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1

2 ( )

3 ( )

4 ( )

5 i (i = 1, 2,, n) x( ) log(a i x + 1) a i > 0 t i (> 0) T

6 i x i z n z = log(a i x i + 1) i=1

7 i t i ( ) x i t i (i = 1, 2, n) T n x i T i=1

8 z = n log(a i x i + 1) i=1 x i t i (i = 1, 2,, n) n x i T i=1

9 5 (1, 2, 3, 4, 5) 5 x y

10 (x, y) i (i = 1, 2, 3, 4, 5) (x i, y i ) i (x xi ) 2 + (y y i ) 2

11 (x, y) i (i = 1, 2, 3, 4, 5) (x i, y i ) z 5 z = (x xi ) 2 + (y y i ) 2 i=1 z x, y

12 f(x) g i (x) = 0 (i = 1, 2,, m) h j (x) 0 (j = 1, 2,, l) x n [x 1, x 2,, x n ] T

13 f(x)x 2 2

14 x f(x ) f(x) x f x x f(x ) f(x) x f

15 1 f(x) f(x) (1 ) ( ) f (x) = df(x) dx = lim x 0 f(x + x) f(x) x 2 f (x) = d2 f(x) f (x + x) f (x) = lim dx 2 x 0 x

16 f (x) x f(x) ( ) x f (x) x f(x) f (x) x f(x) x f(x) x f (x) = 0 f (x) = 0

17

18 f (x) = 0 x f(x) f (x) f(x) ( ) f (x) f(x) ( ) f (x) 0

19 f(x) = 1 4 x x3 1 2 x2 2x + 1

20 f(x) = 1 4 x x3 1 2 x2 2x + 1 f (x) = x 3 + 2x 2 x 2 = (x + 2)(x + 1)(x 1) x = 2, 1, 1 f (x) = 0 f (x) = 3x 2 + 4x 1 x = 2 ± 7 f (x) = 0 3

21 f(x) = 1 4 x x3 1 2 x2 2x x < > 1 f (x) f (x) f(x)

22 f(x) = 1 4 x x3 1 2 x2 2x x < > 1 f (x) f (x) f(x)

23 f(x) = 1 4 x x3 1 2 x2 2x x < > 1 f (x) f (x) f(x)

24 n f(x) (x = [x 1, x 2,, x n ] T ) f(x) f(x = lim 1,, x i + x i, x i+1,, x n ) f(x 1,, x i, x i+1,, x n ) x i x i 0 x i f(x 1,, x i + x i, x i+1,, x n ) f(x 1,, x i, x i+1,, x n ) x i x i

25 f(x) x i, f xi (x)n f(x) = f(x) x 1 f(x) x 2. f(x) x n

26 2 2 f(x) x i x j, f xi x j (x)n n 2 f(x) 2 f(x) x 2 1 x 1 x 2 2 f(x) x 1 x n 2 2 f(x) 2 f(x) f(x) = x 2 x 1 2 f(x) x 2 2 x 2 x n. 2 f(x) x n x 2 2 f(x) x n x 1 2 f(x) x 2 n

27 n n M nx x T Mx 0 M x m 0 mx 2 0

28 n n M nx x T Mx > 0 M 0x m > 0 mx 2 > 0

29 x f(x) = 0 x 1 f(x) = 0 x f(x)

30

31 x f(x) = 0 2 f(x) x 2

32 x f(x) = 0 2 f(x) x 2

33

34 f (x) f(x) x 0 f (x 0 ) > 0 x f (x 0 ) < 0 x f(x)

35 f(x) f(x) x f(x)

36

37 k x (k) x (k+1) x (k) α (k) f(x (k) ) α (k) f(x (k) α (k) f(x (k) )) ( )

38

39

40

41

42 (0) x x (0) k 0 (1) f(x (k) ) = 0 x (k) (2) (2) f(x (k) α (k) f(x (k) )) ( ) α (k) x (k+1) = x (k) α (k) f(x (k) ) x (k) k k + 1 (1)

43 ε f(x (k) ) < ε x

44

45

46 1 f(x) x (k) f(x) =f(x (k) ) + f (x (k) )(x x (k) ) + 1 2! f (x (k) ) ( x x (k)) ! f (x (k) ) ( x x (k))

47 f(x) = x 4 + (x + 2) 2 g(x) = f(0.5) + f (0.5)(x 0.5) f (0.5)(x 0.5) 2

48 f(x) x (k) 2 g(x) g(x) = f(x (k) ) + f (x (k) )(x x (k) ) f (x (k) ) > f (x (k) )(x x (k) ) 2 g (x) = 0 x g(x)

49 x g (x) = f (x (k) ) + f (x (k) )(x x (k) ) = 0 x = x (k) f (x (k) )/f (x (k) ) x f(x) f(x)

50

51

52 f(x)x (k) f(x) g(x) = f(x (k) ) + f(x (k) ) T (x x (k) ) (x x(k) ) T 2 f(x (k) )(x x (k) ) +

53 f(x) x (k) 2 g(x) g(x) = f(x (k) ) + f(x (k) ) T (x x (k) ) (x x(k) ) T 2 f(x (k) )(x x (k) ) 2 f(x (k) ) 1 g(x) = 0 x g(x)

54 x g(x) = f(x (k) ) + 2 f(x (k) )(x x (k) ) = 0 x = x (k) 2 f(x (k) ) 1 f(x (k) ) x f(x) f(x)

55 (0) x x (0) k 0 (1) f(x (k) ) = 0 x (k) (2) (2) x (k+1) x (k) 2 f(x (k) ) 1 f(x (k) ) x (k) k k + 1 (1)

56 f(x) = (x1 0.4) 2 + (x 2 1 x 2 ) 2 x [ 2(x1 0.4) + 4x f(x) = 1 (x 2 1 x ] 2) 2(x 2 1 x, 2) [ ] 2 12x 2 f(x) = 1 4x x 1 4x 1 2

57

58

59

60 BFGS B (k+1) = B (k) + y(k) y (k)t y (k)t s B (k) s (k) s (k)t B (k)t (k) s (k)t B (k) s (k) B (0) = I, s (k) = x (k+1) x (k), y (k) = f(x (k+1) ) f(x (k) )

[] x < T f(x), x < T f(x), < x < f(x) f(x) f(x) f(x + nt ) = f(x) x < T, n =, 1,, 1, (1.3) f(x) T x 2 f(x) T 2T x 3 f(x), f() = f(t ), f(x), f() f(t )

[] x < T f(x), x < T f(x), < x < f(x) f(x) f(x) f(x + nt ) = f(x) x < T, n =, 1,, 1, (1.3) f(x) T x 2 f(x) T 2T x 3 f(x), f() = f(t ), f(x), f() f(t ) 1 1.1 [] f(x) f(x + T ) = f(x) (1.1), f(x), T f(x) x T 1 ) f(x) = sin x, T = 2 sin (x + 2) = sin x, sin x 2 [] n f(x + nt ) = f(x) (1.2) T [] 2 f(x) g(x) T, h 1 (x) = af(x)+ bg(x) 2 h 2 (x) = f(x)g(x)

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, 1 ( f n (x))dx d dx ( f n (x)) 1 f n (x)dx d dx f n(x) lim f n (x) = [, 1] x f n (x) = n x x 1 f n (x) = x f n (x) = x 1 x n n f n(x) = [, 1] f n (x 1 1.1 4n 2 x, x 1 2n f n (x) = 4n 2 ( 1 x), 1 x 1 n 2n n, 1 x n n 1 1 f n (x)dx = 1, n = 1, 2,.. 1 lim 1 lim 1 f n (x)dx = 1 lim f n(x) = ( lim f n (x))dx = f n (x)dx 1 ( lim f n (x))dx d dx ( lim f d

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