最优化理论与算法(12)

y²(3.2.16) d(3.2.15) "ey(3.2.15)"

d¥ ½n§é?¿α>0§k

f(x(k)+αd(k))=f(x(k))+α f(x(k)+tkαd(k))Td(k)

=f(x(k))+α f(x(k))Td(k)

+α[ f(x(k)+tkαd(k)) f(x(k))]Td(k)≤f(x(k))+α f(x(k))Td(k)

+α f(x(k)+tkαd(k)) f(x(k)) d(k) ≤f(x(k))+α f(x(k))Td(k)+Lα2 d(k) 2,

Ù¥tk∈(0,1)"AO §- f(x(k))Td(k)

α¯k= ,

2L d(k) 2

k

2

f(x(k)+α¯kd(k)) f(x(k))≤α¯k f(x(k))Td(k)+Lα¯k d(k) 2

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

= .

d d°( |¢ Ú αk÷v

f(x(k+1)) f(x(k))=f(x(k)+αkd(k)) f(x(k))

≤f(x(k)+α¯kd(k)) f(x(k))1( f(x(k))Td(k))2

=

d 1

= f(x(k)) 2cos2θk,

=

f(x(k)) f(x(k+1))≥

dfke.§ (3.2.15)¤á"

1

f(x(k)) 2cos2θk.4L

(3.2.17)

½n3.2.9 ½n3.2.8 ^ ¤á§{x(k)}dæ^Wolfe-Powell. |¢ {3.2.1 )§=αk÷v(3.2.13)"K½n3.2.8 (ؤá"

y²d(3.2.13) 1 Ø ª9 f LipschitzëY5§

(1 σ2) f(x(k))Td(k)≤( f(x(k+1) f(x(k))Td(k)≤αkL d(k) 2,

1 σ2 f(x(k))Td(k) f(x(k))Td(k)

αk≥ c1,

L d(k) 2 d(k) 2

(3.2.18)