最优化理论与算法(13)

Ù¥c1=(1 σ2)L 1"l d(3.2.13) 1 Ø ª

f(x

(k+1)

) f(x

(k)

( f(x(k))Td(k))2

)≤ σ1c1

d(k) 2= σ1c1 f(x(k)) 2cos2θk.

(3.2.19)

aqu½n3.2.8 (3.2.15).

½n3.2.10 ½n3.2.8 ^ ¤á§{x(k)}dæ^Armijo. |¢ {3.2.1 )§=αkd {3.2.5 )§2b 3~êC>0§¦

f(x(k)) ≤C d(k) .

K½n3.2.8 (ؤá"

y²Ñ"

3½n3.2.8–3.2.10 Ä:þ§e¡‰Ñeü {3.2.1 ÛÂñ ¿©^ "½n3.2.11 f(x)ëY ke.§ fLipschitzëY§= 3L>0¦

f(x) f(y) ≤L x y ,

x,y∈Rn.

(3.2.20)

{x(k)}d {3.2.1 )§Ù¥Ú αkd°( |¢(½§½dWolfe-Powell.|¢(½§½dArmijo. |¢(½§…(3.2.20)¤á"e 3~êη>0§¦

k 1 i=0

cosθi≥ηk,

(3.2.21)

K

liminf f(x(k)) =0.

k→∞

(3.2.22)

y² (3.2.22)ؤá§K 3~êε>0§¦ f(x(k)) ≥ε, k"d(3.2.15) § 3~

êM>0§¦ é¤kk>0k

ε

2k 1 i=0

cosθi≤

2

k 1 i=0

f(x(i)) 2cos2θi≤M.

þªüàÓرk§¿|^AÛØ ªÚ(3.2.21)

εη≤ε(

22

2k 1 i=0

cosθi)

21/k

≤ε

21

k 1 i=0

k

cos2θi≤

M

.k

-k→∞§ gñØ ªε2η2≤0"