最优化理论与算法(18)

Ù¥f(x) gëY "

x(k)´ cS“:§Kf(x)3x(k)? TaylorЪ

1

f(x)=f(x(k))+ f(x(k))T(x x(k))+(x x(k))T 2f(x(k))(x x(k))+o( (x x(k)) 2),

Ä g5y¯K

1

minf(x)= f(x(k))T(x x(k))+(x x(k))T 2f(x(k))(x x(k)),x2

2f(x(k)) ½ §þã¯K `)

1

x =x(k) 2f(x(k)) f(x(k)).

òx # S“:x(k+1)§

x(k+1)

eP

d(k)

K

x(k+1)=x(k)+d(k).

d(k)

1

= 2f(x(k)) f(x(k))¡ f(x)3x(k)? Newton "Ïd§Newton{Ò´±Newton

1

= 2f(x(k)) f(x(k)),

(3.4.1)

1=x(k) 2f(x(k)) f(x(k)).

|¢ §±1 Ú £={ü ‘|¢¤?1S“ {"e¡‰ÑNewton{ äNS“Ú

½"

{3.4.1£Newton{¤

Ú1.À½Ð©êâ"‰½Ð©:x(0)∈Rn§°Ýëêε≥0"-k=0"Ú2.ª O"e f(x(k)) ≤ε§Ê § x =x(k)§ÄK=Ú3"Ú3. E|¢ "¦) 5 §| 2f(x(k))d= f(x(k)) d(k)"Ú4.(½# S“:"-x(k+1)=x(k)+d(k),k:=k+1§=Ú2"

~3.4.2^Newton{¦)Ã å`z¯K

2

minf(x)=x21+2x2,x

‰½Ð©:x(0)

1

§°Ýëêε=10 6"=2