最优化理论与算法(6)

y² kyf ÛÜ4 : ´ Û :" x ´f ÛÜ4 :§K 3x + U(x )¦

f(x)≥f(x ), x∈U(x ).é?¿x∈Rn§ α>0¿© §x +α(x x )∈U(x ).df à5

f(x )≤f(x +α(x x ))≤αf(x)+(1 α)f(x ).

f(x )≤f(x)§=x ´f Û4 :"

eyx ´¯K(3.1.1) ) ¿ ^ ´ f(x )=0.Ï f´Rnþ à¼ê, f(x )=0, k

f(x) f(x )≥ f(x )(x x )=0, x∈D.ùL²x ´D¥f Û4 :.

§3.2 { Ú½9Âñ5

à å`z eü { Ä g µl, Щ:x(0)Ñu§ E: {x(k)}¦ f(xk+1)<f(x(k)),k=0,1,···. { 8I´: {x(k)}¥ , :½, 4 :´¯K(3.1.1) )½-½:"

!eü {

d(k)´f3x(k)? eü §

f(x(k))Td(k)<0.

K α>0¿© §f(x(k)+αd(k))<f(x(k)).Ïd x(k+1)=x(k)+αkd(k)§Ù¥αk>0¦ f(x(k)+αkd(k))<f(x(k)).

{3.2.1Ú1.‰½Ð©:x(0)∈Rn§°Ýε≥0"-k=0"

Ú2.e f(x(k)) ≤ε§Ê § )x(k)"ÄK=Ú3"Ú3.(½eü d(k)§¦

f(x(k))Td(k)<0.

Ú4.(½Ú αk>0§¦

f(x(k)+αkd(k))<f(x(k)).

Ú5.-x(k+1)=x(k)+αkd(k),k:=k+1§=Ú2"

5µÚ2¥ Ø ª f(x(k)) ≤ε { ª OK§Ù¥°Ýε â¢S¯K I (½" 3nØ©Û §þ ε=0"Ú4¥ αk¡ Ú "(½Ú ~^ {´ ‘ 5|¢"