最优化理论与算法(9)

µi+1=ai+1+α(bi+1 ai+1)=ai+α2(bi ai),

(3.2.9)

' (3.2.8)Ú(3.2.7)§âα∈(0,1) λi+1Ø U λi § kµi+1=λi§dd α2=1 α§=

√

1α=≈0.6180339887418948.(3.2.10)

23 ¹2e§Ó λi+1=µiÚ(3.2.10)"

√

1

· ¡α= ‘7© X꧑7© {Ò´|^‘7© Xêéü «m?1

2

{"e¡‰Ñ‘7© { äNS“Ú½" {3.2.3£‘7© {¤

Ú1.À½Ð©êâ"‰½Ð©ü «m[a,b]§°Ýëêε>0§‘7© Xêα"eb a≤

=(b+a)/2§ÄK=2"ε§Ê § t

Ú2.Щz"-[a0,b0]=[a,b]§λ0=a0+(1 α)(b0 a0)§µ0=a0+α(b0 a0)§

¦ (λ0)Ú (µ0)"-i=0§=3"

Ú3.' 8I "e (λi)≤ (µi)§=4§ÄK=5"

Ú4. |¢"-ai+1=ai§bi+1=µi§µi+1=λi§ (µi+1)= (λi)§λi+1=ai+1+(bi+1 µi+1)§¦ (λi+1)§=6"

Ú5. m|¢"-ai+1=λi§bi+1=bi§λi+1=µi§ (λi+1)= (µi)§µi+1=bi+1 (λi+1 ai+1)§¦ (µi+1)§=6"

=(bi+1 ai+1)/2§ÄK=7"Ú6.ª O"ebi+1 ai+1≤ε§Ê § t

Ú7.-#m© O"eλi+1<µi+1§Ki=i+1§=3"ÄKa=ai+1§b=bi+1§=2"~3.2.4^‘7© {¦ ‘`z¯K

t∈[0,3]

mint3 t+1

`)"

£ ¤ °( |¢

°( |¢ 8 ´ ¼ `Ú §§ I O þ" °( |¢ 8

K´¦ ¦8I¼ê ½eüþ )§= ÉÚ §§ I O þ"

°( |¢Ò´¦ ‘|¢¯K

minφ(t)s.t.t∈[a,b]

(3.2.11)

§Ù¥φ(t)=f(x(k)+td(k))§x(k)´S“:§d(k)´¯K(3.1.1)3x(k)? |¢ É)t

§K# S“:x(k+1)=x(k)+tkd(k)§e §φ (0)= f(x(k))Td(k)<0"- ÉÚ tk=t

üþ

).D=f(x(k)) f(x(k)+tkd(k))=φ(0) φ(t