. p.1/34

Size: px

Start display at page:

Download ". p.1/34"

のぶのすけゆきしげ
5 years ago
Views:

1 . p.1/34

2 (Optimization) (Mathematical Programming),,. p.2/34

3 p.3/34

4 p.4/34

5 4x + 2y 6, 2x + y 6, x 0, y 0 x, yx + yx, y x + y x + 2y 6, 2x + y 6, x 0, y 0. 2x, y x + y 4x 1, x 2, x 3, x 4 3x 1 + 4x 2 + 2x 3 2x 4 2x 1 + 3x 2 + 4x 3 + x 4 10, 3x 1 + 2x 2 x 3 + 6x 4 8, x 1, x 2, x 3, x 4 :(0). 100, 1000, p.5/34

6 : f(x) (or),: x = (x 1, x 2,..., x n ) S. x j, S : f : x = (x 1,..., x n ), S x f(x) n = 2 S =x, f(x) : f(x) : x* f(x*) f(x) :x (). p.6/34

7 : f(x) (or),: x = (x 1, x 2,..., x n ) S. x j, S : f : x = (x 1,..., x n ), 1.2 (a) (b) (c) =. p.7/34

8 : f(x) (or),: x = (x 1, x 2,..., x n ) S. x j, S : f : x = (x 1,..., x n ), 1.3 S { } S = x R n g j (x) 0 (j = 1, 2,..., p), : h k (x) = 0 (k = 1, 2,..., q), g j h k f,,,. p.8/34

9 (). p.9/34

10 3x, y, z 1, () 3x + 4y + 2z 2x + 3y + 4z 10, 3x + 2y z 8, x 0, y 0, z 0., (), 3x + 4y + 2z 2x + 3y + 4z 10, 3x + 2y z 8, x, y, z :. x + 3x 2 4xy 2y + y 2 + 2z 2 + 4z 2x + 3xy 5y + 4z 2 10, 3x + 2y 2 z 8, x 2 + 3y 2 + 5z p.10/34

11 3x, y, z 2 (), 3x + 4 cos y + 2 log z 2x + 3y 2 + 4z 10, 3e x + 2y z 8, x 0, y 0, z 0.. p.11/34

12 p.12/34

13 LSI. p.13/34

14 2.1 = = mn a ij :j1i c j :jb i :i x j :j : : c 1 x 1 + c 2 x c n x n () a i1 x 1 + a i2 x a in x n b i (i = 1, 2,..., m), x j 0 (j = 1, 2,..., n). =. p.14/34

15 a i :i(/) (i = 1, 2,..., 29) b j :j(/) (j = 1, 2,..., 283) c ij :ij(/) f i :i+(/) S {10,..., 29}: x ij :i j(/) a i f 10 x ij c b ij j (0-1 ) y i = { 1 0. p.15/34

16 p.16/34

17 p.17/34

18 . p.18/34

19 = 32. p.19/34

20 1, x 1 = (x 1 1, x1 2 ), x4 = (x 4 1, x4 2 ) = x i x j = x i x j x 2, x 3, x 4, x 1, x 2, x 3, x 4 x 1 x 1 x 4 = (x 1 1 x 4 1) 2 + (x 1 2 x 4 2) 2 = = 5 (). x 1 1, x 1 2. p.20/34

21 1 = x i x j = x i x j x 2, x 3, x 4, x 1, x 2, x 3, x 4 x 1 n + m+ m. p.21/34

22 x 1 x 2 x 3 a 4 = a 5 = a 6 = a 7 =??? (0,0) (0,1) (1,0) (1,1) x 1 x 2 2 = (x 1 1 x 2 1) 2 + (x 1 2 x 2 2) 2 = x 1 a 4 2 = (x 1 1) 2 + (x 1 2) 2 = p.22/34

23 x 1 x 2 x 3 a 4 = a 5 = a 6 = a 7 =??? (0,0) (0,1) (1,0) (1,1) x 1 x 2 2 = (x 1 1 x 2 1) 2 + (x 1 2 x 2 2) 2 = , x 1 x 3 2 = , x 2 x 3 2 = , x 1 a 4 2 = , x 2 a 5 2 = , x 2 a 7 2 = , x 3 a 6 2 = x 1, x 2, x 3 x 1 x x 3 a p.23/34

24 % : : : :30. p.24/34

25 % : : : :60. p.25/34

26 p.26/34

27 (Linear Program, LP) : 4x 1 + 7x 2 : 5x 1 + 3x 2 21 (1) x 1 + 3x 2 9 (2) x 1 0, x 2 0. nm n = 2, m = 4. (n, m 10, 000). p.27/34

28 (Linear Program, LP) : 4x 1 + 7x 2 : 5x 1 + 3x 2 21 (1) x 1 + 3x 2 9 (2) x 1 0, x x 2 (3,2) 0 0 S (1) (2) 4.2 (a) S= (b) = Dantzig, 1947) x 1. p.28/34

29 3.2 (1947) (1984) S ). p.29/34

30 3.3 (nx 1,..., x n m) : c 1 x c n x n : a i1 x a in x n b i or = b i (i = 1,..., m). n: m:, cont1 40, , , 991 rail4284 1, 092, 610 4, , 372, 358 spal , , , 167, 908 a ij CPLEX xpress-mp CPLEX xpress-mp cont ,389 2,174 2,229 rail ,235 8,928 spal004 4,669 > 200,000. p.30/34

31 p.31/34

32 globallib : 6.3x 5 x x x 3 + x x 6 : 0.820x 2 + x x 6 = 0, 0.98x 4 x 7 (0.01x 5 x 10 + x 4 ) = 0, x 1 x 11 3x 8 = 1.33, x 2 x x 3 + x 6 = 0, x 10 x x 11 = 35.82, x 5 x 12 x 2 ( x x 2 9) = 0, x 8 x x 9 ( x 9 ) 0.325x 7 = 0.574, lbd i x i ubd i (i = 1, 2,..., 14). x 1,..., x n n = 14. x 1,..., x n Sum of Squares of Polynomials). p.32/34

33 globallib : 6.3x 5 x x x 3 + x x 6 : 0.820x 2 + x x 6 = 0, 0.98x 4 x 7 (0.01x 5 x 10 + x 4 ) = 0, x 1 x 11 3x 8 = 1.33, x 2 x x 3 + x 6 = 0, x 10 x x 11 = 35.82, x 5 x 12 x 2 ( x x 2 9) = 0, x 8 x x 9 ( x 9 ) 0.325x 7 = 0.574, lbd i x i ubd i (i = 1, 2,..., 14). 14 ɛ obj ɛ feas ɛ obj ɛ feas 5.6e e ɛ obj = max{1, } () ɛ feas =. p.33/34

34 =() =. p.34/34

AHPを用いた大相撲の新しい番付編成

AHPを用いた大相撲の新しい番付編成 5304050 2008/2/15 1 2008/2/15 2 42 2008/2/15 3 2008/2/15 4 195 2008/2/15 5 2008/2/15 6 i j ij >1 ij ij1/>1 i j i 1 ji 1/ j ij 2008/2/15 7 1 =2.01/=0.5 =1.51/=0.67 2008/2/15 8 1 2008/2/15 9 () u ) i i i