybrid Composition Function
( )
f (x): Ackley's Function
1 1
f (x) = -20exp -0.2 x - exp cos 2 x + 20 + e
D D
f (x): Rastrigin's Function
f (x) = x - 10cos 2 x + 10
f (x): Weierstrass Function
f (x) = a cos 2 b x + 0.5 - D a cos 2 b 0.5 ,
M are all rotation matrices. condition number are [2 3 2 3 2 3 20 30 200 300]
o [0,0, ,0]
σ
λ ∗ ∗ ∗ ∗ ∗
…
19. F19: Rotated Hybrid Composition Function with a narrow basin for the global optimum
[ ]
[ ]
All setting are the same as F except18
= 0.1, 2, 1.5, 1.5, 1, 1, 1.5, 1.5, 2, 2 ;
= 0.1 5/32; 5/32; 2 1; 1; 2 5/100; 5/100; 2 10; 10; 2 5/60; 5/60 σ
λ ∗ ∗ ∗ ∗ ∗
20. F20: Rotated Hybrid Composition Function with the Global Optimum on the Bounds
18 1(2j)
All settings are the same as F except after load the data file, set o =5, for j=1,2, , D/2…
21. F21: Rotated Hybrid Composition Function
( )
f (x): Rotated Expanded Scaffer's F6 Function sin x +y - 0.5 F(x,y)=0.5+
1 + 0.001 x + y
f (x)=F(x ,x )+F(x ,x )+ +F(x ,x )+F(x ,x ) f (x): Rastrigin's Function
f f (x): F8F2 Function
x x
f (x): Weierstrass Function
f (x)= a cos 2 b x +0.5 - D a cos 2 b 0.5 ,
9-10
2 D
D i i
i
i=1 i=1
f (x): Griewank's Function
x x M are all orthogonal matrix
σ
λ ∗ ∗ ∗ ∗ ∗
22. F22: Rotated Hybrid Composition Function with High Condition Number Matrix
[
21 i]
All settings are the same as F except M 's condition number are 10 20 50 100 200 1000 2000 3000 4000 5000
23. F23: Non-Continuous Rotated Hybrid Composition Function
21
j j 1j
j
j j 1j
All settings are the same as F
x x - o <1/2
where a is x's integral part and b is x's decial part All "round" operators use the same schedule.
⎧⎪
⎨⎪ ≥
⎩
24. F24: Rotated Hybrid Composition Function
( )
f (x): Weierstrass Function
f (x)= a cos 2 b x +0.5 - D a cos 2 b 0.5 , a=0.5, b=3, k =20
f (x): Rotated Expanded Scaffer's F6 Function F(x,y
( ) ( )
f (x): F8F2 Function
x x
f (x): Rastrigin's Function
f (x)= x - 10cos 2 x + 10
f (x): Griewank's Function
x x
f (x)= - cos + 1
4000 i
f (x): Non-Continuous Expanded Scaffer's F6 Function sin x +y -0.5 f (x): Non-Continuous Rastrigin's Function f(x)= y - 10cos 2 y + 10π f (x): High Conditioned Elliptic Function f(x)= 10 x
f (x): Sphere Function with N
⎧⎪⎨
[ ]
[ ]
i
i
=2, for i=1,2, ,D
= 10; 5/20; 1; 5/32; 1; 5/100; 5/50; 1; 5/100; 5/100
M are all rotation matrices, condition numbers are 100 50 30 10 5 5 4 3 2 2 ; σ
λ
…
25. F25: Rotated Hybrid Composition Function without Bounds
All settings are the same as F except no exact search range set for this test function.24
附錄二
CEC’05 25 個 test functions 整體最佳解的個
體目標變數與其適應值
附錄三
PSGES 所達到的最佳解之個體目標變數與
其適應值
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