最优化理论与算法(10)

e¡0 ü« °( |¢ {"1!Armijo. |¢

d(k)´f3x(k)? eü §÷v f(x(k))Td(k)<0"Armijo. |¢µ‰½σ1∈(0,1)§ αk>0§¦

f(x(k)+αkd(k))≤f(x(k))+σ1αk f(x(k))Td(k),

=

φ(αk)≤φ(0)+σ1αkφ (0).

(3.2.12)

´ §(3.2.12)é¿© αkþ¤á"Ï~F"αk¦ U " β>0,ρ∈(0,1)" αk 8Ü{βρi,i=0,1,···}¥¦ (3.2.12)¤á " {3.2.5£Armijo. |¢¤

Ú1.eαk=1÷v(3.2.12)§K αk=1"ÄK=Ú2"Ú2.‰½~êβ>0,ρ∈(0,1)"-αk=β"

Ú3.eα÷v(3.2.12)§Kª O §¿ Ú αk"ÄK=Ú4"Ú4.-αk:=ραk§=Ú3"

5µ£1¤ü Ú αk=1´é- Ú §§3 { Âñ Ý©Û¥å ©- ^"£2¤Á&ÚU'~ρ §eρ∈(0,1) §K ügÁ&Ú UC §I õg|¢âU αk"eρ∈(0,1) §K ügÁ&Ú UC é §d ²L é |¢Ò αk§ αk Ué "~3.2.6 Äà å¯K

minf(x)=

12

x1+x22.2

x(0)=(1,1)T" yd(0)=(1, 1)T f3x(0)? eü "¿^Armijo |¢(½Ú α0=

0.5i§¦

f(x(0)+α0d(0))≤f(x(0))+0.9α0 f(x(0))Td(0).

2!Wolfe–Powell. |¢

ÑArmijo. |¢ "€§ æ^Wolfe–Powell. |¢"

1

Wolfe–Powell. |¢µ‰½~êσ1,σ2÷v0<σ1<,σ1<σ2<1" αk>0¦

f(x(k)+αkd(k))≤f(x(k))+σ1αk f(x(k))Td(k),

f(x(k)+αkd(k))Td(k)≥σ2 f(x(k))Td(k),

(3.2.13)

=

φ(αk)≤φ(0)+σ1αkφ (0),φ (αk)≥σ2φ (0).