( )

( )

( )

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

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

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

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

5 (1, 2, 3, 4, 5) 5 x y 1 1 12 2 4 14 3 15 10 4 11 2 5 5 5

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

(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

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

f(x)x 2 2

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

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

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

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

f(x) = 1 4 x4 + 2 3 x3 1 2 x2 2x + 1

f(x) = 1 4 x4 + 2 3 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

f(x) = 1 4 x4 + 2 3 x3 1 2 x2 2x + 1 2 7 2+ 7 x < 2 2 3 1 3 1 > 1 f (x) 0 + + + 0 0 + f (x) + + + 0 0 + + + f(x) 5 3 25 12 7 12

f(x) = 1 4 x4 + 2 3 x3 1 2 x2 2x + 1 2 7 2+ 7 x < 2 2 3 1 3 1 > 1 f (x) 0 + + + 0 0 + f (x) + + + 0 0 + + + f(x) 5 3 25 12 7 12

f(x) = 1 4 x4 + 2 3 x3 1 2 x2 2x + 1 2 7 2+ 7 x < 2 2 3 1 3 1 > 1 f (x) 0 + + + 0 0 + f (x) + + + 0 0 + + + f(x) 5 3 25 12 7 12

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

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

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

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

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

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

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

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

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

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

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

(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)

ε f(x (k) ) < ε x

1 f(x) x (k) f(x) =f(x (k) ) + f (x (k) )(x x (k) ) + 1 2! f (x (k) ) ( x x (k)) 2 + 1 3! f (x (k) ) ( x x (k)) 3 + 1 2

f(x) = x 4 + (x + 2) 2 g(x) = f(0.5) + f (0.5)(x 0.5) + 1 2 f (0.5)(x 0.5) 2

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

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)

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

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

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)

(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)

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 2 + 2 4x 1 4x 1 2

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) )