最优化理论与算法(17)

dd {f(x(k))}´î üNeüS §2|^f(x)ke. {f(x(k))}Âñ"u´§

k→∞

lim(f(x(k+1)) f(x(k)))=0.

(3.3.2)

|^^ (3.3.1)Úd(k)= f(x(k))=0"

f(x(k)) f(x(k+1))≥δ f(x(k)) 2,

3þª¥-k→∞§Kd(3.3.2) lim f(x(k)) =0"

k→∞

53.3.6|^½n3.3.5êþ §e^ eü{¦)(UNP)§Ù¥ε>0§@o { ½k Úª u(UNP) Cq²-:x(k)§÷v^ f(x(k)) ≤ε"

l±þ©Û ±wѧ eü{äk {{B! ;þ `:§¿äk ÛÂñ5" ´§duFÝ== N¼ê ÛÜ5 §KFÝ é, : ÛÜ5`´eüþ ¯ §é N5`¿Ø ½´eü ¯ "Ó §duzgS“¥tk´ (t)=mintf(x(k)+td(k)) 4 :§Ïd

0= (tk)= f(x(k)+tkd(k))Td(k)= f(x(k+1))Td(k)= (d(k+1))Td(k),

=c ü |¢ p R § Ò´` eü{±ç¸.´ %C(UNP) `)§¿… C `)Ú " ±y²§ eü{=äk 5Âñ Ý"

½n3.3.7 Ý Q∈Rn×né¡ ½§q∈Rn"PλmaxÚλmin©O´Q Ú A §κ=λmax/λmin"éXe g¼ê4 z¯K

minf(x)=

1T

xQx+qTx.2

{x(k)}´æ^°( |¢ eü{ ) : §Ké¤k kk

κ 1

x(k) x Q, x(k+1) x Q≤

κ+1Ù¥x ´¯K )§ x Q=(xTQx)1/2"

(3.3.3)

§3.4Úî{

¦)à å 55y¯K Newton{´|^8I¼ê gTaylorЪ E|¢ {" !k ÑNewton §¿ïáNewton{§, ‰ÑNewton{ ? {—{ZNewton§ y²ÙÂñ5"

!Newton{

ÄÃ å 55y¯K(UNP)§=

x∈Rn

minf(x)(UNP)