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2 染色体の多様性調整可能な遺伝的アルゴリズムと 多種の最適化問題への応用 2017 年 3 月 前川 廣太郎

3 染色体の多様性調整可能な遺伝的アルゴリズムと 多種の最適化問題への応用 前川廣太郎 システム情報工学研究科 筑波大学 2017 年 3 月

4 WS MPGA

5 2.1 GA PDGA MPGA MPGA WS GA SGA MPGA WS GA Rastrign Rosenbrock Griewank Griewank Rosenbrock SGA Rosenbrock MPGA Rosenbrock WSGA p = Rosenbrock WSGA p = Rosenbrock WSGA p = GA A F

6 WS GA

7 2.1 GA,MPGA WSGA σ GA

8 1 2 1 [1, 2, 3, 4, 5] 2 1 [6, 7, 8, 9] [10, 11, 12, 13, 14] 6

9 [15, 16] [17, 18] [19, 20] 1 ([21, 22, 23, 24, 25, 26] GA GA (Massively Parallel model of Genetic Algorithm[27, 28] MPGA ) MPGA MPGA 7

10 Watts and Strogatz [29, 30] WS SGA MPGA WS GA 3 SGA MPGA GA 1 WS GA SGA MPGA WS GA 2 WS GA MPGA WS WS 8

11 GA 3 SGA MPGA WS GA 3 WS GA A 2 GA GA WS GA 9

12 2 2.1, [15, 16, 17, 18, 31, 32, 33, 34, 19, 20] Genetic Algorithm, GA GA 1 10

13 (Parallel Distributed Genetic Algorithm[35, 36] PDGA ) (Massively Parallel model of Genetic Algorithm[27, 28] MPGA ) PDGA GA GA GA MPGA PDGA MPGA GA PDGA MPGA MPGA 11

14 WS WS MPGA WS GA WS 2.1 GA,MPGA WSGA SGA MPGA WSGA 2.2 PDGA 2.3 MPGA WS GA

15 2.2 PDGA [35] PDGA GA 1 GA 1 MPGA [27] MPGA (Cellular GA) Asynchronous Massively Paralled Genetic Algorithm AMPGA AMPGA 1 0 MPGA Perego [37] MPGA MPGA GA 13

16 [38] Parameter-free Genetic Algorithm PfGA ) PfGA GA GA MPGA WS GA 14

17 2.3 WS MPGA.2.1 GA GA GA [39, 40] 15

18 [41, 42, 43] GA PDGA 2.2 PDGA 2.2 PDGA 16

19 PDGA GA MPGA PDGA MPGA (1) S I S i (i = 1, 2,..., I,S i S j, i I, S i ) (2) S i 2.3 ) (3) S i (4) (5) (3)(4) 17

20 2.3 MPGA MPGA MPGA 18

21 2 MPGA, MPGA MPGA WS 19

22 2.3.2 Watts and Strogatz GA MPGA WS [29] MPGA 2.2 V E G V E) V = {S 1, S 2,..., S I } (2.1) E S i MPGA WS 2.1 V E E S i 2.4 E p 0 < p < 1 0 MPGA 1 20

23 MPGA WS GA 2.4 WS WS Step1 2 Step2 Step3 MPGA MPGA WS GA WS GA 21

24 SGA MPGA GA 0 MPGA p = 0 SGA SGA WS 2.5 SGA MPGA WS GA 22

25 2.4 WS GA 3 SGA MPGA Rastrign Rosenbrock Griewank Rastrigin Griewank min(f (x) ) = F (0, 0,..., 0) = 0 Rosenbrock min(f (x) ) = F (1, 1,..., 1) = Rastrign function Rosenbrock function Griewank function Interdependence of variables none have have Shape multimodal unimodal multimodal Rastrign n ( F Rastrigin (x) = 10n + x 2 i 10 cos(2πx i ) ) (2.2) i=1 ( 5.12 x i < 5.12) 23

26 2.6 Rastrign Rastrign Rosenbrock n 1 ( F Rosenbrock (x) = 100(xi+1 x 2 i ) 2 + (1 x i ) 2) (2.3) i=1 ( x i < 2.048) Rastrign 24

27 2.7 Rosenbrock 2.7 Rosenbrock Griewank F Griewank (x) = 1 + n x 2 n i 4000 ( cos ( x i ) ) (2.4) i i=1 i=1 ( 512 x i < 512) Rosenbrock GA MPGA WS GA

28 2.8 Griewank 2.9 Griewank

29 2.3 SGA MPGA WSGA Number of experiments 50 Number of chromosomes 256 Number of Generation 3000 Dimensions of X 20 Crossover ratio 70% Mutation ratio 1% Number of group Short cut ratio %,1%,10%

30 2.4 SGA MPGA WSGA p = WSGA p = 0.01 WSGA p = 0.1 Rastrign function Rosenbrock function Griewank function SGA MPGA WSGA p = WSGA p = 0.01 WSGA p = 0.1 Rastrign function Rosenbrock function Griewank function Rastrign WS GAp = 0.1 Rosenbrock Griewank p = 0.01 WS GA WS GA 2 Rastrign Griewank Rosenbrock WS GAp = 0.1 SGA MPGA SGA MPGA SGA MPGA 28

31 2.10 Rosenbrock SGA 2.11 Rosenbrock MPGA 2.12 Rosenbrock WSGA p = Rosenbrock WSGA p = 0.01 Rosenbrock MPGA SGA ,2.11,2.12,2.13,2.14 SGA MPGA WS 29

32 2.14 Rosenbrock WSGA p = Rosenbrock D=2 Rosenbrock D=3 Conventional method Proposed method GA Zhu [44] Rosenbrock 2.6 D

33 2.5 GA WS MPGA WS GA MPGA MPGA 3 Rastrign WS GAp = 0.1 Rosenbrock Griewank WS GAp = 0.01 SGA WS MPGA WS GA 31

34 GA WS GA WS GA GA GA GA WS GA SGA MPGA 32

35 3.1 WS GA WS GA A 2 GA 3.1 GA WS GA

36

37 GA GA 2 SGA MPGA WS GA SGA SGA MPGA WS GA 2.3 GA [ ] M M N N N N) n C n C n = (G n 1, G n 2,, G n M ) (3.1) G n m = (S n m, D n m, R n m) (3.2) S n m {1, 2,, S max } (3.3) D n m {1, 2,, D max } (3.4) 35

38 R n m {1, 2,, R max } (3.5) S max = 6 D max = 2 R max = 8 G n m n m S n m Dn m R n m 3 n C n 3 M [ ] C n F (C n ) N (3.6) C n

39 3.1 Agent1 Agent2 Agent3 Agent4 1st period class1 class2 class3 class3 2nd period class4 class5 class3 3rd period class6 class6 class7 4th period class6 class6 class8 class8 5th period class9 class8 class8 6th period class10 class

40 3.2.2 GA N C n X 1 X 32Y Y 3 4 [ ] (6) F (C i ) min(f (C i )) 38

41 3.2 GA [ ] 2 C i = (G i 1, G i 2,, G i M) (3.7) C j = (G j 1, Gj 2,, Gj M ) (3.8)

42 m m C i = (G i 1, G i 2,, G i m, G i m+1, G i m+2,, G i M) (3.9) C j = (G j 1, Gj 2,, Gj m, G j m+1, Gj m+2,, Gj M ) (3.10) (3.9), (3.10) m m + 2 C i = (G i 1, G i 2,, G j m, G j m+1, Gj m+2,, Gi M ) (3.11) C j = (G j 1, Gj 2,, Gi m, G i m+1, G i m+2,, G j M ) (3.12) [ ] n G i n = (1, 1, 0) (1, 1, 6) 40

43

44 (1) (2) (3) (1) (2) (3),. GA 42

45 3.5 Step1 Step2 Step3 GA

46 [45, 46] GA [ ]

47 : X Y 4 X 4 Y 2 X 2/3 Y 1/3 2 : 45

48 3 : 46

49 3.4 [47, 48] 2 47

50 Frequency Frequency Frequency σ σ σ Distance Distance Distance σ σ σ 48

51 3.4.1 N V (= 1, 2, 3,..., N) E V V {(i, j) i V, j V } V, E G G = (V, E), (3.13) (i, j) = (j, i) i, j l i,j l i,j j = 2,..., n j L j P L : L 1 = 0 (3.14) j = 2,..., n T L : L j = l 1j (3.15) PL = {1}, TL = {2, 3,..., n} (3.16) 49

52 TL L k k L k = L k TL k PL TL = L j TL j L j = min L j, L k + l kj k N N 2 x 1 ± a x i x N ± a, y 1 ± b y i y N ± b, (3.17) (a, b ) E make(i) = {j l ij < F } F (i, j) V (3.17) σ 50

53

54

55 [49] A GA

56 σ 3.6 σ σ σ σ σ = 0.6 σ σ = 0.9 σ = 0.6 σ = 0.7 σ = 0.6 σ 54

57 3.14 GA WS GA GA 3.2 WS GA SGA MPGA 55

58 3.7 GA SGA MPGA WSGA minimum GA GA GA MPGA WS GA 3.7 WSGA WSGA GA 56

59 3.18,

60 3.6 2 WS GA SGA MPGA WS GA 20 5 σ = GA WSGA 90 58

61 3.10 A F

62

63

64 3.17 WS GA

65

66 4 2 WS GA MPGA WS WS GA 3 SGA MPGA WS GA WS GA SGA 3 WS GA A 2 GA GA WS GA WSGA 10 64

67 20 5 σ = 0.6 GA WS GA WS WS 65

68 WS GA 1 66

69 67

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