最优化理论与算法(15)

{3.3.1( eü{)

Ú1.À½Ð©êâ"‰ÑЩ:x(0)∈Rn§°Ýëêε≥0"-k=0"

Ú2.ª O"¦ f(x(k))"e f(x(k)) ≤ε§Ê § x =x(k)§ÄK=3"Ú3. E|¢ "-d(k)= f(x(k))"

Ú4.?1 ‘|¢"¦minf(x(k)+td(k)) `Ú ½ ÉÚ tk"

t≥0

Ú5.(½# S“:"-x(k+1)=x(k)+tkd(k)§k:=k+1§=2"

53.3.2tk `Ú ½ ÉÚ § 3~êδ>0§¦

D=f(x(k)) f(x(k)+tkd(k))≥δ f(x(k))Td(k)

2

/ d(k) 2

(3.3.1)

é k¤á= "dþ ! §°( |¢!Armijo |¢9Wolfe–Powell |¢ Ú þ÷vþª"

53.3.3k § O B§ d(k)= α f(x(k))§Ù¥α>0´, ~ê"

~3.3.4^ eü{¦)Ã å 55y¯K

2

minf(x)=x21+2x2,x

‰½Ð©:x(0)

1=§°Ýëêε=0.1"

1

2x14x2

"

Äk§ f(x)=

1 gS“µ

√ 121

"§ f(x(0)) =2≥ε§ d(0)==2 f(x(0))=

224

1 t

§ (t)=f(x(0)+td(0))=(1 t)2+2(1 2t)2§ (t)= ‘|¢µx(0)+td(0)=

1 2t

2(1 t) 8(1 2t)= 10+18t"- (t)=0 (t) ²-:t0=5/9"du (t0)=18>0§Ïdt0=5/9´min (t)=f(x(0)+td(0)) `)"

t≥0

1 5/94/9

# S“:µx(1)=x(0)+t0d(0)=="

1 10/9 1/9

1 gS“µ

f(x(1))=

8/9 4/9

=49

√24

§ f(x(1)) =≥ε§ d(1)= 1

2

"1