最优化理论与算法(7)

! ‘|¢

‘|¢kü« ª§°( |¢Ú °( |¢"°( |¢ÏL¦) ‘ `z¯K

minf(x(k)+αd(k)) φ(α)

α>0

(3.2.1)

Ú αk§Kk

f(x(k)+αkd(k))Td(k)=0.

= αk=argminφ(α)=f(x(k)+αd(k))§ù αk¡ `Ú "ù« {Ø=U y÷veü^ § …3d þ¦eüþD=f(x(k)) f(x(k)+αkd(k)) § I O þ"Ù¢§3¢SO L§¥§duO Åi ÚO Ø Ï§nØþ `Ú ´¦Ø § ´ Cq `)"

éu g¼ê4 z¯K

minf(x)=

1T

xQx+qTx,2

α>0(k)

Ù¥Q∈Rn×né¡ ½" d(k) f3x(k)? eü §…÷v f(x(k))Td(k)<0"-φ(α)=f(x(k)+αd(k))

1

=(x(k)+αd(k))TQ(x(k)+αd(k))+qT(x(k)+αd(k))21

=α2d(k)TQd(k)+α f(x(k))Td(k)+f(x(k)),2

Kdφ (α)=0 `Ú

f(x(k))Td(k)

αk= .

dQd

(3.2.2)

°( |¢(½ Ú αk I÷vf(x(k)+αkd(k)) f(x(k))k ½§Ý eü= "=

αk>0§¦eüþD=f(x(k)) f(x(k+1)) É þ§ù αk¡ ÉÚ "ù« {Ø= y÷veü^ § … I O þ"¢ L²§ù« ‘|¢ { éÐ ê O J"

£ ¤°( |¢¨‘7© {£0.618{¤

‘7© {´(½°( |¢Ú « {§§·^u¦ ü¸¼ê 4 ¯K"kÚ?ü «m Vg"

½Â3.2.1 :[a,b]→R"e 3t ∈[a,b]§¦ (t)3[a,t ]þî 4~§3[t ,b]þî 4O§K¡[a,b]´ (t) ü «m"

w, (t)3Ùü «mþk 4 :t "y3y²ü «m ~k^ 5 "½n3.2.2 [a,b]´ (t) ü «m§λ,µ∈[a,b]§λ<µ"

£1¤e (λ)≤ (µ)§K[a,µ]´ (t) ü «m£2¤e (λ)> (µ)§K[λ,b]´ (t) ü «m"