最优化理论与算法(5)

÷v f(x )=0 :x ¡ ¼êf -½:½7:.XJ f(x )=0,Kx U´4 :, U´4 :, UØ´4 :.QØ´4 : Ø´4 : -½: ‰¼ê Q:.Xx =0 f(x)=x2 4 :§ Ø´f(x)=x3 4 :"

n!¿©^

½n3.1.4( ¿©^ ) f:D Rn→R3m8Dþ ëY ,Kx ∈D´f î ÛÜ4 : ¿©^ ´

f(x )=0… 2f(x ) 0.(3.1.4)y² (3.1.4)¤á,KdTaylorЪ,é?¿ þd,

1

f(x +εd)=f(x )+ε2dT 2f(x +θεd)d.

2

du 2f(x ) ½,f∈C2, ÀJε,¦ x +εd∈Nδ(x ),l dT 2f(x +θεd)d>0,ù

f(x +εd)>f(x ),

=x ´î ÛÜ4 :.

4

5µ½n3.1.4 ^ Ø´7 ^ "~X§x =(0,0)T´f(x)=x41+x2 î ÛÜ4 :§ 2f(x )Ø ½"

~3.1.5|^4 ^ ¦)¯K

minf(x)=

) O µ

f(x)=

x21 2x1x22 1

,

2f(x)=

2x1 20

02x2

.

1313

x1+x2 x21 x2.33

d f(x)=0 -½:µ

0022

x(1)=,x(2)=,x(3)=,x(4)=,

1 11 1q

2f(x(3))=

x(3) 4 :"

/,8I¼ê -½:Ø ½´4 :. e8I¼ê´à¼ê,KÙ-½:Ò´Ù4

:,… Û4 :.

½n3.1.6(à¿©5½n) f:Rn→R´ëY à¼ê.Kf ÛÜ4 : ´Ù Û4 :"…x ´¯K(3.1.1) ) ¿©7 ^ ´ f(x )=0.

2002

,