最优化理论与算法(4)

y²£2¤´£1¤ AÏ /§ y£1¤" f(x)Td<0"dTaylorЪ§ α>0¿© §

f(x+αd)=f(x)+α f(x)Td+o(α)<f(x),=d´f3x? eü "

n=1 §ex ´¯K(3.1.1) )§Kf (x )=0…f (x )≥0., ¡§ex ÷vf (x )=0,f (x )>0§Kx ´¯K(3.1.1) î ÛÜ4 :"þã^ í2 n>1 /"

!7 ^

½n3.1.2( 7 ^ ) f:D Rn→R3m8DþëY .ex ∈D´(3.1.1) ÛÜ4 :,K

f(x )=0.(3.1.2)y² x ´ ÛÜ4 :, ÄS

x(k)=x αk f(x ).

|^TaylorЪ,éu¿© k,k

0≤f(x(k)) f(x )= αk f(x )T f(ηk),

Ù¥ηk´x(k)Úx à|Ü.ü>Óرαk,¿ 4 .duf∈C1, k

0≤ f(x ) 2.

w,,= f(x )=0 ,þª¤á.

½n3.1.3( 7 ^ ) f:D Rn→R3m8Dþ ëY ,ex ∈D´(3.1.1)

ÛÜ4 :,K

f(x )=0, 2f(x ) 0,(3.1.3)y² (3.1.3)¥1 ª3½n3.1.2¥®²y², Ly²1 ª. ÄS x(k)=x +αkd,

d?¿.duf∈C2Ú f(x )=0, dTaylorЪ,éu¿© k,k

0≤f(x(k)) f(x )=

12T2

αd f(ηk)d,2k

12

α,¿ 42k

Ù¥ηk´x(k)Úx à|Ü.dux ´ÛÜ4 :,f∈C2,Kþªü>Óر ,

dT 2f(x )d≥0,

l (2.1.3)1 ª .

d∈Rn.